Documentation Verification Report

Basic

📁 Source: Mathlib/Algebra/Star/Basic.lean

Statistics

MetricCount
DefinitionstermConj, instStarRing, InvolutiveStar, toStar, instInvolutiveStar, instStar, instStarAddMonoid, instStarMul, instStarRing, instStarRing, involutiveStar, StarAddMonoid, toInvolutiveStar, StarHomClass, StarMemClass, instStar, StarModule, StarMul, toInvolutiveStar, StarRing, toStarAddMonoid, toStarMul, TrivialStar, instStarMul, starAddEquiv, starMulAut, starMulEquiv, starMulOfComm, starRingAut, starRingEnd, starRingEquiv, starRingOfComm
32
Theoremsstar_left, star_right, star_star, conj_conj, instTrivialStar, star_involutive, instStarModule, op_star, unop_star, instTrivialStar, star_mulSingle, star_single, conj_conj, inverse_star, star_apply, star_def, instStarRingEnd, star_star_star, star_add, toStarModuleInt, toStarModuleNat, map_star, coe_star, star_mem, star_smul, star_mul, toStarModule, star_add, toOpposite_starModule, star_trivial, coe_star, coe_star_inv, instStarModule, commute_star_comm, commute_star_star, conj_trivial, eq_star_iff_eq_star, eq_star_of_eq_star, isLeftRegular_star_iff, isRegular_star_iff, isRightRegular_star_iff, isUnit_star, mem_of_star_mem, semiconjBy_star_star_star, starAddEquiv_apply, starMulAut_apply, starMulEquiv_apply, starRingAut_apply, starRingEnd_apply, starRingEnd_self_apply, starRingEquiv_apply, star_div, star_div₀, star_eq_iff_star_eq, star_eq_zero, star_id_of_comm, star_inj, star_injective, star_intCast, star_inv, star_invOf, star_inv₀, star_mem_iff, star_mul', star_mul_star, star_natCast, star_ne_zero, star_neg, star_nsmul, star_ofNat, star_one, star_pow, star_star, star_star_mul, star_sub, star_zero, star_zpow, star_zpow₀, star_zsmul
79
Total111

Commute

Theorems

NameKindAssumesProvesValidatesDepends On
star_left 📖Commute
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
commute_star_comm
star_right 📖Commute
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
commute_star_comm
star_star 📖mathematicalCommuteStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
commute_star_star

Complex

Theorems

NameKindAssumesProvesValidatesDepends On
conj_conj 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
starRingEnd
starRingEnd_self_apply

ComplexConjugate

Definitions

NameCategoryTheorems
termConj 📖CompOp

Int

Definitions

NameCategoryTheorems
instStarRing 📖CompOp
3 mathmath: instStarOrderedRing, instTrivialStar, StarAddMonoid.toStarModuleInt

Theorems

NameKindAssumesProvesValidatesDepends On
instTrivialStar 📖mathematicalTrivialStar
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
instCommRing
StarRing.toStarAddMonoid
instStarRing

InvolutiveStar

Definitions

NameCategoryTheorems
toStar 📖CompOp
1131 mathmath: unitary.spectrum.unitary_conjugate, range_cfcₙ_nnreal, Matrix.l2_opNorm_toEuclideanCLM, ContinuousMapZero.coe_star, NonUnitalSubalgebra.coe_starClosure, continuous_decomposeProdAdjoint, Matrix.posSemidef_conjTranspose_mul_self, StarModule.decomposeProdAdjointL_symm_apply, NonUnitalStarSubalgebra.prod_toNonUnitalSubalgebra, StarModule.selfAdjointPart_add_skewAdjointPart, cfcₙ_def, star_eq_zero, continuous_cfcₙHomSuperset_left, unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, Unitary.conjStarAlgAut_trans_conjStarAlgAut, norm_cfcₙHom, star_mul_star, cfcₙHom_mono, IsStarNormal.instNonUnitalContinuousFunctionalCalculus, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, Complex.coe_realPart, IsIdempotentElem.star_iff, IsStarNormal.one_sub, DoubleCentralizer.star_snd, StarAddMonoid.toStarModuleNNRat, differentiableOn_star_iff, norm_star, Unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, NonUnitalIsometricContinuousFunctionalCalculus.isometric, cfcₙHom_apply_mem_elemental, StarSubalgebra.inclusion_apply, IsUnit.star, CFC.sqrt_eq_one_iff', CFC.posPart_one, Set.image_star, AlgHomClass.instStarHomClass, NonUnitalCStarAlgebra.toStarModule, NonUnitalStarAlgHom.inl_apply, nnnorm_cfcₙ_nnreal_le_iff, realPart_idem, star_ofNat, Matrix.posSemidef_iff_eq_conjTranspose_mul_self, NonUnitalStarAlgebra.mem_bot, IsUnit.isStrictlyPositive_star_right_conjugate_iff, CStarAlgebra.nonneg_iff_exists_isSelfAdjoint_and_eq_mul_self, NonUnitalStarSubalgebra.centralizer_le, NonUnitalStarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, CFC.posPart_natCast, RingHom.star_apply, IsSelfAdjoint.star_mul_self, Matrix.trace_mul_conjTranspose_self_eq_zero_iff, CFC.tendsto_cfc_rpow_sub_one_log, StarSubalgebra.coe_map, Matrix.cstar_nnnorm_def, NonUnitalStarSubsemiring.instNonUnitalSubsemiringClass, realPart_ofReal, spinGroup.coe_star_mul_self, CFC.sq_sqrt, Unitary.conjStarAlgAut_symm, RCLike.star_def, unitary.map_id, BoundedContinuousFunction.mkOfCompact_star, Matrix.isHermitian_conjTranspose_mul_self, Matrix.conjTranspose_sum, pinGroup.coe_star_mul_self, instTrivialStarReal, CentroidHom.star_centerToCentroidCenter, isRightRegular_star_iff, BoundedContinuousFunction.char_neg, val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe, Unitization.real_cfcₙ_eq_cfc_inr, TensorProduct.star_tmul, CliffordAlgebraQuaternion.toQuaternion_star, lp.coeFn_star, imaginaryPart_I_smul, unitary.inv_mul_mem_iff, NumberField.mem_maximalRealSubfield_iff, Matrix.PosSemidef.isHermitian, isClosedEmbedding_cfcₙAux, Complex.star_def, Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv, Matrix.posSemidef_self_mul_conjTranspose, selfAdjoint.expUnitaryPathToOne_apply, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, CFC.nnnorm_sqrt, Subalgebra.star_adjoin_comm, HasSum.matrix_conjTranspose, LinearMap.intrinsicStar_mulRight, CentroidHom.starCenterIsoCentroid_symm_apply_coe, Unitary.mapEquiv_symm_apply, LE.le.star_eq, Module.End.spectrum_intrinsicStar, NonUnitalStarAlgebra.elemental.isClosedEmbedding_coe, Matrix.conjTranspose_mul_self_mulVec_eq_zero, IsUnit.star_right_conjugate_nonneg_iff, IsCHSHTuple.B₀_sa, NonUnitalStarSubalgebra.subsingleton_of_subsingleton, skewAdjoint.isStarNormal_of_mem, StarSubsemiring.mem_toSubsemiring, commute_star_star, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal, CFC.sqrt_rpow, NonUnitalStarRingHom.coe_zero, Matrix.PosSemidef.sqrt_eq_zero_iff, UnitAddTorus.mFourierSubalgebra_coe, Unitization.starAlgHom_ext_iff, CFC.abs_nnrpow_two, summable_star_iff, instStarModuleNNRealOfReal, RCLike.is_real_TFAE, Pi.star_mulSingle, Matrix.intrinsicStar_toLin', NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, LinearMap.toMatrixOrthonormal_reindex, dotProduct_star_self_eq_zero, Matrix.posDef_iff_dotProduct_mulVec, UnitAddTorus.mFourierSubalgebra_closure_eq_top, LinearMap.intrinsicStar_toSpanSingleton, CentroidHom.starCenterIsoCentroid_apply, StarSubalgebra.subtype_comp_inclusion, isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts, Matrix.PosDef.eigenvalues_pos, CFC.negPart_algebraMap, cfcₙHom_nnreal_eq_restrict, star_dotProduct_toMatrix₂_mulVec, selfAdjointPart_comp_subtype_skewAdjoint, Matrix.conjTranspose_natCast_smul, cfcHom_continuous, polynomialFunctions.starClosure_topologicalClosure, Set.nonempty_star, cfcL_apply, WithCStarModule.inner_def, StarAddMonoid.toStarModuleRat, NonUnitalStarAlgebra.mem_sInf, unitary.coe_star_mul_self, deriv.star, star_star, cfcHom_nonneg_iff, Matrix.isHermitian_mul_conjTranspose_self, gelfandStarTransform_naturality, Pi.star_single, gelfandStarTransform_apply_apply, Unitary.inv_mul_mem_iff, Matrix.IsUnit.posSemidef_star_right_conjugate_iff, StarModule.decomposeProdAdjointL_apply, IsCHSHTuple.B₁_sa, cfc_apply_mkD, Set.star_centralizer, instStarModuleNNRealReal, HasStrictFDerivAt.star, Memℓp.star_mem, Set.star_inv', unitary.spectrum.unitary_conjugate', unitary.mem_iff_self_mul_star, HasFDerivAt.star, Unitization.starLift_symm_apply_apply, unitary.star_mul_self_of_mem, Unitization.starLift_apply_apply, Continuous.cfc_nnreal', StarOrderedRing.lt_iff, NonUnitalSubalgebra.starClosure_le_iff, Matrix.PosDef.fromBlocks₁₁, CStarRing.norm_mul_self_le, skewAdjoint.mem_iff, CompactlySupportedContinuousMap.star_apply, star_involutive, RCLike.im_eq_zero_iff_isSelfAdjoint, Matrix.dotProduct_star_self_pos_iff, cfcHom_isClosedEmbedding, StarAlgebra.adjoin_nonUnitalStarSubalgebra, StarSubalgebra.map_map, Unitization.inr_star, selfAdjointPart_comp_subtype_selfAdjoint, fderivWithin_star, Matrix.conjTranspose_inv_intCast_smul, Matrix.conjTranspose_natCast, LE.le.isSelfAdjoint, IsUnit.cfcSqrt, CFC.continuousOn_log, Complex.instContinuousStar, Matrix.PosSemidef.conjTranspose_mul_mul_same, BoundedContinuousFunction.coe_toContinuousMapStarₐ, ContinuousOn.cfc_nnreal', IsGreatest.nnnorm_cfc_nnreal, NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra, IsSelfAdjoint.add_star_self, NonUnitalStarAlgebra.coe_top, skewAdjoint.instIsStarNormalValMemAddSubgroup, LinearMap.toMatrixOrthonormal_apply_apply, NonUnitalStarAlgebra.coe_sInf, unitary.map_comp, Units.instStarModule, IsSelfAdjoint.mul_star_self, MulChar.star_apply, cfcₙ_apply_mkD, cfcₙHom_id, CStarMatrix.inner_toCLM_conjTranspose_left, Matrix.conjTranspose_sub, Matrix.conjTranspose_eq_diagonal, Matrix.posSemidef_vecMulVec_self_star, Module.End.mem_eigenspace_intrinsicStar_iff, CFC.continuousOn_sqrt, Matrix.star_dotProduct, Matrix.isHermitian_conjTranspose_iff, IsLeftRegular.star, starMulAut_apply, StarOrderedRing.pos_iff, StarAddMonoid.star_add, Pi.single_star, PositiveLinearMap.preGNS_norm_def, IsSelfAdjoint.intCast, star_pos_iff, Matrix.ker_mulVecLin_conjTranspose_mul_self, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, CliffordAlgebra.star_def', Matrix.conjTranspose_permMatrix, starL_apply, unitary.mapEquiv_trans, FreeAlgebra.star_algebraMap, Pi.intrinsicStar_comul_commSemiring, IsStarNormal.cfcₙ_map, CompactlySupportedContinuousMap.coe_star, cfcHom_le_iff, cfcHom_id, Matrix.rank_conjTranspose, Unitary.spectrum_star_left_conjugate, MeasureTheory.MemLp.star, Matrix.PosSemidef.det_sqrt, spinGroup.mul_star_self_of_mem, Set.star_add, range_cfcₙHom, StarSubalgebra.mem_comap, NonUnitalStarSubalgebra.center_toNonUnitalSubalgebra, Matrix.conjTranspose_tsum, Units.mul_inv_mem_unitary, Matrix.PosDef.conjTranspose_mul_mul_same, starRingAut_apply, NonUnitalStarSubalgebra.unitization_range, Matrix.toAlgEquiv_kroneckerTMulStarAlgEquiv, Matrix.dotProduct_star, StarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, star_zsmul, eq_star_iff_eq_star, unitary.mem_iff, NonUnitalStarAlgebra.adjoin_toNonUnitalSubalgebra, Matrix.conjTranspose_kronecker, Unitary.mem_iff_self_mul_star, Matrix.mem_unitaryGroup_iff, HasFDerivWithinAt.star, Matrix.star_dotProduct_star, star_right_conjugate_pos, Commute.cfc_real, CFC.abs_nnrpow, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, selfAdjoint.imaginaryPart_coe, CliffordAlgebra.star_def, nnnorm_star, nnnorm_cfcₙ_nnreal_lt, CFC.sqrt_eq_rpow, Matrix.conjTranspose_zero, Unitary.map_coe, NonUnitalStarSubsemiring.coe_ofClass, differentiableAt_star_iff, CStarAlgebra.nonneg_iff_isSelfAdjoint_and_negPart_eq_zero, NumberField.ComplexEmbedding.IsConj.eq, cfc_real_eq_nnreal, Matrix.PosSemidef.posSemidef_sqrt, cfcₙHom_integral, mul_star_self_nonneg, NonUnitalStarSubsemiring.instStarMemClass, StarAlgHom.realContinuousMapOfNNReal_apply, star_left_conjugate_pos, CStarAlgebra.isStrictlyPositive_iff_exists_isUnit_and_isSelfAdjoint_and_eq_mul_self, BoundedContinuousFunction.instStarModule, unitary.toUnits_comp_map, StarSubsemiring.toSubsemiring_le_iff, continuous_decomposeProdAdjoint_symm, CentroidHom.star_apply, gelfandStarTransform_symm_apply, IsSelfAdjoint.cfc, StarMul.toStarModule, Matrix.star_vecMul, CStarModule.inner_smul_left_complex, Nat.instTrivialStar, Unitary.star_mul_self_of_mem, HasDerivAt.star, Matrix.conjTranspose_inv_natCast_smul, star_isometry, StarRing.star_add, Matrix.eigenvalues_conjTranspose_mul_self_nonneg, StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, CStarModule.star_inner, Matrix.frobenius_norm_conjTranspose, Matrix.toEuclideanCLM_toLp, WeakDual.CharacterSpace.homeoEval_naturality, realPart.norm_le, CFC.posPart_algebraMap_nnreal, range_cfcₙ, unitary.mem_iff_star_mul_self, Matrix.PosSemidef.sqrt_eq_one_iff, CStarRing.norm_star_mul_self', Units.coe_star, Matrix.PosSemidef.sqrt_sq, RCLike.mul_wInner_left, StarMemClass.star_coe_eq, IsStarNormal.neg, NormedStarGroup.to_continuousStar, IsCHSHTuple.A₁_sa, PositiveLinearMap.preGNS_inner_def, IsStarNormal.val_inv, star_natCast_smul, starL'_apply, cfcₙAux_id, star_intCast, IsStarNormal.zero, StarSubalgebra.subtype_apply, CFC.sqrt_eq_one_iff, IsSelfAdjoint.inr, realPart_add_I_smul_imaginaryPart, CStarAlgebra.norm_posPart_le, Matrix.conjTranspose_zsmul, Matrix.det_conjTranspose, IsGreatest.nnnorm_cfcₙ_nnreal, NonUnitalSubalgebra.mem_star_iff, Matrix.PosSemidef.eq_sqrt_iff_sq_eq, isometry_cfcₙHom, Unitization.starMap_id, CFC.continuous_abs, Unitary.spectrum_star_right_conjugate, StarSubalgebra.isClosedEmbedding_inclusion, NonUnitalStarAlgebra.adjoin_mono, StarSubsemiring.toSubsemiring_injective, gelfandTransform_map_star, dotProduct_star_self_nonneg, polynomialFunctions.starClosure_eq_adjoin_X, continuousOn_cfc_nnreal, Matrix.conjTranspose_nonsing_inv, FreeMonoid.star_one, pinGroup.star_mem, star_neg, FreeAlgebra.star_ι, cfcHom_integral, Matrix.l2_opNorm_conjTranspose_mul_self, Matrix.conjTranspose_swap, StarSubsemiring.coe_toSubsemiring, star_ratCast_smul, Subalgebra.mem_starClosure, Unitary.conjStarAlgAut_ext_iff', star_finsuppProd, cfcₙHom_predicate, cfc_nnreal_eq_real, StarSubsemiring.subsemiringClass, Set.star_inv, CStarRing.star_mul_self_eq_zero_iff, selfAdjoint.star_val_eq, StarModule.decomposeProdAdjoint_symm_apply, Unitization.starLift_range, Matrix.posDef_conjTranspose_iff, StarSubsemiring.mem_carrier, Matrix.PosSemidef.eigenvalues_nonneg, IsSelfAdjoint.log, IsStarProjection.one, ContinuousFunctionalCalculus.exists_cfc_of_predicate, Unitization.inl_star, Matrix.PosSemidef.mul_mul_conjTranspose_same, NonUnitalStarAlgebra.mem_iInf, CFC.abs_natCast, Matrix.adjugate_conjTranspose, CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt, TensorProduct.instStarModule, LDL.lower_conj_diag, IsIdempotentElem.isSelfAdjoint_iff_isStarNormal, star_nnrat_smul, MeasureTheory.eLpNorm_star, NonUnitalStarAlgebra.adjoin_eq_span, Matrix.l2_opNNNorm_conjTranspose, Matrix.conjTranspose_inv_ofNat_smul, IsSelfAdjoint.cfc_arg, NonUnitalStarAlgebra.gc, CliffordAlgebraQuaternion.ofQuaternion_star, NonUnitalStarAlgebra.subset_adjoin, summable_matrix_conjTranspose, NonUnitalStarAlgHom.inr_apply, Unitary.map_comp, LinearMap.IntrinsicStar.isSelfAdjoint_iff_map_star, cfc_def, Unitization.inrNonUnitalStarAlgHom_apply, Unitary.mapEquiv_apply, DoubleCentralizer.star_fst, NormedStarGroup.norm_star_le, Unitization.starMap_inl, fourierSubalgebra_separatesPoints, NonUnitalStarSubalgebra.range_val, StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal, LinearMap.toMatrixOrthonormal_symm_apply, NonUnitalSubalgebra.starClosure_mono, LinearMap.toMatrix_innerₛₗ_apply, Unitization.starMap_inr, Matrix.eigenvalues_self_mul_conjTranspose_nonneg, star_lt_one_iff, star_inv_intCast_smul, NonUnitalStarAlgebra.elemental.isClosed, CFC.abs_nnrpow_two_mul, Continuous.cfc_nnreal_of_mem_nhdsSet, Matrix.PosSemidef.toLinearMap₂'_zero_iff, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, Filter.IsIncreasingApproximateUnit.eventually_isSelfAdjoint, NormedSpace.star_exp, IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus, Matrix.blockDiagonal_conjTranspose, Matrix.l2_opNorm_conjTranspose, Complex.selfAdjointEquiv_apply, IsSelfAdjoint.natCast, Matrix.conjTranspose_eq_natCast, unitary.star_mem, CliffordAlgebra.star_smul, Continuous.cfcₙ_nnreal, Matrix.PosDef.conjTranspose_mul_self, CStarAlgebra.isStrictlyPositive_iff_eq_mul_star_self, HasDerivAtFilter.star, CFC.abs_star, Unitization.isSelfAdjoint_inr, NonUnitalStarAlgebra.range_eq_top, CStarAlgebra.linear_combination_nonneg, NonUnitalStarSubalgebra.unitization_injective, NonUnitalStarSubalgebra.unitization_apply, IsUnit.isStrictlyPositive_star_left_conjugate_iff, NonUnitalStarSubalgebra.coe_centralizer, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, ContinuousAt.cfcₙ_nnreal, star_mul_self_add_self_mul_star, nonneg_iff_isSelfAdjoint_and_spectrumRestricts, differentiableWithinAt_star_iff, range_cfcₙ_eq_range_cfcₙHom, ContinuousMap.adjoin_id_eq_span_one_union, star_sum, imaginaryPart.norm_le, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, StarSubalgebra.centralizer_toSubalgebra, CFC.abs_algebraMap, NonUnitalStarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, Matrix.conjTranspose_kroneckerTMul, Matrix.isHermitian_gram, LinearMap.IsSymmetric.isSymmetric_smul_iff, realPart_comp_subtype_selfAdjoint, TrivialStar.isStarNormal, IsCHSHTuple.A₀_sa, Matrix.conjTranspose_ratCast_smul, Matrix.ofLp_toEuclideanCLM, Matrix.conjTranspose_list_sum, IsStrictlyPositive.nnrpow, Matrix.exp_conjTranspose, Circle.star_addChar, Matrix.kroneckerTMulStarAlgEquiv_apply, StarSubalgebra.star_mem', StarSubalgebra.coe_subtype, isSelfAdjoint_algebraMap_iff, RCLike.instContinuousStar, Ring.inverse_star, ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply, Matrix.conjTranspose_eq_ofNat, starRingEnd_apply, Matrix.isDiag_conjTranspose_iff, inr_comp_cfcₙHom_eq_cfcₙAux, StarAlgebra.star_self_mem_adjoin_singleton, CFC.monotoneOn_one_sub_one_add_inv_real, IsUnit.star_left_conjugate_nonneg_iff, skewAdjoint.negISMul_apply_coe, NonUnitalStarSubsemiring.mem_carrier, Matrix.UnitaryGroup.inv_val, NonUnitalStarSubalgebra.mem_centralizer_iff, ContinuousWithinAt.cfcₙ_nnreal, StarAlgHom.ofId_apply, CommCStarAlgebra.toStarModule, QuasispectrumRestricts.isSelfAdjoint, Unitary.mul_inv_mem_iff, NonUnitalStarAlgebra.elemental.self_mem, CFC.continuousOn_nnrpow, isRegular_star_iff, Summable.star, imaginaryPart_realPart, Matrix.isHermitian_inv, StarAlgebra.star_subset_adjoin, starMulEquiv_apply, star_mul_self_nonneg, Matrix.star_mul, imaginaryPart_eq_neg_I_smul_skewAdjointPart, IsUnit.isSelfAdjoint_conjugate_iff, selfAdjoint.isSelfAdjoint, range_cfcₙ_nnreal_eq_image_cfcₙ_real, StarSubalgebra.map_toSubalgebra, Matrix.posDef_iff_eq_conjTranspose_mul_self, skewAdjointPart_comp_subtype_selfAdjoint, star_nsmul, Filter.Tendsto.cfcₙ_nnreal, Unitary.conjStarAlgAut_apply, WeakDual.CharacterSpace.compContinuousMap_apply, NonUnitalStarAlgHom.coe_zero, starₗᵢ_apply, ContinuousLinearMap.instStarModuleId, RCLike.wInner_smul_left, continuous_cfcHomSuperset_left, IsUnit.cfcNNRpow, unitary.mapEquiv_symm, Matrix.conjTranspose_eq_intCast, Unitary.mem_iff_star_mul_self, star_right_conjugate_lt_conjugate, Unitary.toAlgEquiv_conjStarAlgAut, cfcₙHom_isClosedEmbedding, Subalgebra.coe_star, apply_eq_star_dotProduct_toMatrix₂_mulVec, realPart_I_smul, Matrix.vecMul_conjTranspose_mul_self_eq_zero, SpectrumRestricts.isSelfAdjoint, selfAdjoint.realPart_unitarySelfAddISMul, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, ContinuousOn.cfc_nnreal, nnnorm_cfcₙ_nnreal_lt_iff, PositiveLinearMap.gnsStarAlgHom_apply, mul_star_self_pos, Matrix.trace_conjTranspose_mul_self_eq_zero_iff, IsSelfAdjoint.of_nonneg, LinearMap.toMatrix'_intrinsicStar, cfcₙHom_le_iff, Matrix.trace_conjTranspose, nnnorm_cfc_nnreal_lt, isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts, Matrix.diagonal_conjTranspose, Matrix.conjTransposeAddEquiv_apply, star_injective, imaginaryPart_apply_coe, Matrix.posSemidef_vecMulVec_star_self, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply, NonUnitalStarAlgebra.star_subset_adjoin, IsSelfAdjoint.ratCast, IsStrictlyPositive.sqrt, StarSubalgebra.comap_id, CStarRing.norm_star_mul_self, Matrix.PosSemidef.isUnit_sqrt_iff, cfcₙ_real_eq_nnreal, imaginaryPart_comp_subtype_selfAdjoint, nnnorm_cfc_nnreal_le, Subalgebra.starClosure_toSubalgebra, CStarModule.inner_op_smul_left, CStarMatrix.mul_entry_mul_eq_inner_toCLM, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, Matrix.PosSemidef.conjTranspose, one_lt_star_iff, Matrix.rank_self_mul_conjTranspose, star_inv_natCast_smul, Continuous.cfc_nnreal, StarAlgebra.elemental.star_self_mem, cfcHomSuperset_apply, star_lt_star_iff, nnnorm_cfc_nnreal_le_iff, isStarProjection_one_sub_iff, Matrix.conjTransposeRingEquiv_symm_apply, Unitary.coe_isStarNormal, HasFDerivAtFilter.star, LinearMap.toMatrixOrthonormal_apply, CFC.sqrt_sq, cfcₙHom_eq_cfcₙ_extend, IsSelfAdjoint.star_iff, EuclideanSpace.inner_eq_star_dotProduct, NonUnitalSubalgebra.coe_star, NonUnitalCommCStarAlgebra.toStarModule, CFC.rpow_sqrt_nnreal, realPart_surjective, Matrix.conjTranspose_one, star_qsmul, cfc_star_id, ContinuousMapZero.instTrivialStar, Unitary.conjStarAlgAut_symm_apply, spectrum.star_mem_resolventSet_iff, selfAdjoint.star_coe_unitarySelfAddISMul, NonUnitalStarAlgebra.elemental.le_centralizer_centralizer, star_nnqsmul, Unitary.toRingEquiv_conjStarAlgAut, instContinuousStarUnits, NonUnitalStarAlgHom.map_adjoin_singleton, skewAdjoint.star_val_eq, Set.Finite.star, ContinuousMapZero.toContinuousMapHom_toNNReal, Complex.instStarModuleReal, StarSubalgebra.comap_comap, DoubleCentralizer.coeHom_apply, CStarAlgebra.mul_star_le_algebraMap_norm_sq, cfcₙHom_map_quasispectrum, spectrum.map_star, IsSelfAdjoint.instIsometricContinuousFunctionalCalculus, Memℓp.star_iff, fourierSubalgebra_coe, NonUnitalStarSubalgebra.prod_inf_prod, Matrix.conjTranspose_zpow, Matrix.PosDef.fromBlocks₂₂, star_right_conjugate_le_conjugate, star_le_iff, Matrix.toLin_conjTranspose, RCLike.instStarModuleReal, imaginaryPart_surjective, cfcHom_nnreal_eq_restrict, star_eq_iff_star_eq, star_prod, MulChar.star_apply', NonUnitalStarAlgebra.comap_top, NonUnitalStarAlgebra.elemental.star_self_mem, Matrix.PosSemidef.sq_sqrt, unitary.coe_map, spec_cfcₙAux, ContinuousOn.cfcₙ_nnreal', Matrix.norm_conjTranspose, IsRightRegular.star, pinGroup.star_mem_iff, cfcHom_eq_cfc_extend, star_lt_iff, CFC.abs_mul_self, conjugate_lt_conjugate', selfAdjoint.realPart_coe, NonUnitalStarRingHom.zero_apply, HasSum.star, Matrix.isHermitian_add_transpose_self, isLeftRegular_star_iff, star_nnratCast, NNRat.instTrivialStar, LinearMap.instStarModuleId, Unitization.starLift_apply, SpectrumRestricts.starAlgHom_apply, CFC.abs_ofNat, dotProduct_self_star_nonneg, CFC.negPart_one, Matrix.isHermitian_iff_isSymmetric, cfcₙHomSuperset_id, conjugate_lt_conjugate, Matrix.PosSemidef.inv_sqrt, Set.star_mem_center, spinGroup.star_mem, Unitary.toUnits_comp_map, Unitization.starMap_comp, InnerProductSpace.toMatrix_rankOne, NonUnitalStarAlgebra.coe_iInf, CFC.abs_algebraMap_nnreal, StarSubsemiring.starMemClass, CliffordAlgebra.star_ι, UnitAddTorus.mFourierSubalgebra_separatesPoints, Unitary.coe_map_star, Matrix.PosSemidef.dotProduct_mulVec_nonneg, Matrix.isUnit_conjTranspose, CStarRing.norm_self_mul_star, skewAdjointPart_apply_coe, CStarMatrix.ofMatrix_eq_ofMatrixStarAlgEquiv, continuous_selfAdjointPart, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, Matrix.nnnorm_conjTranspose, ContinuousMap.evalStarAlgHom_apply, NonUnitalStarRingHom.map_le_map_of_map_star, Matrix.star_eq_inv, selfAdjointPartL_apply_coe, nnnorm_cfcₙHom, CFC.nnnorm_rpow, Matrix.mulVec_conjTranspose, Matrix.conjTranspose_eq_one, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, cfc_apply, IsStarNormal.instContinuousFunctionalCalculus, LinearMap.intrinsicStar_mulLeft, StarSubalgebra.starModule, one_le_star_iff, CStarAlgebra.star_mul_le_algebraMap_norm_sq, SemiconjBy.star_star_star, NonUnitalStarSubalgebra.mem_center_iff, CStarAlgebra.isStrictlyPositive_iff_isUnit_sqrt_and_eq_sqrt_mul_sqrt, cfc_comp_norm, StarModule.complexToReal, CFC.abs_intCast, apply_le_nnnorm_cfc_nnreal, Matrix.conjTranspose_inj, Matrix.isHermitian_fromBlocks_iff, Matrix.posSemidef_iff_eq_sum_vecMulVec, NonUnitalStarSubalgebra.coe_center, range_cfc_nnreal_eq_image_cfc_real, IsStarNormal.one, Complex.coe_selfAdjointEquiv, Matrix.isHermitian_transpose_add_self, Matrix.vecMul_conjTranspose, StarAlgebra.adjoin_toSubalgebra, Matrix.isHermitian_natCast, isClosedEmbedding_cfcₙHom_of_cfcHom, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective, IsStarNormal.cfc_map, unitary.coe_map_star, Set.Nonempty.star, conjugate_pos', imaginaryPart_ofReal, star_zero, Matrix.self_mul_conjTranspose_mulVec_eq_zero, NonUnitalSubalgebra.star_mono, LinearMap.intrinsicStarModule, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, StarOrderedRing.le_iff, Unitization.starMap_apply, coe_starₗᵢ, Set.star_subset, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, Matrix.star_dotProduct_gram_mulVec, Unitization.instIsStarNormal, IsSelfAdjoint.zero, StarAlgHom.realContinuousMapOfNNReal_injective, Unitary.mapEquiv_symm, MulChar.star_eq_inv, NonUnitalStarAlgebra.map_bot, Subalgebra.topologicalClosure_star_comm, StarSubalgebra.coe_centralizer_centralizer, isSelfAdjoint_conjugate_iff_of_isUnit', cfcHom_isStrictlyPositive_iff, Subalgebra.star_mem_star_iff, Matrix.conjTranspose_pow, toMatrix_innerSL_apply, StarSubsemiring.toSubsemiring_inj, Summable.matrix_conjTranspose, StarSemigroup.toOpposite_starModule, Matrix.PosSemidef.re_dotProduct_nonneg, BoundedContinuousFunction.toContinuousMapStarₐ_apply_apply, IsStarNormal.star, CFC.exists_measure_nnrpow_eq_integral_cfcₙ_rpowIntegrand₀₁, LinearMap.intrinsicStar_smulRight, CFC.spectrum_abs, star_id_of_comm, Matrix.IsDiag.conjTranspose, star_left_conjugate_le_conjugate, NonUnitalStarSubalgebra.coe_centralizer_centralizer, Homeomorph.compStarAlgEquiv'_apply, Filter.IsIncreasingApproximateUnit.eventually_star_eq, Matrix.posSemidef_iff_dotProduct_mulVec, cfc_apply_pi, Unitary.conjStarAlgAut_mul_apply, StarSubalgebra.coe_comap, Matrix.conjTranspose_intCast_smul, CStarAlgebra.nonneg_iff_eq_star_mul_self, StarSubalgebra.toSubalgebra_subtype, pinGroup.star_mul_self_of_mem, DifferentiableAt.star_star, NonUnitalStarAlgebra.mem_adjoin_of_mem, ContinuousMapZero.toContinuousMapHom_apply, NonUnitalStarSubalgebra.centralizer_toNonUnitalSubalgebra, skewAdjoint.conjugate', StarOrderedRing.nonneg_iff, cfc_unitary_iff, unitary.toMonoidHom_mapEquiv, Matrix.PosSemidef.dotProduct_mulVec_zero_iff, HasDerivWithinAt.star, Unitary.map_id, CliffordAlgebra.star_algebraMap, BoundedContinuousFunction.star_mem_range_charAlgHom, Complex.selfAdjointEquiv_symm_apply, Int.instTrivialStar, cfcₙAux_mem_range_inr, ContinuousMap.adjoin_id_eq_span_one_add, cfcₙ_norm_sq_nonneg, StarSubalgebra.mem_centralizer_iff, selfAdjointPart_apply_coe, Unitary.mapEquiv_trans, skewAdjointPartL_apply_coe, BoundedContinuousFunction.star_apply, star_div, RingHom.star_def, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', IsSelfAdjoint.star_add_self, star_left_conjugate_lt_conjugate, Matrix.specialUnitaryGroup.coe_star, NonUnitalStarAlgebra.adjoin_le_iff, conjugate_nonneg', BoundedContinuousFunction.coe_star, cfcHomSuperset_id, unitary_iff_isStarNormal_and_spectrum_subset_unitary, CStarMatrix.conjTranspose_zero, CStarAlgebra.nonneg_iff_eq_mul_star_self, NonUnitalStarAlgebra.span_eq_toSubmodule, CFC.sqrt_eq_cfc, Matrix.conjTranspose_multiset_sum, unitary.coe_star, CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self, Matrix.mem_unitaryGroup_iff', Set.star_singleton, Unitary.coe_map, Unitary.mul_star_self_of_mem, realPart_imaginaryPart, Matrix.conjTranspose_ofNat_smul, instTrivialStarNNReal, Filter.Tendsto.cfc_nnreal, selfAdjoint.mem_iff, IsSelfAdjoint.instContinuousFunctionalCalculus, continuousOn_cfcₙ_nnreal, cfcₙHom_continuous, unitary.mul_inv_mem_iff, CFC.abs_sq, LinearMap.intrinsicStar_apply, Subalgebra.coe_starClosure, IsStarNormal.instNonUnitalIsometricContinuousFunctionalCalculus, FreeMonoid.star_of, DifferentiableAt.star, LinearMap.isSelfAdjoint_iff_map_star, StarMul.star_mul, cfcₙHomSuperset_continuous, skewAdjointPart_eq_I_smul_imaginaryPart, StarSubsemiring.ext_iff, isStarProjection_iff_isIdempotentElem_and_isStarNormal, Matrix.conjTranspose_eq_zero, Set.star_mem_star, Rat.instTrivialStar, NonUnitalStarAlgebra.coe_inf, cfcₙ_apply_pi, Matrix.PosDef.mul_mul_conjTranspose_same, IsStrictlyPositive.isSelfAdjoint, NonUnitalStarAlgebra.map_sup, Set.star_subset_star, NonUnitalStarAlgebra.adjoin_eq, Unitization.inrRangeEquiv_apply_coe, Unitization.instStarModule, star_star_mul, CStarAlgebra.isStrictlyPositive_TFAE, fourierSubalgebra_closure_eq_top, CStarAlgebra.isStrictlyPositive_iff_isSelfAdjoint_and_spectrum_pos, derivWithin.star, Matrix.conjTranspose_neg, NonUnitalStarAlgebra.range_id, Unitization.complex_cfcₙ_eq_cfc_inr, StarSubalgebra.one_mem_toNonUnitalStarSubalgebra, LDL.diag_eq_lowerInv_conj, Matrix.conjTranspose_list_prod, NonUnitalStarAlgebra.map_top, CStarAlgebra.norm_negPart_le, fderiv_star, cfcₙ_star, cfcₙ_nnreal_eq_real, CFC.sqrt_algebraMap, CFC.posPart_algebraMap, star_sub, Matrix.self_mul_conjTranspose_mul_eq_zero, CFC.sqrt_one, IsIdempotentElem.star, NonUnitalStarAlgHom.map_adjoin, unitary.mapEquiv_refl, Matrix.isHermitian_zero, CFC.nnrpow_eq_cfcₙ_real, star_le_one_iff, ZeroAtInftyContinuousMap.star_apply, skewAdjoint.conjugate, imaginaryPart_imaginaryPart, IsSelfAdjoint.ofNat, NonUnitalStarAlgebra.star_self_mem_adjoin_singleton, StarSubalgebra.coe_centralizer, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, NonUnitalStarAlgebra.elemental.le_iff_mem, Matrix.isHermitian_one, StarSubalgebra.inclusion_injective, Unitary.mem_iff, cfcHom_predicate, Subalgebra.mem_star_iff, tsum_star, Matrix.conjTranspose_single, CFC.negPart_def, NonUnitalStarSubalgebra.center_eq_top, spinGroup.star_mul_self_of_mem, NonUnitalStarSubalgebra.coe_prod, star_right_conjugate_nonneg, StarAddMonoid.toStarModuleNat, CFC.sqrt_rpow_nnreal, WeakDual.Complex.instStarHomClass, instContinuousStarReal, isStarNormal_of_mem_unitary, NonUnitalStarAlgebra.eq_top_iff, nnnorm_cfc_nnreal_lt_iff, Matrix.conjTranspose_vecMulVec, Units.embed_product_star, PositiveLinearMap.instStarHomClassOfLinearMapClassComplexOfOrderHomClass, imaginaryPart_smul, norm_cfcHom, StarAlgebra.adjoin_eq_span, Unitization.norm_splitMul_snd_sq, EuclideanSpace.inner_toLp_toLp, IsSelfAdjoint.nnratCast, Matrix.conjTranspose_mul_self_eq_zero, CFC.norm_rpow, realPart_apply_coe, starLinearEquiv_apply, dotProduct_self_star_eq_zero, Matrix.conjTranspose_rat_smul, commute_star_comm, isSelfAdjoint_smul_of_mem_skewAdjoint, star_finsuppSum, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, MonotoneOn.nnnorm_cfc, conjugate_pos, Units.coe_star_inv, DifferentiableOn.star, star_zpow, NonUnitalStarAlgebra.self_mem_adjoin_singleton, Matrix.IsUnit.posSemidef_star_left_conjugate_iff, OrthonormalBasis.toMatrix_orthonormalBasis_self_mul_conjTranspose, NonUnitalStarAlgebra.map_inf, cfcₙL_apply, ContinuousOn.cfcₙ_nnreal, cfcHomSuperset_continuous, cfcₙ_apply_mem_elemental, IsStarNormal.one_add, CFC.posPart_def, NonUnitalStarAlgHom.coe_inr, CFC.nnrpow_eq_rpow, Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv, pinGroup.coe_star, Matrix.conjTransposeRingEquiv_apply, CFC.cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one, ContinuousMap.elemental_id_eq_top, Unitization.starLift_range_le, DoubleCentralizer.instStarModule, WeakDual.CharacterSpace.compContinuousMap_comp, star_nonpos_iff, differentiable_star_iff, Complex.im_eq_zero_iff_isSelfAdjoint, LinearMap.isSymm_iff_basis, star_inj, continuousOn_cfc_nnreal_setProd, CStarRing.nnnorm_self_mul_star, Homeomorph.compStarAlgEquiv'_symm_apply, OrthonormalBasis.toMatrix_orthonormalBasis_conjTranspose_mul_self, starRingEquiv_apply, CFC.abs_one, Matrix.vecMul_self_mul_conjTranspose_eq_zero, CentroidHom.starCenterToCentroid_apply, CStarAlgebra.nonneg_TFAE, nnnorm_cfcHom, Matrix.PosSemidef.sqrt_mul_self, Matrix.isHermitian_diagonal_iff, NonUnitalStarAlgHom.coe_inl, realPart_smul, summable_star_iff', cfcₙ_star_id, CFC.abs_mul_abs, NonUnitalStarSubalgebra.prod_top, star_natCast, ContinuousMapZero.elemental_eq_top, ContinuousMapZero.adjoin_id_dense, Matrix.l2_opNNNorm_conjTranspose_mul_self, lp.star_apply, IsRegular.star, Matrix.cstar_norm_def, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, CStarAlgebra.toStarModule, CFC.isUnit_nnrpow_iff, Unitization.starLift_symm_apply, isUnit_star, star_div₀, Matrix.isHermitian_transpose_mul_self, star_le_star_iff, selfAdjoint.isStarNormal, star_mul', inner_matrix_row_row, CFC.monotoneOn_cfcₙ_rpowIntegrand₀₁, star_zpow₀, conjugate_le_conjugate', NonUnitalStarAlgebra.sInf_toNonUnitalSubalgebra, Matrix.conjTranspose_mul_self_mul_eq_zero, Unitary.coe_star_mul_self, star_mul_self_pos, Matrix.isHermitian_diagonal, StarSubalgebra.mem_map, StarAlgHom.map_adjoin, Unitary.toMonoidHom_mapEquiv, Matrix.conjTranspose_conjTranspose, IsStarNormal.instIsometricContinuousFunctionalCalculus, cfc_star, IsUnit.mem_unitary_iff_mul_star_self, star_ratCast, Unitary.star_mem_iff, cfcHom_mono, star_invOf, IsometricContinuousFunctionalCalculus.isometric, Matrix.conjTranspose_intCast, Matrix.conjTranspose_smul_self, Matrix.conjTranspose_add, ContinuousAt.cfc_nnreal, NonUnitalSubalgebra.starClosure_toNonUnitalSubalgebra, Unitary.argSelfAdjoint_coe, Matrix.isHermitian_intCast, Matrix.conjTranspose_kronecker', skewAdjoint.I_smul_neg_I, LinearMap.toMatrix_adjoint, CommMonoid.isStarNormal, NonUnitalStarAlgebra.adjoin_le_starAlgebra_adjoin, spinGroup.coe_star, cfcₙHom_of_cfcHom_map_quasispectrum, WeakDual.CharacterSpace.compContinuousMap_id, IsSelfAdjoint.one, ContinuousMapZero.instStarModule, unitary.star_mem_iff, MeasureTheory.AEEqFun.eLpNorm_star, StarSubalgebra.map_id, BoundedContinuousFunction.separatesPoints_charPoly, ContinuousWithinAt.cfc_nnreal, star_nonneg_iff, cfcₙHomSuperset_apply, IsSelfAdjoint.instNonUnitalIsometricContinuousFunctionalCalculus, NonUnitalStarAlgebra.elemental.instCompleteSpaceSubtypeMemNonUnitalStarSubalgebra, starAddEquiv_apply, cfcₙ_apply, nnnorm_cfcₙ_nnreal_le, CFC.sqrt_eq_real_sqrt, Unitary.coe_star, deriv.star', apply_le_nnnorm_cfcₙ_nnreal, Subalgebra.star_mono, skewAdjointPart_comp_subtype_skewAdjoint, HasFDerivAt.star_star, continuousOn_cfcₙ_nnreal_setProd, cfcHom_eq_of_isStarNormal, continuous_skewAdjointPart, Matrix.UnitaryGroup.star_mul_self, Matrix.self_mul_conjTranspose_eq_zero, Matrix.conjTranspose_mul, ContinuousMapZero.coe_toContinuousMapHom, star_mem_iff, CFC.norm_sqrt, Unitary.conjStarAlgAut_star_apply, CFC.continuousOn_rpow, conjugate_nonneg, Matrix.kroneckerStarAlgEquiv_apply, NonUnitalStarAlgHom.subsingleton, cfcHom_map_spectrum, MonotoneOn.nnnorm_cfcₙ, StarSubalgebra.starMemClass, StarSubsemiring.coe_ofClass, Unitization.inrRangeEquiv_symm_apply, NonUnitalStarSubalgebra.mem_prod, StarSubalgebra.isEmbedding_inclusion, cfcₙHom_nonneg_iff, CStarRing.mul_star_self_eq_zero_iff, CFC.nnnorm_nnrpow, Matrix.IsUnit.posDef_star_left_conjugate_iff, IsStarProjection.zero, Unitary.mapEquiv_refl, star_rat_smul, CStarRing.nnnorm_star_mul_self, Matrix.PosDef.posDef_sqrt, Matrix.conjTranspose_injective, star_left_conjugate_nonneg, Matrix.posSemidef_conjTranspose_iff, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, continuousFunctionalCalculus_map_id, CFC.norm_nnrpow, NonUnitalStarSubalgebra.inclusion_self, StarAddMonoid.toStarModuleInt, Differentiable.star, InnerProductSpace.symm_toEuclideanLin_rankOne, MeasureTheory.Lp.coeFn_star, Commute.cfcₙ_real, Matrix.PosDef.mul_conjTranspose_self, Matrix.frobenius_nnnorm_conjTranspose, Matrix.conjTranspose_ofNat, NonUnitalSubalgebra.mem_starClosure, semiconjBy_star_star_star, Unitary.star_mem, Matrix.rank_conjTranspose_mul_self, IsUnit.mem_unitary_iff_star_mul_self, Unitary.path_apply, Unitary.conjStarAlgAut_ext_iff, HasDerivAt.star_conj, star_one, Set.star_mul, Matrix.IsHermitian.instContinuousFunctionalCalculus, Matrix.PosDef.conjTranspose, star_inv, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, cfcHom_apply_mem_elemental, PositiveLinearMap.preGNS_norm_sq, Matrix.kroneckerStarAlgEquiv_symm_apply, Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg, WeakDual.CharacterSpace.instStarHomClass, pinGroup.mul_star_self_of_mem, NonUnitalStarAlgebra.mem_inf, IsUnit.isSelfAdjoint_conjugate_iff', NonUnitalStarAlgHom.zero_apply, Units.inv_mul_mem_unitary, Matrix.blockDiagonal'_conjTranspose, DifferentiableWithinAt.star, Matrix.dotProduct_self_star_pos_iff, CStarModule.isSelfAdjoint_inner_self, star_inv₀, conjugate_le_conjugate, Unitization.isStarNormal_inr, Matrix.PosSemidef.sqrt_eq_iff_eq_sq, CFC.rpow_eq_cfc_real, star_neg_iff, RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict, Matrix.PosDef.isHermitian, Matrix.mul_self_mul_conjTranspose_eq_zero, inner_matrix_col_col, Matrix.PosDef.dotProduct_mulVec_pos, StarModule.decomposeProdAdjoint_apply, algebraMap_star_comm, Continuous.cfcₙ_nnreal', Matrix.UnitaryGroup.inv_apply, Matrix.conjTranspose_invOf, CStarMatrix.inner_toCLM_conjTranspose_right, CFC.isUnit_sqrt_iff, Matrix.IsUnit.posDef_star_right_conjugate_iff, Matrix.PosDef.re_dotProduct_pos, spinGroup.star_mem_iff, ZeroAtInftyContinuousMap.coe_star, isometry_cfcHom, LinearMap.isSymm_iff_isHermitian_toMatrix, unitary.mul_star_self_of_mem, NonUnitalStarAlgebra.map_iInf, NonUnitalStarAlgebra.inf_toNonUnitalSubalgebra, NonUnitalSubalgebra.star_mem_star_iff, NonUnitalStarAlgebra.adjoin_le_centralizer_centralizer, Matrix.mul_conjTranspose_mul_self_eq_zero, Matrix.conjTranspose_nsmul, isSelfAdjoint_conjugate_iff_of_isUnit, Matrix.star_mulVec, Commute.star_star, HasStrictDerivAt.star, CFC.rpow_sqrt, star_pow, IsSelfAdjoint.cfcₙ, star_intCast_smul, NNReal.instContinuousStar, NonUnitalSubalgebra.star_adjoin_comm

Theorems

NameKindAssumesProvesValidatesDepends On
star_involutive 📖mathematicalFunction.Involutive
Star.star
toStar

MulOpposite

Definitions

NameCategoryTheorems
instInvolutiveStar 📖CompOp
instStar 📖CompOp
6 mathmath: op_star, unop_star, StarSemigroup.toOpposite_starModule, instStarModule, Units.embed_product_star, instContinuousStarMulOpposite
instStarAddMonoid 📖CompOp
instStarMul 📖CompOp
instStarRing 📖CompOp
2 mathmath: instStarOrderedRing, CStarRing.MulOpposite.instMulOpposite

Theorems

NameKindAssumesProvesValidatesDepends On
instStarModule 📖mathematicalStarModule
MulOpposite
instStar
instSMul
unop_injective
StarModule.star_smul
op_star 📖mathematicalop
Star.star
MulOpposite
instStar
unop_star 📖mathematicalunop
Star.star
MulOpposite
instStar

Nat

Definitions

NameCategoryTheorems
instStarRing 📖CompOp
3 mathmath: instTrivialStar, instStarOrderedRing, StarAddMonoid.toStarModuleNat

Theorems

NameKindAssumesProvesValidatesDepends On
instTrivialStar 📖mathematicalTrivialStar
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
instNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
instStarRing

Pi

Theorems

NameKindAssumesProvesValidatesDepends On
star_mulSingle 📖mathematicalStar.star
instStarForall
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
mulSingle
MulOne.toOne
apply_mulSingle
star_one
star_single 📖mathematicalStar.star
instStarForall
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
single
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
apply_single
star_zero

RCLike

Theorems

NameKindAssumesProvesValidatesDepends On
conj_conj 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
starRingEnd
starRingEnd_self_apply

Ring

Theorems

NameKindAssumesProvesValidatesDepends On
inverse_star 📖mathematicalinverse
Semiring.toMonoidWithZero
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
StarRing.toStarAddMonoid
inverse_unit
Units.coe_star
Units.coe_star_inv
inverse_non_unit
isUnit_star
star_zero

RingHom

Definitions

NameCategoryTheorems
involutiveStar 📖CompOp
2 mathmath: star_apply, star_def

Theorems

NameKindAssumesProvesValidatesDepends On
star_apply 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
instFunLike
Star.star
InvolutiveStar.toStar
involutiveStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
star_def 📖mathematicalStar.star
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
InvolutiveStar.toStar
involutiveStar
comp
starRingEnd

RingHomInvPair

Theorems

NameKindAssumesProvesValidatesDepends On
instStarRingEnd 📖mathematicalRingHomInvPair
CommSemiring.toSemiring
starRingEnd
RingHom.ext
star_star

SemiconjBy

Theorems

NameKindAssumesProvesValidatesDepends On
star_star_star 📖mathematicalSemiconjByStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
semiconjBy_star_star_star

StarAddMonoid

Definitions

NameCategoryTheorems
toInvolutiveStar 📖CompOp
956 mathmath: range_cfcₙ_nnreal, Matrix.l2_opNorm_toEuclideanCLM, ContinuousMapZero.coe_star, NonUnitalSubalgebra.coe_starClosure, Matrix.posSemidef_conjTranspose_mul_self, NonUnitalStarSubalgebra.prod_toNonUnitalSubalgebra, cfcₙ_def, star_eq_zero, continuous_cfcₙHomSuperset_left, unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, norm_cfcₙHom, cfcₙHom_mono, IsStarNormal.instNonUnitalContinuousFunctionalCalculus, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, Complex.coe_realPart, IsStarNormal.one_sub, DoubleCentralizer.star_snd, toStarModuleNNRat, differentiableOn_star_iff, norm_star, Unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, NonUnitalIsometricContinuousFunctionalCalculus.isometric, cfcₙHom_apply_mem_elemental, StarSubalgebra.inclusion_apply, CFC.sqrt_eq_one_iff', CFC.posPart_one, AlgHomClass.instStarHomClass, NonUnitalCStarAlgebra.toStarModule, NonUnitalStarAlgHom.inl_apply, nnnorm_cfcₙ_nnreal_le_iff, realPart_idem, star_ofNat, Matrix.posSemidef_iff_eq_conjTranspose_mul_self, NonUnitalStarAlgebra.mem_bot, IsUnit.isStrictlyPositive_star_right_conjugate_iff, CStarAlgebra.nonneg_iff_exists_isSelfAdjoint_and_eq_mul_self, NonUnitalStarSubalgebra.centralizer_le, NonUnitalStarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, CFC.posPart_natCast, RingHom.star_apply, Matrix.trace_mul_conjTranspose_self_eq_zero_iff, CFC.tendsto_cfc_rpow_sub_one_log, StarSubalgebra.coe_map, Matrix.cstar_nnnorm_def, NonUnitalStarSubsemiring.instNonUnitalSubsemiringClass, realPart_ofReal, spinGroup.coe_star_mul_self, CFC.sq_sqrt, RCLike.star_def, BoundedContinuousFunction.mkOfCompact_star, Matrix.isHermitian_conjTranspose_mul_self, Matrix.conjTranspose_sum, pinGroup.coe_star_mul_self, instTrivialStarReal, CentroidHom.star_centerToCentroidCenter, BoundedContinuousFunction.char_neg, Unitization.real_cfcₙ_eq_cfc_inr, TensorProduct.star_tmul, CliffordAlgebraQuaternion.toQuaternion_star, lp.coeFn_star, imaginaryPart_I_smul, NumberField.mem_maximalRealSubfield_iff, Matrix.PosSemidef.isHermitian, isClosedEmbedding_cfcₙAux, Complex.star_def, Matrix.posSemidef_self_mul_conjTranspose, selfAdjoint.expUnitaryPathToOne_apply, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, CFC.nnnorm_sqrt, Subalgebra.star_adjoin_comm, HasSum.matrix_conjTranspose, LinearMap.intrinsicStar_mulRight, CentroidHom.starCenterIsoCentroid_symm_apply_coe, LE.le.star_eq, NonUnitalStarAlgebra.elemental.isClosedEmbedding_coe, Matrix.conjTranspose_mul_self_mulVec_eq_zero, IsUnit.star_right_conjugate_nonneg_iff, NonUnitalStarSubalgebra.subsingleton_of_subsingleton, skewAdjoint.isStarNormal_of_mem, StarSubsemiring.mem_toSubsemiring, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal, CFC.sqrt_rpow, NonUnitalStarRingHom.coe_zero, Matrix.PosSemidef.sqrt_eq_zero_iff, UnitAddTorus.mFourierSubalgebra_coe, Unitization.starAlgHom_ext_iff, CFC.abs_nnrpow_two, summable_star_iff, instStarModuleNNRealOfReal, RCLike.is_real_TFAE, Matrix.intrinsicStar_toLin', NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, LinearMap.toMatrixOrthonormal_reindex, dotProduct_star_self_eq_zero, Matrix.posDef_iff_dotProduct_mulVec, UnitAddTorus.mFourierSubalgebra_closure_eq_top, LinearMap.intrinsicStar_toSpanSingleton, CentroidHom.starCenterIsoCentroid_apply, StarSubalgebra.subtype_comp_inclusion, isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts, Matrix.PosDef.eigenvalues_pos, CFC.negPart_algebraMap, cfcₙHom_nnreal_eq_restrict, star_dotProduct_toMatrix₂_mulVec, Matrix.conjTranspose_natCast_smul, cfcHom_continuous, polynomialFunctions.starClosure_topologicalClosure, cfcL_apply, WithCStarModule.inner_def, toStarModuleRat, NonUnitalStarAlgebra.mem_sInf, deriv.star, cfcHom_nonneg_iff, Matrix.isHermitian_mul_conjTranspose_self, gelfandStarTransform_naturality, Pi.star_single, gelfandStarTransform_apply_apply, Matrix.IsUnit.posSemidef_star_right_conjugate_iff, cfc_apply_mkD, instStarModuleNNRealReal, HasStrictFDerivAt.star, Memℓp.star_mem, HasFDerivAt.star, Unitization.starLift_symm_apply_apply, Unitization.starLift_apply_apply, Continuous.cfc_nnreal', StarOrderedRing.lt_iff, NonUnitalSubalgebra.starClosure_le_iff, Matrix.PosDef.fromBlocks₁₁, CStarRing.norm_mul_self_le, skewAdjoint.mem_iff, CompactlySupportedContinuousMap.star_apply, RCLike.im_eq_zero_iff_isSelfAdjoint, Matrix.dotProduct_star_self_pos_iff, cfcHom_isClosedEmbedding, StarAlgebra.adjoin_nonUnitalStarSubalgebra, StarSubalgebra.map_map, Unitization.inr_star, fderivWithin_star, Matrix.conjTranspose_inv_intCast_smul, Matrix.conjTranspose_natCast, LE.le.isSelfAdjoint, IsUnit.cfcSqrt, CFC.continuousOn_log, Complex.instContinuousStar, Matrix.PosSemidef.conjTranspose_mul_mul_same, BoundedContinuousFunction.coe_toContinuousMapStarₐ, ContinuousOn.cfc_nnreal', IsGreatest.nnnorm_cfc_nnreal, NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra, IsSelfAdjoint.add_star_self, NonUnitalStarAlgebra.coe_top, skewAdjoint.instIsStarNormalValMemAddSubgroup, LinearMap.toMatrixOrthonormal_apply_apply, NonUnitalStarAlgebra.coe_sInf, MulChar.star_apply, cfcₙ_apply_mkD, cfcₙHom_id, CStarMatrix.inner_toCLM_conjTranspose_left, Matrix.conjTranspose_sub, Matrix.conjTranspose_eq_diagonal, Matrix.posSemidef_vecMulVec_self_star, Module.End.mem_eigenspace_intrinsicStar_iff, CFC.continuousOn_sqrt, Matrix.star_dotProduct, StarOrderedRing.pos_iff, star_add, Pi.single_star, PositiveLinearMap.preGNS_norm_def, IsSelfAdjoint.intCast, star_pos_iff, Matrix.ker_mulVecLin_conjTranspose_mul_self, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, CliffordAlgebra.star_def', Matrix.conjTranspose_permMatrix, starL_apply, FreeAlgebra.star_algebraMap, Pi.intrinsicStar_comul_commSemiring, IsStarNormal.cfcₙ_map, CompactlySupportedContinuousMap.coe_star, cfcHom_le_iff, cfcHom_id, Matrix.rank_conjTranspose, MeasureTheory.MemLp.star, Matrix.PosSemidef.det_sqrt, spinGroup.mul_star_self_of_mem, Set.star_add, range_cfcₙHom, StarSubalgebra.mem_comap, NonUnitalStarSubalgebra.center_toNonUnitalSubalgebra, Matrix.conjTranspose_tsum, Matrix.PosDef.conjTranspose_mul_mul_same, starRingAut_apply, NonUnitalStarSubalgebra.unitization_range, Matrix.toAlgEquiv_kroneckerTMulStarAlgEquiv, Matrix.dotProduct_star, StarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, star_zsmul, NonUnitalStarAlgebra.adjoin_toNonUnitalSubalgebra, Matrix.mem_unitaryGroup_iff, HasFDerivWithinAt.star, Matrix.star_dotProduct_star, star_right_conjugate_pos, Commute.cfc_real, CFC.abs_nnrpow, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, selfAdjoint.imaginaryPart_coe, CliffordAlgebra.star_def, nnnorm_star, nnnorm_cfcₙ_nnreal_lt, CFC.sqrt_eq_rpow, Matrix.conjTranspose_zero, NonUnitalStarSubsemiring.coe_ofClass, LinearMap.intrinsicStar_mul', differentiableAt_star_iff, CStarAlgebra.nonneg_iff_isSelfAdjoint_and_negPart_eq_zero, NumberField.ComplexEmbedding.IsConj.eq, cfc_real_eq_nnreal, Matrix.PosSemidef.posSemidef_sqrt, cfcₙHom_integral, mul_star_self_nonneg, NonUnitalStarSubsemiring.instStarMemClass, StarAlgHom.realContinuousMapOfNNReal_apply, star_left_conjugate_pos, CStarAlgebra.isStrictlyPositive_iff_exists_isUnit_and_isSelfAdjoint_and_eq_mul_self, BoundedContinuousFunction.instStarModule, StarSubsemiring.toSubsemiring_le_iff, CentroidHom.star_apply, gelfandStarTransform_symm_apply, IsSelfAdjoint.cfc, Matrix.star_vecMul, CStarModule.inner_smul_left_complex, Nat.instTrivialStar, HasDerivAt.star, Matrix.conjTranspose_inv_natCast_smul, star_isometry, Matrix.eigenvalues_conjTranspose_mul_self_nonneg, StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, CStarModule.star_inner, Matrix.frobenius_norm_conjTranspose, Matrix.toEuclideanCLM_toLp, WeakDual.CharacterSpace.homeoEval_naturality, realPart.norm_le, CFC.posPart_algebraMap_nnreal, range_cfcₙ, Matrix.PosSemidef.sqrt_eq_one_iff, CStarRing.norm_star_mul_self', LinearMap.IntrinsicStar.starLinearEquiv_eq_arrowCongr, Matrix.PosSemidef.sqrt_sq, RCLike.mul_wInner_left, IsStarNormal.neg, NormedStarGroup.to_continuousStar, PositiveLinearMap.preGNS_inner_def, star_natCast_smul, starL'_apply, cfcₙAux_id, star_intCast, IsStarNormal.zero, StarSubalgebra.subtype_apply, CFC.sqrt_eq_one_iff, IsSelfAdjoint.inr, realPart_add_I_smul_imaginaryPart, CStarAlgebra.norm_posPart_le, Matrix.conjTranspose_zsmul, Matrix.det_conjTranspose, IsGreatest.nnnorm_cfcₙ_nnreal, NonUnitalSubalgebra.mem_star_iff, Matrix.PosSemidef.eq_sqrt_iff_sq_eq, isometry_cfcₙHom, Unitization.starMap_id, CFC.continuous_abs, StarSubalgebra.isClosedEmbedding_inclusion, NonUnitalStarAlgebra.adjoin_mono, StarSubsemiring.toSubsemiring_injective, gelfandTransform_map_star, dotProduct_star_self_nonneg, polynomialFunctions.starClosure_eq_adjoin_X, continuousOn_cfc_nnreal, Matrix.conjTranspose_nonsing_inv, pinGroup.star_mem, star_neg, FreeAlgebra.star_ι, cfcHom_integral, Matrix.l2_opNorm_conjTranspose_mul_self, Matrix.conjTranspose_swap, StarSubsemiring.coe_toSubsemiring, star_ratCast_smul, cfcₙHom_predicate, cfc_nnreal_eq_real, StarSubsemiring.subsemiringClass, CStarRing.star_mul_self_eq_zero_iff, selfAdjoint.star_val_eq, Unitization.starLift_range, Matrix.posDef_conjTranspose_iff, StarSubsemiring.mem_carrier, Matrix.PosSemidef.eigenvalues_nonneg, IsSelfAdjoint.log, ContinuousFunctionalCalculus.exists_cfc_of_predicate, Unitization.inl_star, Matrix.PosSemidef.mul_mul_conjTranspose_same, NonUnitalStarAlgebra.mem_iInf, CFC.abs_natCast, Matrix.adjugate_conjTranspose, CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt, TensorProduct.instStarModule, LDL.lower_conj_diag, IsIdempotentElem.isSelfAdjoint_iff_isStarNormal, star_nnrat_smul, MeasureTheory.eLpNorm_star, NonUnitalStarAlgebra.adjoin_eq_span, Matrix.l2_opNNNorm_conjTranspose, Matrix.conjTranspose_inv_ofNat_smul, IsSelfAdjoint.cfc_arg, NonUnitalStarAlgebra.gc, CliffordAlgebraQuaternion.ofQuaternion_star, NonUnitalStarAlgebra.subset_adjoin, summable_matrix_conjTranspose, NonUnitalStarAlgHom.inr_apply, LinearMap.IntrinsicStar.isSelfAdjoint_iff_map_star, cfc_def, Unitization.inrNonUnitalStarAlgHom_apply, DoubleCentralizer.star_fst, NormedStarGroup.norm_star_le, Unitization.starMap_inl, fourierSubalgebra_separatesPoints, NonUnitalStarSubalgebra.range_val, StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal, LinearMap.toMatrixOrthonormal_symm_apply, LinearMap.intrinsicStar_rTensor, NonUnitalSubalgebra.starClosure_mono, LinearMap.toMatrix_innerₛₗ_apply, Unitization.starMap_inr, Matrix.eigenvalues_self_mul_conjTranspose_nonneg, star_lt_one_iff, star_inv_intCast_smul, NonUnitalStarAlgebra.elemental.isClosed, CFC.abs_nnrpow_two_mul, Continuous.cfc_nnreal_of_mem_nhdsSet, Matrix.PosSemidef.toLinearMap₂'_zero_iff, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, Filter.IsIncreasingApproximateUnit.eventually_isSelfAdjoint, NormedSpace.star_exp, IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus, Matrix.blockDiagonal_conjTranspose, Matrix.l2_opNorm_conjTranspose, TensorProduct.intrinsicStar_map, Complex.selfAdjointEquiv_apply, IsSelfAdjoint.natCast, Matrix.conjTranspose_eq_natCast, CliffordAlgebra.star_smul, Continuous.cfcₙ_nnreal, Matrix.PosDef.conjTranspose_mul_self, CStarAlgebra.isStrictlyPositive_iff_eq_mul_star_self, HasDerivAtFilter.star, CFC.abs_star, Unitization.isSelfAdjoint_inr, NonUnitalStarAlgebra.range_eq_top, CStarAlgebra.linear_combination_nonneg, NonUnitalStarSubalgebra.unitization_injective, NonUnitalStarSubalgebra.unitization_apply, IsUnit.isStrictlyPositive_star_left_conjugate_iff, NonUnitalStarSubalgebra.coe_centralizer, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, ContinuousAt.cfcₙ_nnreal, star_mul_self_add_self_mul_star, nonneg_iff_isSelfAdjoint_and_spectrumRestricts, differentiableWithinAt_star_iff, range_cfcₙ_eq_range_cfcₙHom, ContinuousMap.adjoin_id_eq_span_one_union, star_sum, imaginaryPart.norm_le, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, StarSubalgebra.centralizer_toSubalgebra, CFC.abs_algebraMap, NonUnitalStarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, Matrix.conjTranspose_kroneckerTMul, Matrix.isHermitian_gram, LinearMap.IsSymmetric.isSymmetric_smul_iff, realPart_comp_subtype_selfAdjoint, Matrix.conjTranspose_ratCast_smul, Matrix.ofLp_toEuclideanCLM, Matrix.conjTranspose_list_sum, IsStrictlyPositive.nnrpow, Matrix.exp_conjTranspose, Circle.star_addChar, Matrix.kroneckerTMulStarAlgEquiv_apply, StarSubalgebra.star_mem', StarSubalgebra.coe_subtype, isSelfAdjoint_algebraMap_iff, RCLike.instContinuousStar, Ring.inverse_star, ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply, Matrix.conjTranspose_eq_ofNat, starRingEnd_apply, Matrix.isDiag_conjTranspose_iff, inr_comp_cfcₙHom_eq_cfcₙAux, StarAlgebra.star_self_mem_adjoin_singleton, CFC.monotoneOn_one_sub_one_add_inv_real, IsUnit.star_left_conjugate_nonneg_iff, skewAdjoint.negISMul_apply_coe, NonUnitalStarSubsemiring.mem_carrier, Matrix.UnitaryGroup.inv_val, NonUnitalStarSubalgebra.mem_centralizer_iff, ContinuousWithinAt.cfcₙ_nnreal, StarAlgHom.ofId_apply, CommCStarAlgebra.toStarModule, QuasispectrumRestricts.isSelfAdjoint, NonUnitalStarAlgebra.elemental.self_mem, CFC.continuousOn_nnrpow, Summable.star, imaginaryPart_realPart, Matrix.isHermitian_inv, StarAlgebra.star_subset_adjoin, star_mul_self_nonneg, Matrix.star_mul, imaginaryPart_eq_neg_I_smul_skewAdjointPart, selfAdjoint.isSelfAdjoint, range_cfcₙ_nnreal_eq_image_cfcₙ_real, StarSubalgebra.map_toSubalgebra, Matrix.posDef_iff_eq_conjTranspose_mul_self, star_nsmul, Filter.Tendsto.cfcₙ_nnreal, WeakDual.CharacterSpace.compContinuousMap_apply, NonUnitalStarAlgHom.coe_zero, starₗᵢ_apply, ContinuousLinearMap.instStarModuleId, RCLike.wInner_smul_left, continuous_cfcHomSuperset_left, IsUnit.cfcNNRpow, Matrix.conjTranspose_eq_intCast, star_right_conjugate_lt_conjugate, cfcₙHom_isClosedEmbedding, Subalgebra.coe_star, apply_eq_star_dotProduct_toMatrix₂_mulVec, realPart_I_smul, Matrix.vecMul_conjTranspose_mul_self_eq_zero, SpectrumRestricts.isSelfAdjoint, selfAdjoint.realPart_unitarySelfAddISMul, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, ContinuousOn.cfc_nnreal, nnnorm_cfcₙ_nnreal_lt_iff, PositiveLinearMap.gnsStarAlgHom_apply, mul_star_self_pos, Matrix.trace_conjTranspose_mul_self_eq_zero_iff, IsSelfAdjoint.of_nonneg, LinearMap.toMatrix'_intrinsicStar, cfcₙHom_le_iff, Matrix.trace_conjTranspose, nnnorm_cfc_nnreal_lt, isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts, Matrix.diagonal_conjTranspose, Matrix.conjTransposeAddEquiv_apply, imaginaryPart_apply_coe, Matrix.posSemidef_vecMulVec_star_self, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply, NonUnitalStarAlgebra.star_subset_adjoin, IsSelfAdjoint.ratCast, IsStrictlyPositive.sqrt, StarSubalgebra.comap_id, CStarRing.norm_star_mul_self, Matrix.PosSemidef.isUnit_sqrt_iff, cfcₙ_real_eq_nnreal, imaginaryPart_comp_subtype_selfAdjoint, nnnorm_cfc_nnreal_le, CStarModule.inner_op_smul_left, CStarMatrix.mul_entry_mul_eq_inner_toCLM, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, Matrix.PosSemidef.conjTranspose, one_lt_star_iff, Matrix.rank_self_mul_conjTranspose, star_inv_natCast_smul, Continuous.cfc_nnreal, StarAlgebra.elemental.star_self_mem, cfcHomSuperset_apply, star_lt_star_iff, nnnorm_cfc_nnreal_le_iff, isStarProjection_one_sub_iff, Matrix.conjTransposeRingEquiv_symm_apply, HasFDerivAtFilter.star, LinearMap.toMatrixOrthonormal_apply, CFC.sqrt_sq, cfcₙHom_eq_cfcₙ_extend, EuclideanSpace.inner_eq_star_dotProduct, NonUnitalSubalgebra.coe_star, NonUnitalCommCStarAlgebra.toStarModule, CFC.rpow_sqrt_nnreal, realPart_surjective, Matrix.conjTranspose_one, star_qsmul, cfc_star_id, ContinuousMapZero.instTrivialStar, spectrum.star_mem_resolventSet_iff, selfAdjoint.star_coe_unitarySelfAddISMul, NonUnitalStarAlgebra.elemental.le_centralizer_centralizer, star_nnqsmul, NonUnitalStarAlgHom.map_adjoin_singleton, skewAdjoint.star_val_eq, ContinuousMapZero.toContinuousMapHom_toNNReal, Complex.instStarModuleReal, StarSubalgebra.comap_comap, DoubleCentralizer.coeHom_apply, CStarAlgebra.mul_star_le_algebraMap_norm_sq, cfcₙHom_map_quasispectrum, spectrum.map_star, IsSelfAdjoint.instIsometricContinuousFunctionalCalculus, Memℓp.star_iff, fourierSubalgebra_coe, NonUnitalStarSubalgebra.prod_inf_prod, Matrix.conjTranspose_zpow, Matrix.PosDef.fromBlocks₂₂, star_right_conjugate_le_conjugate, star_le_iff, Matrix.toLin_conjTranspose, RCLike.instStarModuleReal, imaginaryPart_surjective, cfcHom_nnreal_eq_restrict, MulChar.star_apply', NonUnitalStarAlgebra.comap_top, NonUnitalStarAlgebra.elemental.star_self_mem, Matrix.PosSemidef.sq_sqrt, spec_cfcₙAux, ContinuousOn.cfcₙ_nnreal', Matrix.norm_conjTranspose, pinGroup.star_mem_iff, cfcHom_eq_cfc_extend, star_lt_iff, CFC.abs_mul_self, conjugate_lt_conjugate', selfAdjoint.realPart_coe, NonUnitalStarRingHom.zero_apply, HasSum.star, Matrix.isHermitian_add_transpose_self, star_nnratCast, NNRat.instTrivialStar, LinearMap.instStarModuleId, Unitization.starLift_apply, SpectrumRestricts.starAlgHom_apply, CFC.abs_ofNat, dotProduct_self_star_nonneg, CFC.negPart_one, Matrix.isHermitian_iff_isSymmetric, cfcₙHomSuperset_id, conjugate_lt_conjugate, Matrix.PosSemidef.inv_sqrt, spinGroup.star_mem, Unitization.starMap_comp, InnerProductSpace.toMatrix_rankOne, NonUnitalStarAlgebra.coe_iInf, CFC.abs_algebraMap_nnreal, StarSubsemiring.starMemClass, CliffordAlgebra.star_ι, UnitAddTorus.mFourierSubalgebra_separatesPoints, Matrix.PosSemidef.dotProduct_mulVec_nonneg, Matrix.isUnit_conjTranspose, CStarRing.norm_self_mul_star, skewAdjointPart_apply_coe, CStarMatrix.ofMatrix_eq_ofMatrixStarAlgEquiv, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, Matrix.nnnorm_conjTranspose, ContinuousMap.evalStarAlgHom_apply, NonUnitalStarRingHom.map_le_map_of_map_star, selfAdjointPartL_apply_coe, nnnorm_cfcₙHom, CFC.nnnorm_rpow, Matrix.mulVec_conjTranspose, Matrix.conjTranspose_eq_one, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, cfc_apply, IsStarNormal.instContinuousFunctionalCalculus, LinearMap.intrinsicStar_mulLeft, StarSubalgebra.starModule, one_le_star_iff, CStarAlgebra.star_mul_le_algebraMap_norm_sq, NonUnitalStarSubalgebra.mem_center_iff, CStarAlgebra.isStrictlyPositive_iff_isUnit_sqrt_and_eq_sqrt_mul_sqrt, cfc_comp_norm, StarModule.complexToReal, CFC.abs_intCast, apply_le_nnnorm_cfc_nnreal, Matrix.posSemidef_iff_eq_sum_vecMulVec, NonUnitalStarSubalgebra.coe_center, range_cfc_nnreal_eq_image_cfc_real, Complex.coe_selfAdjointEquiv, Matrix.isHermitian_transpose_add_self, Matrix.vecMul_conjTranspose, StarAlgebra.adjoin_toSubalgebra, Matrix.isHermitian_natCast, isClosedEmbedding_cfcₙHom_of_cfcHom, LinearMap.intrinsicStar_lTensor, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective, IsStarNormal.cfc_map, conjugate_pos', imaginaryPart_ofReal, star_zero, Matrix.self_mul_conjTranspose_mulVec_eq_zero, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, StarOrderedRing.le_iff, Unitization.starMap_apply, coe_starₗᵢ, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, Matrix.star_dotProduct_gram_mulVec, Unitization.instIsStarNormal, IsSelfAdjoint.zero, StarAlgHom.realContinuousMapOfNNReal_injective, NonUnitalStarAlgebra.map_bot, StarSubalgebra.coe_centralizer_centralizer, cfcHom_isStrictlyPositive_iff, Subalgebra.star_mem_star_iff, Matrix.conjTranspose_pow, toMatrix_innerSL_apply, StarSubsemiring.toSubsemiring_inj, Summable.matrix_conjTranspose, Matrix.PosSemidef.re_dotProduct_nonneg, BoundedContinuousFunction.toContinuousMapStarₐ_apply_apply, CFC.exists_measure_nnrpow_eq_integral_cfcₙ_rpowIntegrand₀₁, LinearMap.intrinsicStar_smulRight, CFC.spectrum_abs, Matrix.IsDiag.conjTranspose, star_left_conjugate_le_conjugate, NonUnitalStarSubalgebra.coe_centralizer_centralizer, Homeomorph.compStarAlgEquiv'_apply, Filter.IsIncreasingApproximateUnit.eventually_star_eq, Matrix.posSemidef_iff_dotProduct_mulVec, cfc_apply_pi, StarSubalgebra.coe_comap, Matrix.conjTranspose_intCast_smul, CStarAlgebra.nonneg_iff_eq_star_mul_self, StarSubalgebra.toSubalgebra_subtype, pinGroup.star_mul_self_of_mem, DifferentiableAt.star_star, NonUnitalStarAlgebra.mem_adjoin_of_mem, ContinuousMapZero.toContinuousMapHom_apply, NonUnitalStarSubalgebra.centralizer_toNonUnitalSubalgebra, skewAdjoint.conjugate', StarOrderedRing.nonneg_iff, cfc_unitary_iff, Matrix.PosSemidef.dotProduct_mulVec_zero_iff, HasDerivWithinAt.star, CliffordAlgebra.star_algebraMap, BoundedContinuousFunction.star_mem_range_charAlgHom, Complex.selfAdjointEquiv_symm_apply, Int.instTrivialStar, cfcₙAux_mem_range_inr, ContinuousMap.adjoin_id_eq_span_one_add, cfcₙ_norm_sq_nonneg, StarSubalgebra.mem_centralizer_iff, selfAdjointPart_apply_coe, skewAdjointPartL_apply_coe, BoundedContinuousFunction.star_apply, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', IsSelfAdjoint.star_add_self, star_left_conjugate_lt_conjugate, Matrix.specialUnitaryGroup.coe_star, NonUnitalStarAlgebra.adjoin_le_iff, conjugate_nonneg', BoundedContinuousFunction.coe_star, cfcHomSuperset_id, unitary_iff_isStarNormal_and_spectrum_subset_unitary, CStarMatrix.conjTranspose_zero, CStarAlgebra.nonneg_iff_eq_mul_star_self, NonUnitalStarAlgebra.span_eq_toSubmodule, CFC.sqrt_eq_cfc, Matrix.conjTranspose_multiset_sum, CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self, Matrix.mem_unitaryGroup_iff', realPart_imaginaryPart, Matrix.conjTranspose_ofNat_smul, instTrivialStarNNReal, Filter.Tendsto.cfc_nnreal, selfAdjoint.mem_iff, IsSelfAdjoint.instContinuousFunctionalCalculus, continuousOn_cfcₙ_nnreal, cfcₙHom_continuous, CFC.abs_sq, LinearMap.intrinsicStar_apply, IsStarNormal.instNonUnitalIsometricContinuousFunctionalCalculus, DifferentiableAt.star, LinearMap.isSelfAdjoint_iff_map_star, cfcₙHomSuperset_continuous, skewAdjointPart_eq_I_smul_imaginaryPart, StarSubsemiring.ext_iff, isStarProjection_iff_isIdempotentElem_and_isStarNormal, Matrix.conjTranspose_eq_zero, Rat.instTrivialStar, NonUnitalStarAlgebra.coe_inf, cfcₙ_apply_pi, Matrix.PosDef.mul_mul_conjTranspose_same, IsStrictlyPositive.isSelfAdjoint, NonUnitalStarAlgebra.map_sup, NonUnitalStarAlgebra.adjoin_eq, Unitization.inrRangeEquiv_apply_coe, Unitization.instStarModule, CStarAlgebra.isStrictlyPositive_TFAE, fourierSubalgebra_closure_eq_top, CStarAlgebra.isStrictlyPositive_iff_isSelfAdjoint_and_spectrum_pos, derivWithin.star, Matrix.conjTranspose_neg, NonUnitalStarAlgebra.range_id, Unitization.complex_cfcₙ_eq_cfc_inr, StarSubalgebra.one_mem_toNonUnitalStarSubalgebra, LDL.diag_eq_lowerInv_conj, Matrix.conjTranspose_list_prod, NonUnitalStarAlgebra.map_top, CStarAlgebra.norm_negPart_le, fderiv_star, cfcₙ_star, cfcₙ_nnreal_eq_real, CFC.sqrt_algebraMap, CFC.posPart_algebraMap, star_sub, Matrix.self_mul_conjTranspose_mul_eq_zero, CFC.sqrt_one, NonUnitalStarAlgHom.map_adjoin, Matrix.isHermitian_zero, CFC.nnrpow_eq_cfcₙ_real, star_le_one_iff, ZeroAtInftyContinuousMap.star_apply, skewAdjoint.conjugate, imaginaryPart_imaginaryPart, IsSelfAdjoint.ofNat, NonUnitalStarAlgebra.star_self_mem_adjoin_singleton, StarSubalgebra.coe_centralizer, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, NonUnitalStarAlgebra.elemental.le_iff_mem, Matrix.isHermitian_one, StarSubalgebra.inclusion_injective, cfcHom_predicate, Subalgebra.mem_star_iff, tsum_star, Matrix.conjTranspose_single, CFC.negPart_def, NonUnitalStarSubalgebra.center_eq_top, spinGroup.star_mul_self_of_mem, NonUnitalStarSubalgebra.coe_prod, star_right_conjugate_nonneg, toStarModuleNat, CFC.sqrt_rpow_nnreal, WeakDual.Complex.instStarHomClass, instContinuousStarReal, NonUnitalStarAlgebra.eq_top_iff, nnnorm_cfc_nnreal_lt_iff, PositiveLinearMap.instStarHomClassOfLinearMapClassComplexOfOrderHomClass, imaginaryPart_smul, norm_cfcHom, StarAlgebra.adjoin_eq_span, Unitization.norm_splitMul_snd_sq, EuclideanSpace.inner_toLp_toLp, IsSelfAdjoint.nnratCast, Matrix.conjTranspose_mul_self_eq_zero, CFC.norm_rpow, realPart_apply_coe, starLinearEquiv_apply, dotProduct_self_star_eq_zero, Matrix.conjTranspose_rat_smul, isSelfAdjoint_smul_of_mem_skewAdjoint, star_finsuppSum, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, MonotoneOn.nnnorm_cfc, conjugate_pos, DifferentiableOn.star, NonUnitalStarAlgebra.self_mem_adjoin_singleton, Matrix.IsUnit.posSemidef_star_left_conjugate_iff, OrthonormalBasis.toMatrix_orthonormalBasis_self_mul_conjTranspose, NonUnitalStarAlgebra.map_inf, cfcₙL_apply, ContinuousOn.cfcₙ_nnreal, cfcHomSuperset_continuous, cfcₙ_apply_mem_elemental, IsStarNormal.one_add, CFC.posPart_def, NonUnitalStarAlgHom.coe_inr, CFC.nnrpow_eq_rpow, pinGroup.coe_star, Matrix.conjTransposeRingEquiv_apply, CFC.cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one, ContinuousMap.elemental_id_eq_top, Unitization.starLift_range_le, DoubleCentralizer.instStarModule, WeakDual.CharacterSpace.compContinuousMap_comp, star_nonpos_iff, differentiable_star_iff, Complex.im_eq_zero_iff_isSelfAdjoint, LinearMap.isSymm_iff_basis, continuousOn_cfc_nnreal_setProd, CStarRing.nnnorm_self_mul_star, Homeomorph.compStarAlgEquiv'_symm_apply, OrthonormalBasis.toMatrix_orthonormalBasis_conjTranspose_mul_self, starRingEquiv_apply, CFC.abs_one, Matrix.vecMul_self_mul_conjTranspose_eq_zero, CentroidHom.starCenterToCentroid_apply, CStarAlgebra.nonneg_TFAE, nnnorm_cfcHom, Matrix.PosSemidef.sqrt_mul_self, Matrix.isHermitian_diagonal_iff, NonUnitalStarAlgHom.coe_inl, realPart_smul, summable_star_iff', cfcₙ_star_id, CFC.abs_mul_abs, NonUnitalStarSubalgebra.prod_top, star_natCast, ContinuousMapZero.elemental_eq_top, ContinuousMapZero.adjoin_id_dense, Matrix.l2_opNNNorm_conjTranspose_mul_self, lp.star_apply, Matrix.cstar_norm_def, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, CStarAlgebra.toStarModule, CFC.isUnit_nnrpow_iff, Unitization.starLift_symm_apply, Matrix.isHermitian_transpose_mul_self, star_le_star_iff, selfAdjoint.isStarNormal, inner_matrix_row_row, CFC.monotoneOn_cfcₙ_rpowIntegrand₀₁, conjugate_le_conjugate', NonUnitalStarAlgebra.sInf_toNonUnitalSubalgebra, Matrix.conjTranspose_mul_self_mul_eq_zero, star_mul_self_pos, Matrix.isHermitian_diagonal, TensorProduct.star_map_apply_eq_map_intrinsicStar, StarSubalgebra.mem_map, StarAlgHom.map_adjoin, IsStarNormal.instIsometricContinuousFunctionalCalculus, cfc_star, star_ratCast, cfcHom_mono, IsometricContinuousFunctionalCalculus.isometric, Matrix.conjTranspose_intCast, Matrix.conjTranspose_add, ContinuousAt.cfc_nnreal, NonUnitalSubalgebra.starClosure_toNonUnitalSubalgebra, Unitary.argSelfAdjoint_coe, Matrix.isHermitian_intCast, skewAdjoint.I_smul_neg_I, LinearMap.toMatrix_adjoint, NonUnitalStarAlgebra.adjoin_le_starAlgebra_adjoin, spinGroup.coe_star, cfcₙHom_of_cfcHom_map_quasispectrum, WeakDual.CharacterSpace.compContinuousMap_id, ContinuousMapZero.instStarModule, MeasureTheory.AEEqFun.eLpNorm_star, StarSubalgebra.map_id, BoundedContinuousFunction.separatesPoints_charPoly, ContinuousWithinAt.cfc_nnreal, star_nonneg_iff, cfcₙHomSuperset_apply, IsSelfAdjoint.instNonUnitalIsometricContinuousFunctionalCalculus, NonUnitalStarAlgebra.elemental.instCompleteSpaceSubtypeMemNonUnitalStarSubalgebra, starAddEquiv_apply, cfcₙ_apply, nnnorm_cfcₙ_nnreal_le, CFC.sqrt_eq_real_sqrt, deriv.star', apply_le_nnnorm_cfcₙ_nnreal, HasFDerivAt.star_star, continuousOn_cfcₙ_nnreal_setProd, cfcHom_eq_of_isStarNormal, Matrix.UnitaryGroup.star_mul_self, Matrix.self_mul_conjTranspose_eq_zero, Matrix.conjTranspose_mul, ContinuousMapZero.coe_toContinuousMapHom, CFC.norm_sqrt, CFC.continuousOn_rpow, conjugate_nonneg, Matrix.kroneckerStarAlgEquiv_apply, NonUnitalStarAlgHom.subsingleton, cfcHom_map_spectrum, MonotoneOn.nnnorm_cfcₙ, StarSubalgebra.starMemClass, StarSubsemiring.coe_ofClass, Unitization.inrRangeEquiv_symm_apply, NonUnitalStarSubalgebra.mem_prod, StarSubalgebra.isEmbedding_inclusion, cfcₙHom_nonneg_iff, CStarRing.mul_star_self_eq_zero_iff, CFC.nnnorm_nnrpow, Matrix.IsUnit.posDef_star_left_conjugate_iff, IsStarProjection.zero, star_rat_smul, CStarRing.nnnorm_star_mul_self, Matrix.PosDef.posDef_sqrt, star_left_conjugate_nonneg, Matrix.posSemidef_conjTranspose_iff, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, continuousFunctionalCalculus_map_id, CFC.norm_nnrpow, NonUnitalStarSubalgebra.inclusion_self, toStarModuleInt, Differentiable.star, InnerProductSpace.symm_toEuclideanLin_rankOne, MeasureTheory.Lp.coeFn_star, Commute.cfcₙ_real, Matrix.PosDef.mul_conjTranspose_self, Matrix.frobenius_nnnorm_conjTranspose, Matrix.conjTranspose_ofNat, NonUnitalSubalgebra.mem_starClosure, Matrix.rank_conjTranspose_mul_self, LinearMap.intrinsicStar_eq_comp, Unitary.path_apply, HasDerivAt.star_conj, Matrix.IsHermitian.instContinuousFunctionalCalculus, Matrix.PosDef.conjTranspose, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, cfcHom_apply_mem_elemental, PositiveLinearMap.preGNS_norm_sq, Matrix.kroneckerStarAlgEquiv_symm_apply, Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg, WeakDual.CharacterSpace.instStarHomClass, pinGroup.mul_star_self_of_mem, NonUnitalStarAlgebra.mem_inf, NonUnitalStarAlgHom.zero_apply, Matrix.blockDiagonal'_conjTranspose, DifferentiableWithinAt.star, Matrix.dotProduct_self_star_pos_iff, CStarModule.isSelfAdjoint_inner_self, conjugate_le_conjugate, Unitization.isStarNormal_inr, Matrix.PosSemidef.sqrt_eq_iff_eq_sq, CFC.rpow_eq_cfc_real, star_neg_iff, RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict, Matrix.PosDef.isHermitian, Matrix.mul_self_mul_conjTranspose_eq_zero, inner_matrix_col_col, Matrix.PosDef.dotProduct_mulVec_pos, algebraMap_star_comm, Continuous.cfcₙ_nnreal', Matrix.UnitaryGroup.inv_apply, Matrix.conjTranspose_invOf, CStarMatrix.inner_toCLM_conjTranspose_right, CFC.isUnit_sqrt_iff, Matrix.IsUnit.posDef_star_right_conjugate_iff, Matrix.PosDef.re_dotProduct_pos, spinGroup.star_mem_iff, ZeroAtInftyContinuousMap.coe_star, isometry_cfcHom, LinearMap.isSymm_iff_isHermitian_toMatrix, NonUnitalStarAlgebra.map_iInf, NonUnitalStarAlgebra.inf_toNonUnitalSubalgebra, NonUnitalSubalgebra.star_mem_star_iff, NonUnitalStarAlgebra.adjoin_le_centralizer_centralizer, Matrix.mul_conjTranspose_mul_self_eq_zero, Matrix.conjTranspose_nsmul, Matrix.star_mulVec, HasStrictDerivAt.star, CFC.rpow_sqrt, IsSelfAdjoint.cfcₙ, star_intCast_smul, NNReal.instContinuousStar, NonUnitalSubalgebra.star_adjoin_comm

Theorems

NameKindAssumesProvesValidatesDepends On
star_add 📖mathematicalStar.star
InvolutiveStar.toStar
toInvolutiveStar
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
toStarModuleInt 📖mathematicalStarModule
InvolutiveStar.toStar
toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Int.instCommRing
StarRing.toStarAddMonoid
Int.instStarRing
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
SubNegMonoid.toZSMul
star_zsmul
toStarModuleNat 📖mathematicalStarModule
InvolutiveStar.toStar
toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
Nat.instNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
Nat.instStarRing
AddMonoid.toNatSMul
star_nsmul

StarHomClass

Theorems

NameKindAssumesProvesValidatesDepends On
map_star 📖mathematicalDFunLike.coe
Star.star

StarMemClass

Definitions

NameCategoryTheorems
instStar 📖CompOp
45 mathmath: StarAlgEquiv.ofInjective_symm_apply, StarAlgEquiv.ofLeftInverse'_apply, NonUnitalStarAlgHom.coe_codRestrict, StarSubalgebra.inclusion_apply, CentroidHom.star_centerToCentroidCenter, CentroidHom.starCenterIsoCentroid_symm_apply_coe, NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, CentroidHom.starCenterIsoCentroid_apply, StarSubalgebra.subtype_comp_inclusion, StarAlgEquiv.ofInjective_apply, NonUnitalStarAlgHom.injective_codRestrict, NonUnitalStarSubalgebra.val_inclusion, StarAlgHom.coe_codRestrict, StarSubalgebra.subtype_apply, StarSubalgebra.isClosedEmbedding_inclusion, NonUnitalStarSubalgebra.toNonUnitalSubalgebra_subtype, NonUnitalStarSubalgebra.range_val, NonUnitalStarSubalgebra.inclusion_mk, NonUnitalStarSubalgebra.unitization_injective, NonUnitalStarSubalgebra.unitization_apply, NonUnitalStarSubalgebraClass.coe_subtype, StarSubalgebra.coe_subtype, coe_star, StarSubalgebra.starModule, NonUnitalStarSubalgebra.inclusion_right, NonUnitalStarSubalgebraClass.subtype_injective, StarSubalgebra.toSubalgebra_subtype, NonUnitalStarSubalgebra.toSubring_subtype, NonUnitalStarAlgHom.subtype_comp_codRestrict, Unitization.inrRangeEquiv_apply_coe, StarAlgHom.subtype_comp_codRestrict, StarSubalgebra.inclusion_injective, StarAlgEquiv.ofInjective'_apply, NonUnitalStarSubalgebra.inclusion_injective, CentroidHom.starCenterToCentroid_apply, StarAlgEquiv.ofLeftInverse'_symm_apply, cfcHom_eq_of_isStarNormal, StarAlgHom.injective_codRestrict, NonUnitalStarSubalgebraClass.subtype_apply, Unitization.inrRangeEquiv_symm_apply, StarSubalgebra.isEmbedding_inclusion, continuousFunctionalCalculus_map_id, NonUnitalStarSubalgebra.inclusion_self, NonUnitalStarSubalgebra.inclusion_inclusion, instStarModule

Theorems

NameKindAssumesProvesValidatesDepends On
coe_star 📖mathematicalSetLike.instMembership
Star.star
instStar
star_mem 📖mathematicalSetLike.instMembershipStar.star

StarModule

Theorems

NameKindAssumesProvesValidatesDepends On
star_smul 📖mathematicalStar.star

StarMul

Definitions

NameCategoryTheorems
toInvolutiveStar 📖CompOp
177 mathmath: unitary.spectrum.unitary_conjugate, continuous_decomposeProdAdjoint, StarModule.decomposeProdAdjointL_symm_apply, StarModule.selfAdjointPart_add_skewAdjointPart, Unitary.conjStarAlgAut_trans_conjStarAlgAut, star_mul_star, IsIdempotentElem.star_iff, IsUnit.star, NonUnitalCStarAlgebra.toStarModule, IsSelfAdjoint.star_mul_self, Unitary.conjStarAlgAut_symm, unitary.map_id, isRightRegular_star_iff, val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe, unitary.inv_mul_mem_iff, Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv, Unitary.mapEquiv_symm_apply, IsCHSHTuple.B₀_sa, commute_star_star, Pi.star_mulSingle, selfAdjointPart_comp_subtype_skewAdjoint, unitary.coe_star_mul_self, IsSelfAdjoint.selfAdjointPart_apply, Unitary.inv_mul_mem_iff, StarModule.decomposeProdAdjointL_apply, IsCHSHTuple.B₁_sa, Set.star_centralizer, Set.star_inv', unitary.spectrum.unitary_conjugate', unitary.mem_iff_self_mul_star, unitary.star_mul_self_of_mem, selfAdjointPart_comp_subtype_selfAdjoint, unitary.map_comp, Units.instStarModule, IsSelfAdjoint.mul_star_self, MulChar.star_apply, IsLeftRegular.star, starMulAut_apply, unitary.mapEquiv_trans, Unitary.spectrum_star_left_conjugate, Units.mul_inv_mem_unitary, NonUnitalStarSubalgebra.unitization_range, unitary.mem_iff, Matrix.conjTranspose_kronecker, Unitary.mem_iff_self_mul_star, Unitary.map_coe, unitary.toUnits_comp_map, continuous_decomposeProdAdjoint_symm, toStarModule, Unitary.star_mul_self_of_mem, StarRing.star_add, unitary.mem_iff_star_mul_self, Units.coe_star, IsCHSHTuple.A₁_sa, IsStarNormal.val_inv, Unitary.spectrum_star_right_conjugate, IsSelfAdjoint.skewAdjointPart_apply, FreeMonoid.star_one, Unitary.conjStarAlgAut_ext_iff', star_finsuppProd, Set.star_inv, StarModule.decomposeProdAdjoint_symm_apply, IsStarProjection.one, Unitary.map_comp, Unitary.mapEquiv_apply, unitary.star_mem, TrivialStar.isStarNormal, Matrix.IsHermitian.cfcAux_apply, IsCHSHTuple.A₀_sa, isSelfAdjoint_algebraMap_iff, IsSelfAdjoint.algebraMap, StarAlgHom.ofId_apply, CommCStarAlgebra.toStarModule, Unitary.mul_inv_mem_iff, isRegular_star_iff, starMulEquiv_apply, imaginaryPart_eq_neg_I_smul_skewAdjointPart, IsUnit.isSelfAdjoint_conjugate_iff, skewAdjointPart_comp_subtype_selfAdjoint, Unitary.conjStarAlgAut_apply, unitary.mapEquiv_symm, Unitary.mem_iff_star_mul_self, Unitary.toAlgEquiv_conjStarAlgAut, Unitary.coe_isStarNormal, NonUnitalCommCStarAlgebra.toStarModule, Unitary.conjStarAlgAut_symm_apply, Unitary.toRingEquiv_conjStarAlgAut, instContinuousStarUnits, Matrix.IsHermitian.spectral_theorem, star_prod, unitary.coe_map, IsRightRegular.star, isLeftRegular_star_iff, SpectrumRestricts.starAlgHom_apply, Set.star_mem_center, Unitary.toUnits_comp_map, Unitary.coe_map_star, skewAdjointPart_apply_coe, continuous_selfAdjointPart, Matrix.star_eq_inv, selfAdjointPartL_apply_coe, SemiconjBy.star_star_star, IsStarNormal.one, unitary.coe_map_star, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, IsSelfAdjoint.coe_selfAdjointPart_apply, Unitary.mapEquiv_symm, MulChar.star_eq_inv, isSelfAdjoint_conjugate_iff_of_isUnit', StarSemigroup.toOpposite_starModule, IsStarNormal.star, star_id_of_comm, Unitary.conjStarAlgAut_mul_apply, unitary.toMonoidHom_mapEquiv, Unitary.map_id, selfAdjointPart_apply_coe, Unitary.mapEquiv_trans, skewAdjointPartL_apply_coe, star_div, Matrix.specialUnitaryGroup.coe_star, unitary.coe_star, Unitary.coe_map, Unitary.mul_star_self_of_mem, unitary.mul_inv_mem_iff, FreeMonoid.star_of, star_mul, skewAdjointPart_eq_I_smul_imaginaryPart, star_star_mul, IsIdempotentElem.star, unitary.mapEquiv_refl, Unitary.mem_iff, isStarNormal_of_mem_unitary, Matrix.conjTranspose_vecMulVec, Units.embed_product_star, Matrix.IsHermitian.star_mul_self_mul_eq_diagonal, commute_star_comm, Units.coe_star_inv, star_zpow, Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv, IsRegular.star, CStarAlgebra.toStarModule, isUnit_star, star_div₀, star_mul', star_zpow₀, Unitary.coe_star_mul_self, Unitary.toMonoidHom_mapEquiv, IsUnit.mem_unitary_iff_mul_star_self, Unitary.star_mem_iff, star_invOf, Matrix.conjTranspose_smul_self, Matrix.IsHermitian.conjStarAlgAut_star_eigenvectorUnitary, Matrix.conjTranspose_kronecker', CommMonoid.isStarNormal, IsSelfAdjoint.one, unitary.star_mem_iff, Unitary.coe_star, skewAdjointPart_comp_subtype_skewAdjoint, continuous_skewAdjointPart, Unitary.conjStarAlgAut_star_apply, Unitary.mapEquiv_refl, semiconjBy_star_star_star, Unitary.star_mem, IsUnit.mem_unitary_iff_star_mul_self, Unitary.conjStarAlgAut_ext_iff, star_one, Set.star_mul, star_inv, IsUnit.isSelfAdjoint_conjugate_iff', Units.inv_mul_mem_unitary, star_inv₀, StarModule.decomposeProdAdjoint_apply, algebraMap_star_comm, unitary.mul_star_self_of_mem, isSelfAdjoint_conjugate_iff_of_isUnit, Commute.star_star, star_pow

Theorems

NameKindAssumesProvesValidatesDepends On
star_mul 📖mathematicalStar.star
InvolutiveStar.toStar
toInvolutiveStar
toStarModule 📖mathematicalStarModule
InvolutiveStar.toStar
toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
Monoid.toMulAction
star_mul'

StarRing

Definitions

NameCategoryTheorems
toStarAddMonoid 📖CompOp
838 mathmath: Unitary.openPartialHomeomorph_source, range_cfcₙ_nnreal, Matrix.l2_opNorm_toEuclideanCLM, ContinuousMapZero.coe_star, NonUnitalSubalgebra.coe_starClosure, Matrix.posSemidef_conjTranspose_mul_self, NonUnitalStarSubalgebra.prod_toNonUnitalSubalgebra, cfcₙ_def, continuous_cfcₙHomSuperset_left, unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, norm_cfcₙHom, cfcₙHom_mono, IsStarNormal.instNonUnitalContinuousFunctionalCalculus, Matrix.kroneckerTMulStarAlgEquiv_symm_apply, Complex.coe_realPart, IsStarNormal.one_sub, DoubleCentralizer.star_snd, selfAdjoint.val_div, StarAddMonoid.toStarModuleNNRat, Unitary.norm_expUnitary_smul_argSelfAdjoint_sub_one_le, NonUnitalIsometricContinuousFunctionalCalculus.isometric, cfcₙHom_apply_mem_elemental, StarSubalgebra.inclusion_apply, CFC.sqrt_eq_one_iff', CFC.posPart_one, AlgHomClass.instStarHomClass, NonUnitalCStarAlgebra.toStarModule, nnnorm_cfcₙ_nnreal_le_iff, realPart_idem, star_ofNat, Matrix.posSemidef_iff_eq_conjTranspose_mul_self, NonUnitalStarAlgebra.mem_bot, IsUnit.isStrictlyPositive_star_right_conjugate_iff, CStarAlgebra.nonneg_iff_exists_isSelfAdjoint_and_eq_mul_self, NonUnitalStarSubalgebra.centralizer_le, NonUnitalStarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, CFC.posPart_natCast, RingHom.star_apply, Matrix.trace_mul_conjTranspose_self_eq_zero_iff, CFC.tendsto_cfc_rpow_sub_one_log, StarSubalgebra.coe_map, Matrix.cstar_nnnorm_def, NonUnitalStarSubsemiring.instNonUnitalSubsemiringClass, realPart_ofReal, spinGroup.coe_star_mul_self, CFC.sq_sqrt, RCLike.star_def, Matrix.isHermitian_conjTranspose_mul_self, pinGroup.coe_star_mul_self, Unitary.mem_pathComponentOne_iff, instTrivialStarReal, CentroidHom.star_centerToCentroidCenter, BoundedContinuousFunction.char_neg, Unitization.real_cfcₙ_eq_cfc_inr, CliffordAlgebraQuaternion.toQuaternion_star, imaginaryPart_I_smul, NumberField.mem_maximalRealSubfield_iff, Matrix.PosSemidef.isHermitian, isClosedEmbedding_cfcₙAux, Complex.star_def, Matrix.posSemidef_self_mul_conjTranspose, selfAdjoint.expUnitaryPathToOne_apply, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, Unitary.continuousOn_argSelfAdjoint, CFC.nnnorm_sqrt, Subalgebra.star_adjoin_comm, LinearMap.intrinsicStar_mulRight, CentroidHom.starCenterIsoCentroid_symm_apply_coe, LE.le.star_eq, NonUnitalStarAlgebra.elemental.isClosedEmbedding_coe, Matrix.conjTranspose_mul_self_mulVec_eq_zero, IsUnit.star_right_conjugate_nonneg_iff, NonUnitalStarSubalgebra.subsingleton_of_subsingleton, StarSubsemiring.mem_toSubsemiring, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply_comp_toReal, CFC.sqrt_rpow, Matrix.PosSemidef.sqrt_eq_zero_iff, UnitAddTorus.mFourierSubalgebra_coe, Unitization.starAlgHom_ext_iff, CFC.abs_nnrpow_two, instStarModuleNNRealOfReal, RCLike.is_real_TFAE, selfAdjoint.val_mul, Matrix.intrinsicStar_toLin', NonUnitalStarSubalgebra.unitizationStarAlgEquiv_apply_coe, LinearMap.toMatrixOrthonormal_reindex, dotProduct_star_self_eq_zero, Matrix.posDef_iff_dotProduct_mulVec, UnitAddTorus.mFourierSubalgebra_closure_eq_top, CentroidHom.starCenterIsoCentroid_apply, StarSubalgebra.subtype_comp_inclusion, isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts, Matrix.PosDef.eigenvalues_pos, CFC.negPart_algebraMap, cfcₙHom_nnreal_eq_restrict, star_dotProduct_toMatrix₂_mulVec, cfcHom_continuous, polynomialFunctions.starClosure_topologicalClosure, cfcL_apply, WithCStarModule.inner_def, StarAddMonoid.toStarModuleRat, NonUnitalStarAlgebra.mem_sInf, cfcHom_nonneg_iff, Matrix.isHermitian_mul_conjTranspose_self, gelfandStarTransform_naturality, gelfandStarTransform_apply_apply, Matrix.IsUnit.posSemidef_star_right_conjugate_iff, cfc_apply_mkD, instStarModuleNNRealReal, selfAdjoint.val_one, Unitization.starLift_symm_apply_apply, Unitization.starLift_apply_apply, Continuous.cfc_nnreal', StarOrderedRing.lt_iff, NonUnitalSubalgebra.starClosure_le_iff, Matrix.PosDef.fromBlocks₁₁, CStarRing.norm_mul_self_le, RCLike.im_eq_zero_iff_isSelfAdjoint, Matrix.dotProduct_star_self_pos_iff, cfcHom_isClosedEmbedding, StarAlgebra.adjoin_nonUnitalStarSubalgebra, StarSubalgebra.map_map, Matrix.conjTranspose_natCast, LE.le.isSelfAdjoint, IsUnit.cfcSqrt, CFC.continuousOn_log, Complex.instContinuousStar, Matrix.PosSemidef.conjTranspose_mul_mul_same, ContinuousOn.cfc_nnreal', IsGreatest.nnnorm_cfc_nnreal, NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra, NonUnitalStarAlgebra.coe_top, skewAdjoint.instIsStarNormalValMemAddSubgroup, LinearMap.toMatrixOrthonormal_apply_apply, NonUnitalStarAlgebra.coe_sInf, MulChar.star_apply, cfcₙ_apply_mkD, cfcₙHom_id, CStarMatrix.inner_toCLM_conjTranspose_left, Matrix.posSemidef_vecMulVec_self_star, CFC.continuousOn_sqrt, Matrix.star_dotProduct, StarOrderedRing.pos_iff, PositiveLinearMap.preGNS_norm_def, IsSelfAdjoint.intCast, star_pos_iff, Matrix.ker_mulVecLin_conjTranspose_mul_self, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, CliffordAlgebra.star_def', Matrix.conjTranspose_permMatrix, FreeAlgebra.star_algebraMap, Pi.intrinsicStar_comul_commSemiring, IsStarNormal.cfcₙ_map, cfcHom_le_iff, cfcHom_id, Matrix.rank_conjTranspose, Matrix.PosSemidef.det_sqrt, spinGroup.mul_star_self_of_mem, range_cfcₙHom, StarSubalgebra.mem_comap, NonUnitalStarSubalgebra.center_toNonUnitalSubalgebra, Unitary.openPartialHomeomorph_symm_apply, Matrix.PosDef.conjTranspose_mul_mul_same, starRingAut_apply, NonUnitalStarSubalgebra.unitization_range, Matrix.toAlgEquiv_kroneckerTMulStarAlgEquiv, Matrix.dotProduct_star, StarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, NonUnitalStarAlgebra.adjoin_toNonUnitalSubalgebra, Matrix.mem_unitaryGroup_iff, BoundedContinuousFunction.mem_charPoly, Matrix.star_dotProduct_star, star_right_conjugate_pos, selfAdjoint.instNontrivialSubtypeMemAddSubgroup, Commute.cfc_real, CFC.abs_nnrpow, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, selfAdjoint.imaginaryPart_coe, CliffordAlgebra.star_def, nnnorm_cfcₙ_nnreal_lt, CFC.sqrt_eq_rpow, selfAdjoint.val_zpow, NonUnitalStarSubsemiring.coe_ofClass, LinearMap.intrinsicStar_mul', CStarAlgebra.nonneg_iff_isSelfAdjoint_and_negPart_eq_zero, NumberField.ComplexEmbedding.IsConj.eq, cfc_real_eq_nnreal, Matrix.PosSemidef.posSemidef_sqrt, cfcₙHom_integral, mul_star_self_nonneg, NonUnitalStarSubsemiring.instStarMemClass, StarAlgHom.realContinuousMapOfNNReal_apply, star_left_conjugate_pos, CStarAlgebra.isStrictlyPositive_iff_exists_isUnit_and_isSelfAdjoint_and_eq_mul_self, BoundedContinuousFunction.instStarModule, StarSubsemiring.toSubsemiring_le_iff, CentroidHom.star_apply, gelfandStarTransform_symm_apply, Unitary.expUnitary_eq_mul_inv, IsSelfAdjoint.cfc, Matrix.star_vecMul, CStarModule.inner_smul_left_complex, Nat.instTrivialStar, Matrix.eigenvalues_conjTranspose_mul_self_nonneg, StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, CStarModule.star_inner, Matrix.toEuclideanCLM_toLp, WeakDual.CharacterSpace.homeoEval_naturality, realPart.norm_le, CFC.posPart_algebraMap_nnreal, selfAdjoint.expUnitary_zero, range_cfcₙ, Matrix.PosSemidef.sqrt_eq_one_iff, CStarRing.norm_star_mul_self', LinearMap.IntrinsicStar.starLinearEquiv_eq_arrowCongr, Matrix.PosSemidef.sqrt_sq, RCLike.mul_wInner_left, PositiveLinearMap.preGNS_inner_def, LinearMap.IsSymmetric.coe_toSelfAdjoint, cfcₙAux_id, star_intCast, StarSubalgebra.subtype_apply, CFC.sqrt_eq_one_iff, realPart_add_I_smul_imaginaryPart, CStarAlgebra.norm_posPart_le, Matrix.det_conjTranspose, IsGreatest.nnnorm_cfcₙ_nnreal, NonUnitalSubalgebra.mem_star_iff, Matrix.PosSemidef.eq_sqrt_iff_sq_eq, isometry_cfcₙHom, Unitization.starMap_id, StarAlgebra.elemental.bijective_characterSpaceToSpectrum, CFC.continuous_abs, StarSubalgebra.isClosedEmbedding_inclusion, NonUnitalStarAlgebra.adjoin_mono, StarSubsemiring.toSubsemiring_injective, gelfandTransform_map_star, dotProduct_star_self_nonneg, polynomialFunctions.starClosure_eq_adjoin_X, continuousOn_cfc_nnreal, Matrix.conjTranspose_nonsing_inv, selfAdjoint.continuous_expUnitary, pinGroup.star_mem, FreeAlgebra.star_ι, cfcHom_integral, Matrix.l2_opNorm_conjTranspose_mul_self, Matrix.conjTranspose_swap, StarSubsemiring.coe_toSubsemiring, cfcₙHom_predicate, cfc_nnreal_eq_real, StarSubsemiring.subsemiringClass, CStarRing.star_mul_self_eq_zero_iff, Unitization.starLift_range, Matrix.posDef_conjTranspose_iff, StarSubsemiring.mem_carrier, Matrix.PosSemidef.eigenvalues_nonneg, IsSelfAdjoint.log, ContinuousFunctionalCalculus.exists_cfc_of_predicate, Matrix.PosSemidef.mul_mul_conjTranspose_same, NonUnitalStarAlgebra.mem_iInf, CFC.abs_natCast, Matrix.adjugate_conjTranspose, CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt, TensorProduct.instStarModule, LDL.lower_conj_diag, IsIdempotentElem.isSelfAdjoint_iff_isStarNormal, NonUnitalStarAlgebra.adjoin_eq_span, Matrix.l2_opNNNorm_conjTranspose, IsSelfAdjoint.cfc_arg, NonUnitalStarAlgebra.gc, CliffordAlgebraQuaternion.ofQuaternion_star, NonUnitalStarAlgebra.subset_adjoin, LinearMap.IsSymmetric.toSelfAdjoint_apply, cfc_def, DoubleCentralizer.star_fst, Unitization.starMap_inl, fourierSubalgebra_separatesPoints, NonUnitalStarSubalgebra.range_val, StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal, LinearMap.toMatrixOrthonormal_symm_apply, LinearMap.intrinsicStar_rTensor, NonUnitalSubalgebra.starClosure_mono, LinearMap.toMatrix_innerₛₗ_apply, Unitization.starMap_inr, Matrix.eigenvalues_self_mul_conjTranspose_nonneg, star_lt_one_iff, NonUnitalStarAlgebra.elemental.isClosed, CFC.abs_nnrpow_two_mul, Continuous.cfc_nnreal_of_mem_nhdsSet, Matrix.PosSemidef.toLinearMap₂'_zero_iff, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, Filter.IsIncreasingApproximateUnit.eventually_isSelfAdjoint, NormedSpace.star_exp, IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus, Matrix.l2_opNorm_conjTranspose, TensorProduct.intrinsicStar_map, Complex.selfAdjointEquiv_apply, IsSelfAdjoint.natCast, Matrix.conjTranspose_eq_natCast, CliffordAlgebra.star_smul, Continuous.cfcₙ_nnreal, Matrix.PosDef.conjTranspose_mul_self, CStarAlgebra.isStrictlyPositive_iff_eq_mul_star_self, CFC.abs_star, NonUnitalStarAlgebra.range_eq_top, CStarAlgebra.linear_combination_nonneg, NonUnitalStarSubalgebra.unitization_injective, NonUnitalStarSubalgebra.unitization_apply, IsUnit.isStrictlyPositive_star_left_conjugate_iff, NonUnitalStarSubalgebra.coe_centralizer, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, ContinuousAt.cfcₙ_nnreal, star_mul_self_add_self_mul_star, nonneg_iff_isSelfAdjoint_and_spectrumRestricts, range_cfcₙ_eq_range_cfcₙHom, ContinuousMap.adjoin_id_eq_span_one_union, imaginaryPart.norm_le, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, StarSubalgebra.centralizer_toSubalgebra, CFC.abs_algebraMap, NonUnitalStarSubsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallStar, Matrix.isHermitian_gram, LinearMap.IsSymmetric.isSymmetric_smul_iff, realPart_comp_subtype_selfAdjoint, Matrix.ofLp_toEuclideanCLM, IsStrictlyPositive.nnrpow, Matrix.exp_conjTranspose, Circle.star_addChar, Matrix.kroneckerTMulStarAlgEquiv_apply, StarSubalgebra.star_mem', StarSubalgebra.coe_subtype, isSelfAdjoint_algebraMap_iff, RCLike.instContinuousStar, Ring.inverse_star, ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply, Matrix.conjTranspose_eq_ofNat, starRingEnd_apply, Matrix.isDiag_conjTranspose_iff, inr_comp_cfcₙHom_eq_cfcₙAux, StarAlgebra.star_self_mem_adjoin_singleton, CFC.monotoneOn_one_sub_one_add_inv_real, IsUnit.star_left_conjugate_nonneg_iff, unitary.norm_argSelfAdjoint, skewAdjoint.negISMul_apply_coe, NonUnitalStarSubsemiring.mem_carrier, Matrix.UnitaryGroup.inv_val, NonUnitalStarSubalgebra.mem_centralizer_iff, ContinuousWithinAt.cfcₙ_nnreal, StarAlgHom.ofId_apply, CommCStarAlgebra.toStarModule, QuasispectrumRestricts.isSelfAdjoint, NonUnitalStarAlgebra.elemental.self_mem, CFC.continuousOn_nnrpow, imaginaryPart_realPart, Matrix.isHermitian_inv, StarAlgebra.star_subset_adjoin, star_mul_self_nonneg, Matrix.star_mul, imaginaryPart_eq_neg_I_smul_skewAdjointPart, range_cfcₙ_nnreal_eq_image_cfcₙ_real, StarSubalgebra.map_toSubalgebra, Matrix.posDef_iff_eq_conjTranspose_mul_self, Filter.Tendsto.cfcₙ_nnreal, WeakDual.CharacterSpace.compContinuousMap_apply, ContinuousLinearMap.instStarModuleId, RCLike.wInner_smul_left, continuous_cfcHomSuperset_left, IsUnit.cfcNNRpow, Matrix.conjTranspose_eq_intCast, star_right_conjugate_lt_conjugate, cfcₙHom_isClosedEmbedding, Subalgebra.coe_star, apply_eq_star_dotProduct_toMatrix₂_mulVec, realPart_I_smul, Matrix.vecMul_conjTranspose_mul_self_eq_zero, SpectrumRestricts.isSelfAdjoint, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, ContinuousOn.cfc_nnreal, nnnorm_cfcₙ_nnreal_lt_iff, PositiveLinearMap.gnsStarAlgHom_apply, mul_star_self_pos, Matrix.trace_conjTranspose_mul_self_eq_zero_iff, IsSelfAdjoint.of_nonneg, LinearMap.toMatrix'_intrinsicStar, cfcₙHom_le_iff, nnnorm_cfc_nnreal_lt, isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts, imaginaryPart_apply_coe, Unitary.openPartialHomeomorph_target, Matrix.posSemidef_vecMulVec_star_self, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_apply, NonUnitalStarAlgebra.star_subset_adjoin, IsSelfAdjoint.ratCast, ContinuousMap.instCStarRing, IsStrictlyPositive.sqrt, StarSubalgebra.comap_id, CStarRing.norm_star_mul_self, Matrix.PosSemidef.isUnit_sqrt_iff, cfcₙ_real_eq_nnreal, imaginaryPart_comp_subtype_selfAdjoint, nnnorm_cfc_nnreal_le, CStarModule.inner_op_smul_left, CStarMatrix.mul_entry_mul_eq_inner_toCLM, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Matrix.coe_toEuclideanCLM_eq_toEuclideanLin, Matrix.PosSemidef.conjTranspose, one_lt_star_iff, Matrix.rank_self_mul_conjTranspose, Continuous.cfc_nnreal, StarAlgebra.elemental.star_self_mem, cfcHomSuperset_apply, star_lt_star_iff, nnnorm_cfc_nnreal_le_iff, isStarProjection_one_sub_iff, Matrix.conjTransposeRingEquiv_symm_apply, LinearMap.toMatrixOrthonormal_apply, CFC.sqrt_sq, cfcₙHom_eq_cfcₙ_extend, selfAdjoint.val_inv, ZeroAtInftyContinuousMap.instCStarRing, expUnitary_argSelfAdjoint, EuclideanSpace.inner_eq_star_dotProduct, NonUnitalSubalgebra.coe_star, NonUnitalCommCStarAlgebra.toStarModule, CFC.rpow_sqrt_nnreal, realPart_surjective, Matrix.conjTranspose_one, cfc_star_id, ContinuousMapZero.instTrivialStar, spectrum.star_mem_resolventSet_iff, NonUnitalStarAlgebra.elemental.le_centralizer_centralizer, NonUnitalStarAlgHom.map_adjoin_singleton, ContinuousMapZero.toContinuousMapHom_toNNReal, Complex.instStarModuleReal, StarSubalgebra.comap_comap, DoubleCentralizer.coeHom_apply, CStarAlgebra.mul_star_le_algebraMap_norm_sq, cfcₙHom_map_quasispectrum, spectrum.map_star, IsSelfAdjoint.instIsometricContinuousFunctionalCalculus, fourierSubalgebra_coe, NonUnitalStarSubalgebra.prod_inf_prod, Matrix.conjTranspose_zpow, Matrix.PosDef.fromBlocks₂₂, star_right_conjugate_le_conjugate, star_le_iff, Matrix.toLin_conjTranspose, RCLike.instStarModuleReal, imaginaryPart_surjective, cfcHom_nnreal_eq_restrict, MulChar.star_apply', NonUnitalStarAlgebra.comap_top, NonUnitalStarAlgebra.elemental.star_self_mem, Matrix.PosSemidef.sq_sqrt, spec_cfcₙAux, selfAdjoint.val_pow, ContinuousOn.cfcₙ_nnreal', pinGroup.star_mem_iff, cfcHom_eq_cfc_extend, star_lt_iff, CFC.abs_mul_self, conjugate_lt_conjugate', selfAdjoint.realPart_coe, star_nnratCast, NNRat.instTrivialStar, LinearMap.instStarModuleId, Unitization.starLift_apply, SpectrumRestricts.starAlgHom_apply, CFC.abs_ofNat, dotProduct_self_star_nonneg, CFC.negPart_one, Matrix.isHermitian_iff_isSymmetric, cfcₙHomSuperset_id, conjugate_lt_conjugate, Matrix.PosSemidef.inv_sqrt, spinGroup.star_mem, Unitization.starMap_comp, InnerProductSpace.toMatrix_rankOne, NonUnitalStarAlgebra.coe_iInf, CFC.abs_algebraMap_nnreal, StarSubsemiring.starMemClass, CliffordAlgebra.star_ι, UnitAddTorus.mFourierSubalgebra_separatesPoints, Matrix.PosSemidef.dotProduct_mulVec_nonneg, Matrix.isUnit_conjTranspose, CStarRing.norm_self_mul_star, CStarMatrix.ofMatrix_eq_ofMatrixStarAlgEquiv, selfAdjoint.val_qsmul, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, ContinuousMap.evalStarAlgHom_apply, NonUnitalStarRingHom.map_le_map_of_map_star, nnnorm_cfcₙHom, CFC.nnnorm_rpow, Matrix.mulVec_conjTranspose, Matrix.conjTranspose_eq_one, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, cfc_apply, IsStarNormal.instContinuousFunctionalCalculus, LinearMap.intrinsicStar_mulLeft, StarSubalgebra.starModule, one_le_star_iff, CStarAlgebra.star_mul_le_algebraMap_norm_sq, NonUnitalStarSubalgebra.mem_center_iff, CStarAlgebra.isStrictlyPositive_iff_isUnit_sqrt_and_eq_sqrt_mul_sqrt, cfc_comp_norm, StarModule.complexToReal, CFC.abs_intCast, apply_le_nnnorm_cfc_nnreal, Matrix.posSemidef_iff_eq_sum_vecMulVec, NonUnitalStarSubalgebra.coe_center, range_cfc_nnreal_eq_image_cfc_real, Complex.coe_selfAdjointEquiv, Matrix.vecMul_conjTranspose, StarAlgebra.adjoin_toSubalgebra, Matrix.isHermitian_natCast, isClosedEmbedding_cfcₙHom_of_cfcHom, LinearMap.intrinsicStar_lTensor, NonUnitalStarAlgHom.realContinuousMapZeroOfNNReal_injective, IsStarNormal.cfc_map, conjugate_pos', imaginaryPart_ofReal, Matrix.self_mul_conjTranspose_mulVec_eq_zero, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, StarOrderedRing.le_iff, Unitization.starMap_apply, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, Matrix.star_dotProduct_gram_mulVec, StarAlgHom.realContinuousMapOfNNReal_injective, NonUnitalStarAlgebra.map_bot, StarSubalgebra.coe_centralizer_centralizer, cfcHom_isStrictlyPositive_iff, Subalgebra.star_mem_star_iff, Matrix.conjTranspose_pow, toMatrix_innerSL_apply, StarSubsemiring.toSubsemiring_inj, Matrix.PosSemidef.re_dotProduct_nonneg, CFC.exists_measure_nnrpow_eq_integral_cfcₙ_rpowIntegrand₀₁, CFC.spectrum_abs, selfAdjoint.val_ratCast, Matrix.IsDiag.conjTranspose, star_left_conjugate_le_conjugate, NonUnitalStarSubalgebra.coe_centralizer_centralizer, Homeomorph.compStarAlgEquiv'_apply, Filter.IsIncreasingApproximateUnit.eventually_star_eq, Matrix.posSemidef_iff_dotProduct_mulVec, cfc_apply_pi, StarSubalgebra.coe_comap, CStarAlgebra.nonneg_iff_eq_star_mul_self, StarSubalgebra.toSubalgebra_subtype, pinGroup.star_mul_self_of_mem, NonUnitalStarAlgebra.mem_adjoin_of_mem, ContinuousMapZero.toContinuousMapHom_apply, NonUnitalStarSubalgebra.centralizer_toNonUnitalSubalgebra, StarOrderedRing.nonneg_iff, cfc_unitary_iff, Matrix.PosSemidef.dotProduct_mulVec_zero_iff, CliffordAlgebra.star_algebraMap, BoundedContinuousFunction.star_mem_range_charAlgHom, Complex.selfAdjointEquiv_symm_apply, Int.instTrivialStar, cfcₙAux_mem_range_inr, ContinuousMap.adjoin_id_eq_span_one_add, cfcₙ_norm_sq_nonneg, StarSubalgebra.mem_centralizer_iff, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', star_left_conjugate_lt_conjugate, Matrix.specialUnitaryGroup.coe_star, NonUnitalStarAlgebra.adjoin_le_iff, conjugate_nonneg', cfcHomSuperset_id, unitary_iff_isStarNormal_and_spectrum_subset_unitary, CStarAlgebra.nonneg_iff_eq_mul_star_self, NonUnitalStarAlgebra.span_eq_toSubmodule, CFC.sqrt_eq_cfc, CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self, Matrix.mem_unitaryGroup_iff', realPart_imaginaryPart, instTrivialStarNNReal, Filter.Tendsto.cfc_nnreal, IsSelfAdjoint.instContinuousFunctionalCalculus, continuousOn_cfcₙ_nnreal, ContinuousMapZero.instCStarRing, cfcₙHom_continuous, CFC.abs_sq, IsStarNormal.instNonUnitalIsometricContinuousFunctionalCalculus, cfcₙHomSuperset_continuous, skewAdjointPart_eq_I_smul_imaginaryPart, StarSubsemiring.ext_iff, isStarProjection_iff_isIdempotentElem_and_isStarNormal, Rat.instTrivialStar, NonUnitalStarAlgebra.coe_inf, cfcₙ_apply_pi, Matrix.PosDef.mul_mul_conjTranspose_same, IsStrictlyPositive.isSelfAdjoint, NonUnitalStarAlgebra.map_sup, NonUnitalStarAlgebra.adjoin_eq, Unitization.instStarModule, selfAdjoint.val_nnratCast, CStarAlgebra.isStrictlyPositive_TFAE, fourierSubalgebra_closure_eq_top, selfAdjoint.val_re_map_spectrum, CStarAlgebra.isStrictlyPositive_iff_isSelfAdjoint_and_spectrum_pos, NonUnitalStarAlgebra.range_id, Unitization.complex_cfcₙ_eq_cfc_inr, StarSubalgebra.one_mem_toNonUnitalStarSubalgebra, LDL.diag_eq_lowerInv_conj, Matrix.conjTranspose_list_prod, NonUnitalStarAlgebra.map_top, CStarAlgebra.norm_negPart_le, cfcₙ_star, cfcₙ_nnreal_eq_real, CFC.sqrt_algebraMap, CFC.posPart_algebraMap, Matrix.self_mul_conjTranspose_mul_eq_zero, CFC.sqrt_one, NonUnitalStarAlgHom.map_adjoin, unitary.expUnitary_eq_mul_inv, CFC.nnrpow_eq_cfcₙ_real, star_le_one_iff, imaginaryPart_imaginaryPart, IsSelfAdjoint.ofNat, CStarRing.to_normedStarGroup, NonUnitalStarAlgebra.star_self_mem_adjoin_singleton, StarSubalgebra.coe_centralizer, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, NonUnitalStarAlgebra.elemental.le_iff_mem, Matrix.isHermitian_one, StarSubalgebra.inclusion_injective, cfcHom_predicate, Subalgebra.mem_star_iff, CFC.negPart_def, NonUnitalStarSubalgebra.center_eq_top, spinGroup.star_mul_self_of_mem, NonUnitalStarSubalgebra.coe_prod, star_right_conjugate_nonneg, StarAddMonoid.toStarModuleNat, CFC.sqrt_rpow_nnreal, WeakDual.Complex.instStarHomClass, instContinuousStarReal, NonUnitalStarAlgebra.eq_top_iff, nnnorm_cfc_nnreal_lt_iff, PositiveLinearMap.instStarHomClassOfLinearMapClassComplexOfOrderHomClass, imaginaryPart_smul, norm_cfcHom, StarAlgebra.adjoin_eq_span, Unitization.norm_splitMul_snd_sq, EuclideanSpace.inner_toLp_toLp, IsSelfAdjoint.nnratCast, Unitary.norm_argSelfAdjoint, Matrix.conjTranspose_mul_self_eq_zero, CFC.norm_rpow, realPart_apply_coe, dotProduct_self_star_eq_zero, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, MonotoneOn.nnnorm_cfc, conjugate_pos, NonUnitalStarAlgebra.self_mem_adjoin_singleton, Matrix.IsUnit.posSemidef_star_left_conjugate_iff, OrthonormalBasis.toMatrix_orthonormalBasis_self_mul_conjTranspose, NonUnitalStarAlgebra.map_inf, cfcₙL_apply, ContinuousOn.cfcₙ_nnreal, cfcHomSuperset_continuous, cfcₙ_apply_mem_elemental, IsStarNormal.one_add, CFC.posPart_def, CFC.nnrpow_eq_rpow, pinGroup.coe_star, Matrix.conjTransposeRingEquiv_apply, CFC.cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one, ContinuousMap.elemental_id_eq_top, Unitization.starLift_range_le, DoubleCentralizer.instStarModule, WeakDual.CharacterSpace.compContinuousMap_comp, star_nonpos_iff, Complex.im_eq_zero_iff_isSelfAdjoint, LinearMap.isSymm_iff_basis, continuousOn_cfc_nnreal_setProd, CStarRing.nnnorm_self_mul_star, Homeomorph.compStarAlgEquiv'_symm_apply, OrthonormalBasis.toMatrix_orthonormalBasis_conjTranspose_mul_self, starRingEquiv_apply, CFC.abs_one, Matrix.vecMul_self_mul_conjTranspose_eq_zero, CentroidHom.starCenterToCentroid_apply, CStarAlgebra.nonneg_TFAE, nnnorm_cfcHom, Matrix.PosSemidef.sqrt_mul_self, IsSelfAdjoint.coe_realPart, realPart_smul, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, cfcₙ_star_id, CFC.abs_mul_abs, unitary.norm_argSelfAdjoint_le_pi, NonUnitalStarSubalgebra.prod_top, star_natCast, ContinuousMapZero.elemental_eq_top, ContinuousMapZero.adjoin_id_dense, Matrix.l2_opNNNorm_conjTranspose_mul_self, Unitary.norm_argSelfAdjoint_le_pi, Matrix.cstar_norm_def, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, CStarAlgebra.toStarModule, CFC.isUnit_nnrpow_iff, Unitization.starLift_symm_apply, Matrix.isHermitian_transpose_mul_self, star_le_star_iff, selfAdjoint.isStarNormal, inner_matrix_row_row, CFC.monotoneOn_cfcₙ_rpowIntegrand₀₁, conjugate_le_conjugate', NonUnitalStarAlgebra.sInf_toNonUnitalSubalgebra, Matrix.conjTranspose_mul_self_mul_eq_zero, star_mul_self_pos, TensorProduct.star_map_apply_eq_map_intrinsicStar, StarSubalgebra.mem_map, StarAlgHom.map_adjoin, IsStarNormal.instIsometricContinuousFunctionalCalculus, cfc_star, star_ratCast, cfcHom_mono, IsometricContinuousFunctionalCalculus.isometric, Matrix.conjTranspose_intCast, ContinuousAt.cfc_nnreal, Unitary.openPartialHomeomorph_apply, NonUnitalSubalgebra.starClosure_toNonUnitalSubalgebra, Unitary.argSelfAdjoint_coe, Matrix.isHermitian_intCast, skewAdjoint.I_smul_neg_I, LinearMap.toMatrix_adjoint, NonUnitalStarAlgebra.adjoin_le_starAlgebra_adjoin, spinGroup.coe_star, cfcₙHom_of_cfcHom_map_quasispectrum, WeakDual.CharacterSpace.compContinuousMap_id, ContinuousMapZero.instStarModule, StarSubalgebra.map_id, BoundedContinuousFunction.separatesPoints_charPoly, ContinuousWithinAt.cfc_nnreal, star_nonneg_iff, cfcₙHomSuperset_apply, IsSelfAdjoint.instNonUnitalIsometricContinuousFunctionalCalculus, NonUnitalStarAlgebra.elemental.instCompleteSpaceSubtypeMemNonUnitalStarSubalgebra, cfcₙ_apply, nnnorm_cfcₙ_nnreal_le, CFC.sqrt_eq_real_sqrt, apply_le_nnnorm_cfcₙ_nnreal, continuousOn_cfcₙ_nnreal_setProd, cfcHom_eq_of_isStarNormal, Matrix.UnitaryGroup.star_mul_self, Matrix.self_mul_conjTranspose_eq_zero, Matrix.conjTranspose_mul, ContinuousMapZero.coe_toContinuousMapHom, CFC.norm_sqrt, CFC.continuousOn_rpow, IsSelfAdjoint.imaginaryPart, conjugate_nonneg, Matrix.kroneckerStarAlgEquiv_apply, NonUnitalStarAlgHom.subsingleton, cfcHom_map_spectrum, MonotoneOn.nnnorm_cfcₙ, StarSubalgebra.starMemClass, StarSubsemiring.coe_ofClass, unitary.mem_pathComponentOne_iff, NonUnitalStarSubalgebra.mem_prod, StarSubalgebra.isEmbedding_inclusion, cfcₙHom_nonneg_iff, CStarRing.mul_star_self_eq_zero_iff, CFC.nnnorm_nnrpow, Matrix.IsUnit.posDef_star_left_conjugate_iff, CStarRing.nnnorm_star_mul_self, Matrix.PosDef.posDef_sqrt, star_left_conjugate_nonneg, Matrix.posSemidef_conjTranspose_iff, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, continuousFunctionalCalculus_map_id, CFC.norm_nnrpow, NonUnitalStarSubalgebra.inclusion_self, StarAddMonoid.toStarModuleInt, InnerProductSpace.symm_toEuclideanLin_rankOne, Commute.cfcₙ_real, Matrix.PosDef.mul_conjTranspose_self, Matrix.conjTranspose_ofNat, NonUnitalSubalgebra.mem_starClosure, Matrix.rank_conjTranspose_mul_self, LinearMap.intrinsicStar_eq_comp, Unitary.path_apply, unitary.two_mul_one_sub_cos_norm_argSelfAdjoint, Matrix.IsHermitian.instContinuousFunctionalCalculus, Matrix.PosDef.conjTranspose, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, cfcHom_apply_mem_elemental, PositiveLinearMap.preGNS_norm_sq, Matrix.kroneckerStarAlgEquiv_symm_apply, Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg, WeakDual.CharacterSpace.instStarHomClass, pinGroup.mul_star_self_of_mem, NonUnitalStarAlgebra.mem_inf, Matrix.dotProduct_self_star_pos_iff, CStarModule.isSelfAdjoint_inner_self, conjugate_le_conjugate, selfAdjoint.expUnitary_coe, Matrix.PosSemidef.sqrt_eq_iff_eq_sq, selfAdjoint.val_nnqsmul, CFC.rpow_eq_cfc_real, star_neg_iff, RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict, Matrix.PosDef.isHermitian, Matrix.mul_self_mul_conjTranspose_eq_zero, inner_matrix_col_col, BoundedContinuousFunction.char_mem_charPoly, Matrix.PosDef.dotProduct_mulVec_pos, algebraMap_star_comm, Continuous.cfcₙ_nnreal', Matrix.UnitaryGroup.inv_apply, Matrix.conjTranspose_invOf, CStarMatrix.inner_toCLM_conjTranspose_right, CFC.isUnit_sqrt_iff, Matrix.IsUnit.posDef_star_right_conjugate_iff, Matrix.PosDef.re_dotProduct_pos, spinGroup.star_mem_iff, isometry_cfcHom, LinearMap.isSymm_iff_isHermitian_toMatrix, NonUnitalStarAlgebra.map_iInf, selfAdjoint.joined_one_expUnitary, NonUnitalStarAlgebra.inf_toNonUnitalSubalgebra, NonUnitalSubalgebra.star_mem_star_iff, NonUnitalStarAlgebra.adjoin_le_centralizer_centralizer, Matrix.mul_conjTranspose_mul_self_eq_zero, unitary.continuousOn_argSelfAdjoint, Matrix.star_mulVec, CFC.rpow_sqrt, IsSelfAdjoint.cfcₙ, NNReal.instContinuousStar, Unitary.two_mul_one_sub_cos_norm_argSelfAdjoint, NonUnitalSubalgebra.star_adjoin_comm
toStarMul 📖CompOp
111 mathmath: Unitary.openPartialHomeomorph_source, CStarRing.norm_mul_coe_unitary, Complex.coe_realPart, CStarRing.norm_unitary_smul, NonUnitalCStarAlgebra.toStarModule, realPart_ofReal, Unitary.mem_pathComponentOne_iff, selfAdjoint.expUnitaryPathToOne_apply, Unitary.continuousOn_argSelfAdjoint, selfAdjoint.norm_sq_expUnitary_sub_one, UnitAddTorus.mFourierSubalgebra_coe, Matrix.intrinsicStar_toLin', UnitAddTorus.mFourierSubalgebra_closure_eq_top, polynomialFunctions.starClosure_topologicalClosure, Pell.is_pell_solution_iff_mem_unitary, LinearIsometryEquiv.trans_smul, Matrix.toAlgEquiv_kroneckerStarAlgEquiv, QuadraticAlgebra.norm_eq_one_iff_mem_unitary, Pi.intrinsicStar_comul_commSemiring, Zsqrtd.mker_norm_eq_unitary, Unitary.openPartialHomeomorph_symm_apply, polynomialFunctions.starClosure_le_equalizer, Pell.Solution₁.coe_mk, NonUnitalStarSubalgebra.unitization_range, BoundedContinuousFunction.mem_charPoly, LinearIsometryEquiv.smul_apply, unitary.norm_sub_eq, star_add, selfAdjoint.expUnitary_zero, Unitary.instLocPathConnectedSpace, pinGroup.mem_unitary, polynomialFunctions.starClosure_eq_adjoin_X, selfAdjoint.continuous_expUnitary, selfAdjoint.unitarySelfAddISMul_coe, QuadraticAlgebra.mem_unitary, LinearIsometryEquiv.conjStarAlgEquiv_ext_iff, fourierSubalgebra_separatesPoints, Pell.isPell_iff_mem_unitary, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, Matrix.IntrinsicStar.isSelfAdjoint_toLin'_iff, ContinuousMap.adjoin_id_eq_span_one_union, realPart_comp_subtype_selfAdjoint, Matrix.IsHermitian.cfcAux_apply, Matrix.UnitaryGroup.inv_val, unitary.spectrum_subset_circle, CommCStarAlgebra.toStarModule, imaginaryPart_eq_neg_I_smul_skewAdjointPart, LinearIsometryEquiv.toLinearEquiv_smul, pinGroup.units_mem_iff, selfAdjoint.realPart_unitarySelfAddISMul, LinearMap.toMatrix'_intrinsicStar, Unitary.openPartialHomeomorph_target, Matrix.det_of_mem_unitary, imaginaryPart_comp_subtype_selfAdjoint, NonUnitalCommCStarAlgebra.toStarModule, selfAdjoint.star_coe_unitarySelfAddISMul, fourierSubalgebra_coe, Matrix.IsHermitian.spectral_theorem, QuadraticAlgebra.mker_norm_eq_unitary, SpectrumRestricts.starAlgHom_apply, UnitAddTorus.mFourierSubalgebra_separatesPoints, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, imaginaryPart_ofReal, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, Commute.expUnitary_add, Unitary.isPathConnected_ball, cfc_unitary_iff, ContinuousMap.adjoin_id_eq_span_one_add, IsStarProjection.two_mul_sub_one_mem_unitary, pinGroup.mem_iff, LinearIsometryEquiv.smul_trans, LinearMap.IntrinsicStar.isSelfAdjoint_iff_toMatrix', Unitary.norm_sub_eq, unitary_iff_isStarNormal_and_spectrum_subset_unitary, skewAdjointPart_eq_I_smul_imaginaryPart, fourierSubalgebra_closure_eq_top, LinearIsometryEquiv.symm_units_smul, LinearIsometryEquiv.symm_smul_apply, IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary, Unitary.coe_neg, Matrix.IsHermitian.star_mul_self_mul_eq_diagonal, CStarRing.norm_coe_unitary_mul, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, ContinuousMap.elemental_id_eq_top, unitary.isPathConnected_ball, Zsqrtd.norm_eq_one_iff_mem_unitary, ContinuousMapZero.elemental_eq_top, ContinuousMapZero.adjoin_id_dense, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, CStarAlgebra.toStarModule, Unitary.openPartialHomeomorph_apply, Matrix.IsHermitian.conjStarAlgAut_star_eigenvectorUnitary, Unitary.argSelfAdjoint_coe, BoundedContinuousFunction.separatesPoints_charPoly, CStarAlgebra.exists_sum_four_unitary, NormedSpace.exp_mem_unitary_of_mem_skewAdjoint, Matrix.kroneckerStarAlgEquiv_apply, unitary.coe_neg, LinearIsometryEquiv.toContinuousLinearEquiv_smul, unitary.mem_pathComponentOne_iff, CStarRing.norm_coe_unitary, CStarAlgebra.span_unitary, Unitary.spectrum_subset_circle, Matrix.kroneckerStarAlgEquiv_symm_apply, Commute.expUnitary, selfAdjoint.expUnitary_coe, RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict, BoundedContinuousFunction.char_mem_charPoly, Matrix.UnitaryGroup.inv_apply, selfAdjoint.joined_one_expUnitary, unitary.continuousOn_argSelfAdjoint

Theorems

NameKindAssumesProvesValidatesDepends On
star_add 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
toStarMul
Distrib.toAdd

StarSemigroup

Theorems

NameKindAssumesProvesValidatesDepends On
toOpposite_starModule 📖mathematicalStarModule
MulOpposite
MulOpposite.instStar
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
SemigroupAction.toSMul
Monoid.toSemigroup
MulOpposite.instMonoid
MulAction.toSemigroupAction
Monoid.toOppositeMulAction
star_mul'

TrivialStar

Theorems

NameKindAssumesProvesValidatesDepends On
star_trivial 📖mathematicalStar.star

Units

Definitions

NameCategoryTheorems
instStarMul 📖CompOp
11 mathmath: val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe, val_unitarySubgroupUnitsEquiv_symm_apply_coe, instStarModule, mul_inv_mem_unitary, unitarySubgroupUnitsEquiv_apply_coe, coe_star, unitary_eq, instContinuousStarUnits, embed_product_star, coe_star_inv, inv_mul_mem_unitary

Theorems

NameKindAssumesProvesValidatesDepends On
coe_star 📖mathematicalval
Star.star
Units
InvolutiveStar.toStar
StarMul.toInvolutiveStar
instMul
instStarMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
coe_star_inv 📖mathematicalval
Units
instInv
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
instMul
instStarMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
instStarModule 📖mathematicalStarModule
Units
InvolutiveStar.toStar
StarMul.toInvolutiveStar
instMul
instStarMul
instSMul
StarModule.star_smul

(root)

Definitions

NameCategoryTheorems
InvolutiveStar 📖CompData
StarAddMonoid 📖CompData
StarHomClass 📖CompData
10 mathmath: AlgHomClass.instStarHomClass, NonUnitalStarAlgHom.instStarHomClass, StarAlgHom.instStarHomClass, NonUnitalStarRingHomClass.toStarHomClass, StarMonoidHom.instStarHomClass, WeakDual.Complex.instStarHomClass, PositiveLinearMap.instStarHomClassOfLinearMapClassComplexOfOrderHomClass, StarMulEquiv.instStarHomClass, StarRingEquivClass.instStarHomClass, WeakDual.CharacterSpace.instStarHomClass
StarMemClass 📖CompData
5 mathmath: NonUnitalStarSubsemiring.instStarMemClass, StarSubsemiring.starMemClass, VonNeumannAlgebra.instStarMemClass, StarSubalgebra.starMemClass, NonUnitalStarSubalgebra.instStarMemClass
StarModule 📖CompData
34 mathmath: StarAddMonoid.toStarModuleNNRat, NonUnitalCStarAlgebra.toStarModule, instStarModuleNNRealOfReal, StarAddMonoid.toStarModuleRat, instStarModuleNNRealReal, CompactlySupportedContinuousMap.instStarModule, ContinuousMap.instStarModule, Units.instStarModule, BoundedContinuousFunction.instStarModule, StarMul.toStarModule, TensorProduct.instStarModule, Matrix.instStarModule, CommCStarAlgebra.toStarModule, ContinuousLinearMap.instStarModuleId, NonUnitalCommCStarAlgebra.toStarModule, Complex.instStarModuleReal, RCLike.instStarModuleReal, ZeroAtInftyContinuousMap.instStarModule, LinearMap.instStarModuleId, StarSubalgebra.starModule, StarModule.complexToReal, LinearMap.intrinsicStarModule, StarSemigroup.toOpposite_starModule, MulOpposite.instStarModule, Unitization.instStarModule, StarAddMonoid.toStarModuleNat, DoubleCentralizer.instStarModule, CStarMatrix.instStarModule, CStarAlgebra.toStarModule, Unitary.instStarModuleSubtypeMemSubmonoidUnitary, ContinuousMapZero.instStarModule, Prod.instStarModule, StarAddMonoid.toStarModuleInt, StarMemClass.instStarModule
StarMul 📖CompData
StarRing 📖CompData
TrivialStar 📖CompData
13 mathmath: instTrivialStarReal, Nat.instTrivialStar, Set.instTrivialStar, MeasureTheory.Lp.instTrivialStarSubtypeAEEqFunMemAddSubgroup, CompactlySupportedContinuousMap.instTrivialStar, ContinuousMapZero.instTrivialStar, Prod.instTrivialStar, NNRat.instTrivialStar, Int.instTrivialStar, instTrivialStarNNReal, Rat.instTrivialStar, MeasureTheory.AEEqFun.instTrivialStar, ContinuousMap.instTrivialStar
starAddEquiv 📖CompOp
4 mathmath: starLinearEquiv_symm_apply, starL_symm_apply, starL'_symm_apply, starAddEquiv_apply
starMulAut 📖CompOp
1 mathmath: starMulAut_apply
starMulEquiv 📖CompOp
1 mathmath: starMulEquiv_apply
starMulOfComm 📖CompOp
1 mathmath: star_id_of_comm
starRingAut 📖CompOp
1 mathmath: starRingAut_apply
starRingEnd 📖CompOp
428 mathmath: Pi.comul_eq_adjoint, Complex.norm_conj, LinearMap.IsSymmetric.clm_adjoint_eq, RCLike.conj_re, Complex.log_conj_eq_ite, Orientation.kahler_map_complex, MeasureTheory.lpNorm_conj, LinearMap.IsSymmetric.conj_eigenvalue_eq_self, ContinuousLinearMap.isPositive_iff_eq_sum_rankOne, continuousOn_stereoToFun, InnerProductSpace.isPositive_rankOne_self, jacobiTheta₂_conj, Orthonormal.inner_left_finsupp, Complex.sinh_conj, LinearMap.adjoint_adjoint, InnerProductSpace.toLinearIsometry_toDual, SchwartzMap.integral_sesq_fourier_fourier, hasFDerivAt_iff_hasGradientAt, Complex.lim_conj, InnerProductSpace.rankOne_one_left_eq_innerSL, Complex.kahler, Complex.areaForm, Complex.re_eq_add_conj, CliffordAlgebraComplex.ofComplex_conj, CStarModule.innerₛₗ_apply, LinearMap.isPositive_adjoint_comp_self, InnerProductSpace.isIdempotentElem_rankOne_self, RCLike.star_def, RCLike.inner_apply', Complex.conj_inv, flip_innerSL_real, RCLike.sub_conj, innerSL_apply_apply, Complex.star_def, LinearMap.adjoint_innerₛₗ_apply, EuclideanSpace.inner_single_left, ContinuousLinearMap.innerSL_apply_comp_of_isSymmetric, LinearMap.adjoint_eq_toCLM_adjoint, Complex.nndist_self_conj, starLinearEquiv_symm_apply, MeasureTheory.charFun_neg, Circle.ofConjDivSelf_coe, OrthonormalBasis.orthogonalProjection_eq_sum_rankOne, InnerProductSpace.Core.inner_smul_left, InnerProductSpace.toDual_symm_apply, RCLike.conj_nat_cast, Complex.conjLIE_apply, Circle.starRingEnd_addChar, MeasureTheory.charFun_toDual_symm_eq_charFunDual, Complex.cosh_conj, PreInnerProductSpace.Core.smul_left, Differentiable.fderiv_norm_rpow, Complex.oangle, RCLike.is_real_TFAE, inner_gradientWithin_right, ContinuousLinearMap.adjoint_id, InnerProductSpace.smul_left, ContinuousLinearMap.isStarNormal_iff_norm_eq_adjoint, CStarModule.innerSL_apply, conjneg_conj, inner_smul_left_eq_star_smul, RCLike.mul_conj, RCLike.norm_conj, InnerProductSpace.inner_left_rankOne_apply, star_dotProduct_toMatrix₂_mulVec, HasFDerivAt.norm_sq, innerₛₗ_apply_apply, ContinuousLinearMap.eq_adjoint_iff, InnerProductSpace.rankOne_apply, Complex.isCauSeq_conj, HasDerivAt.hasGradientAt, IsSelfAdjoint.adjoint_eq, TensorProduct.adjoint_map, jacobiTheta₂'_conj, ContinuousLinearMap.isPositive_self_comp_adjoint, LinearMap.adjoint_toContinuousLinearMap, ContinuousLinearMap.toSesqForm_apply_coe, Complex.cos_conj, LinearIsometryEquiv.adjoint_eq_symm, Complex.conj_rootsOfUnity, RCLike.conjCLE_apply, Orthonormal.inner_left_fintype, SchwartzMap.integral_sesq_fourier_eq, ContinuousLinearMap.adjointAux_norm, RingHomIsometric.starRingEnd, Complex.isometry_conj, Pi.counit_eq_adjoint, Complex.summable_conj, Complex.log_inv_eq_ite, Complex.mul_conj', Real.fourier_iteratedFDeriv, LinearIsometryEquiv.adjoint_toLinearMap_eq_symm, InnerProductSpace.toDualMap_apply_apply, LinearMap.eq_adjoint_iff_basis_left, starL_apply, Orientation.kahler_swap, coe_innerSL_apply, Complex.UnitDisc.coe_conj, InnerProductSpace.nnnorm_rankOne, PreInnerProductSpace.Core.conj_inner_symm, Matrix.conjTransposeLinearEquiv_apply, InnerProductSpace.isSymmetricProjection_rankOne_self, hasGradientWithinAt_iff_hasFDerivWithinAt, ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left, Complex.Gamma_conj, Complex.ringHom_eq_id_or_conj_of_continuous, LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf, Complex.dist_self_conj, RCLike.conj_ofReal, Submodule.adjoint_subtypeL, LinearMap.isSymmetric_adjoint_mul_self, RCLike.conjLIE_apply, RCLike.conj_mul, InnerProductSpace.trace_rankOne, fourier_neg', hasFDerivAt_norm_rpow, ContinuousLinearMap.IsPositive.conj_adjoint, contDiffOn_stereoToFun, RCLike.conj_conj, LinearMap.adjoint_rTensor, LinearMap.adjoint_inner_right, InnerProductSpace.adjoint_rankOne, InnerProductSpace.rankOne_eq_zero, LinearMap.IntrinsicStar.starLinearEquiv_eq_arrowCongr, Complex.normSq_add, RCLike.re_eq_norm_of_mul_conj, InnerProductSpace.inner_right_rankOne_apply, HasFDerivWithinAt.hasGradientWithinAt, LinearMap.isPosSemidef_iff_posSemidef_toMatrix, innerSL_inj, RCLike.conj_eq_iff_re, InnerProductSpace.rankOne_eq_rankOne_iff_comm, InnerProductSpace.toDual_apply, UpperHalfPlane.sigma_J, Complex.inv_cpow_eq_ite', RCLike.conj_I_ax, InnerProductSpace.comp_rankOne, OrthonormalBasis.starProjection_eq_sum_rankOne, InnerProductSpace.toDualMap_apply, DifferentiableAt.conj_conj, ContinuousLinearMap.adjoint_inner_left, InnerProductSpace.AlgebraOfCoalgebra.mul_def, ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_right, integral_conj, InnerProductSpace.rank_rankOne, UnitAddTorus.mFourier_neg, InnerProductSpace.toDual_apply_apply, ContinuousLinearMap.adjointAux_apply, ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self, Complex.normSq_sub, ProperCone.innerDual_singleton, RCLike.add_conj, RCLike.im_eq_conj_sub, InnerProductSpace.rankOne_def, RCLike.conj_im_ax, stereoToFun_apply, ContinuousLinearMap.adjointAux_inner_left, inner_self_conj, Circle.coe_inv_eq_conj, ContinuousLinearMap.IsStarNormal.ker_adjoint_eq_ker, Complex.sub_conj, ClosedSubmodule.orthogonal_eq_inter, Orientation.kahler_neg_orientation, Complex.conj_im, conj_trivial, RCLike.conj_div, HasDerivAtFilter.hasGradientAtFilter, RCLike.summable_conj, LinearMap.toMatrix_innerₛₗ_apply, ContinuousLinearMap.IsStarNormal.adjoint_apply_eq_zero_iff, Complex.continuous_conj, Matrix.conjTransposeLinearEquiv_symm, Complex.conj_cpow, LinearMap.isSelfAdjoint_iff', NumberField.IsCMField.complexEmbedding_complexConj, innerSLFlip_apply_apply, Complex.inv_cpow_eq_ite, Complex.inv_def, InnerProductSpace.enorm_rankOne, Orthonormal.inner_finsupp_eq_sum_right, InnerProductSpace.continuousLinearMapOfBilin_zero, ContinuousLinearMap.toLinearMap_innerSL_apply, Complex.arg_conj, inner_map_complex, LinearMap.IsSymmetric.adjoint_eq, HasGradientAt.hasFDerivAt, Real.fourier_fderiv, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, HasFDerivWithinAt.norm_sq, AnalyticAt.harmonicAt_conj, GaussianInt.toComplex_star, Complex.conjAe_coe, ContinuousLinearMap.toPMap_adjoint_eq_adjoint_toPMap_of_dense, ProbabilityTheory.covarianceBilin_eq_covarianceBilinDual, starRingEnd_apply, Complex.conj_mul', RingHomInvPair.instStarRingEnd, LinearMap.star_eq_adjoint, Complex.sin_conj, Complex.conj_eq_iff_re, Polynomial.aeval_conj, RCLike.re_eq_add_conj, IsSelfAdjoint.conj_adjoint, RCLike.wInner_const_left, RCLike.wInner_cWeight_const_left, hasStrictFDerivAt_norm_sq, LinearMap.adjoint_toSpanSingleton, starL_symm_apply, RCLike.inner_apply, IsSelfAdjoint.conj_eq, Complex.conj_cpow_eq_ite, LinearMap.re_inner_adjoint_mul_self_nonneg, InnerProductSpace.nullSubmodule_le_ker_toDualMap_left, LinearMap.isPositive_self_comp_adjoint, starLinearEquiv_tensor, RCLike.hasSum_conj, starₗᵢ_apply, Complex.normSq_conj, ContinuousLinearMap.adjointAux_adjointAux, apply_eq_star_dotProduct_toMatrix₂_mulVec, Complex.add_conj, Complex.tanh_conj, LinearMap.eq_adjoint_iff, LinearMap.norm_extendTo𝕜'_apply_sq, ContinuousLinearMap.isSelfAdjoint_iff', UpperHalfPlane.J_smul, RCLike.conj_tsum, AddChar.starComp_apply, MulChar.starComp_apply, RCLike.wInner_const_right, InnerProductSpace.rankOne_def', Complex.cot_conj, SchwartzMap.integral_sesq_fourierIntegral_eq, ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left, LinearMap.adjoint_lTensor, RCLike.conj_eq_iff_real, starL'_symm_apply, Orthonormal.inner_left_sum, Orientation.inner_mul_areaForm_sub', UpperHalfPlane.petersson_symm, InnerProductSpace.innerSL_norm, hasGradientAt_iff_hasFDerivAt, Complex.inner, Matrix.toLin_conjTranspose, Orthonormal.inner_finsupp_eq_sum_left, RCLike.continuous_conj, UpperHalfPlane.coe_J_smul, InnerProductSpace.isStarProjection_rankOne_self, NumberField.conj_basisMatrix, InnerProductSpace.rankOne_comp_rankOne, InnerProductSpace.rankOne_comp, Real.fourierIntegral_fderiv, InnerProductSpace.toMatrix_rankOne, RCLike.conj_wInner_symm, InnerProductSpace.isSymmetric_rankOne_self, AddChar.starComp_eq_inv, Matrix.toEuclideanLin_conjTranspose_eq_adjoint, LinearMap.eq_adjoint_iff_basis, WStarAlgebra.exists_predual, Orientation.kahler_rotation_left, Complex.hasSum_conj, ContinuousLinearMap.adjoint_toSpanSingleton, RCLike.nnnorm_conj, starₗᵢ_toContinuousLinearEquiv, RCLike.wInner_one_const_right, InnerProductSpace.rankOne_one_right_eq_toSpanSingleton, Complex.dist_conj_conj, RCLike.norm_sq_re_conj_add, Complex.conjCLE_apply, innerSLFlip_apply, Complex.arg_conj_coe_angle, InnerProductSpace.toContinuousLinearMap_toDualMap, Complex.nndist_conj_self, UpperHalfPlane.cosh_half_dist, RCLike.conj_im, RCLike.conj_I, coe_starₗᵢ, innerSL_apply_norm, InnerProductSpace.toDual_apply_eq_toDualMap_apply, LinearMap.adjoint_inner_left, RCLike.inv_def, UpperHalfPlane.exp_half_dist, innerSL_real_flip, Complex.uniformContinuous_ringHom_eq_id_or_conj, MeasureTheory.charFun_eq_charFunDual_toDualMap, fourier_neg, toMatrix_innerSL_apply, Complex.conj_re, Complex.im_eq_sub_conj, ContinuousLinearMap.orthogonal_range, Complex.conj_ofNat, UnitAddTorus.hasSum_prod_mFourierCoeff, ProbabilityTheory.covarianceBilin_map, Complex.hasSum_conj', RingHom.star_def, RCLike.conj_eq_iff_im, Complex.exp_conj, ContinuousLinearMap.adjoint_comp, Pi.conj_apply, InnerProductSpace.toLinearMap_rankOne, ContinuousLinearMap.adjointAux_inner_right, RCLike.conj_re_ax, Complex.conj_conj, UpperHalfPlane.sinh_half_dist_add_dist, conjneg_apply, LinearIsometry.im_apply_eq_im, Submodule.adjoint_orthogonalProjection, hasFDerivWithinAt_iff_hasGradientWithinAt, Complex.mul_conj, LinearMap.adjoint_comp, Complex.nndist_conj_comm, Complex.conj_natCast, Complex.nndist_conj_conj, LinearMap.isSymmetric_iff_sesqForm, ContinuousLinearMap.IsPositive.adjoint_conj, Orientation.areaForm_map_complex, RCLike.conj_to_real, Complex.cpow_conj, ContinuousLinearMap.star_eq_adjoint, RCLike.wInner_one_const_left, ProperCone.hyperplane_separation_of_notMem, RCLike.norm_sq_re_add_conj, innerₛₗ_apply, innerSL_apply, InnerProductSpace.isIdempotentElem_rankOne_self_iff, Complex.conj_eq_iff_real, InnerProductSpaceable.innerProp, RCLike.conj_inv, IsSelfAdjoint.adjoint_conj, Complex.normSq_eq_conj_mul_self, OrthonormalBasis.sum_rankOne_eq_id, Complex.conj_eq_iff_im, MeasureTheory.BoundedContinuousFunction.inner_toLp, ContinuousLinearMap.adjoint_inner_right, Complex.dist_conj_comm, Complex.GammaIntegral_conj, RCLike.enorm_conj, CliffordAlgebraComplex.toComplex_involute, RCLike.conjAe_coe, signedDist_apply, HasFDerivAt.hasGradientAt, LinearMap.IsSymmetric.conj_inner_sym, starLinearEquiv_apply, conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj, ContinuousLinearMap.isAdjointPair_inner, HurwitzZeta.jacobiTheta₂''_conj, RCLike.wInner_cWeight_const_right, RCLike.inv_eq_conj, LinearMap.eq_adjoint_iff_basis_right, UpperHalfPlane.σ_conj, innerₛₗ_apply_coe, NumberField.canonicalEmbedding.conj_apply, LinearMap.isSymm_iff_basis, Complex.conj_ofReal, AddChar.map_neg_eq_conj, EuclideanSpace.inner_single_right, Submodule.orthogonal_eq_inter, RCLike.normSq_add, LinearMap.IsPositive.adjoint_eq, Orientation.oangle_map_complex, fderiv_norm_rpow, ContinuousLinearMap.norm_map_iff_adjoint_comp_self, Complex.inv_eq_conj, ContinuousLinearMap.norm_adjoint_comp_self, LinearPMap.mem_adjoint_domain_iff, Complex.conj_neg_I, HasDerivAt.conj_conj, coe_innerₛₗ_apply, HasGradientAt.hasDerivAt, InnerProductSpace.continuousLinearMapOfBilin_apply, UpperHalfPlane.tanh_half_dist, RCLike.conj_eq_re_sub_im, Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero, RCLike.conj_ofNat, Complex.conj_tsum, NumberField.ComplexEmbedding.conjugate_coe_eq, ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right, innerSL_apply_coe, MeasureTheory.ContinuousMap.inner_toLp, gradient_eq_deriv, Complex.tan_conj, LinearMap.toMatrix_adjoint, RCLike.conj_smul, LinearMap.isSymmetric_iff_inner_map_self_real, ContinuousLinearMap.isPositive_adjoint_comp_self, LinearMap.im_inner_adjoint_mul_self_eq_zero, norm_innerSL_le, Complex.UnitDisc.coe_star, HasFDerivAt.star_star, HasGradientWithinAt.hasFDerivWithinAt, RCLike.hasSum_conj', InnerProductSpace.conj_inner_symm, LinearMap.isAdjointPair_inner, LinearMap.IsPositive.adjoint_conj, Complex.log_conj, HasGradientAtFilter.hasDerivAtFilter, MeasureTheory.eLpNorm_conj, inner_gradient_right, Real.fourierIntegral_iteratedFDeriv, Complex.conj_I, inner_conj_symm, AddChar.inv_apply_eq_conj, InnerProductSpace.symm_toEuclideanLin_rankOne, ContinuousLinearMap.adjoint_innerSL_apply, RCLike.normSq_sub, Orthonormal.inner_sum, starRingEnd_self_apply, LinearMap.intrinsicStar_eq_comp, Complex.dist_conj_self, HasDerivAt.star_conj, InnerProductSpaceable.inner_.conj_symm, fderiv_norm_sq_apply, RCLike.conj_neg_I, InnerProductSpace.Core.inner_conj_symm, fderiv_norm_sq, Orientation.inner_mul_inner_add_areaForm_mul_areaForm', ContinuousLinearMap.innerSL_apply_comp, ContinuousLinearMap.toSesqForm_apply_norm_le, LinearMap.adjoint_id, inner_smul_left, InnerProductSpace.nullSubmodule_le_ker_toDualMap_right, ContinuousLinearMap.adjoint_adjoint, RCLike.normSq_conj, ContinuousLinearMap.orthogonal_ker, InnerProductSpace.norm_rankOne, LinearMap.IsPositive.conj_adjoint, LinearMap.isSymm_iff_isHermitian_toMatrix, sum_conjneg
starRingEquiv 📖CompOp
1 mathmath: starRingEquiv_apply
starRingOfComm 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
commute_star_comm 📖mathematicalCommute
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
commute_star_star
star_star
commute_star_star 📖mathematicalCommute
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
semiconjBy_star_star_star
conj_trivial 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
starRingEnd
TrivialStar.star_trivial
eq_star_iff_eq_star 📖mathematicalStar.star
InvolutiveStar.toStar
eq_star_of_eq_star
eq_star_of_eq_star 📖Star.star
InvolutiveStar.toStar
star_star
isLeftRegular_star_iff 📖mathematicalIsLeftRegular
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
IsRightRegular
IsLeftRegular.star
star_star
IsRightRegular.star
isRegular_star_iff 📖mathematicalIsRegular
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
isRegular_iff
isRightRegular_star_iff
isLeftRegular_star_iff
isRightRegular_star_iff 📖mathematicalIsRightRegular
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
IsLeftRegular
IsRightRegular.star
star_star
IsLeftRegular.star
isUnit_star 📖mathematicalIsUnit
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
IsUnit.star
star_star
mem_of_star_mem 📖SetLike.instMembership
Star.star
InvolutiveStar.toStar
star_star
StarMemClass.star_mem
semiconjBy_star_star_star 📖mathematicalSemiconjBy
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
starAddEquiv_apply 📖mathematicalDFunLike.coe
AddEquiv
AddSemigroup.toAdd
AddMonoid.toAddSemigroup
EquivLike.toFunLike
AddEquiv.instEquivLike
starAddEquiv
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
starMulAut_apply 📖mathematicalDFunLike.coe
MulEquiv
CommMagma.toMul
CommSemigroup.toCommMagma
EquivLike.toFunLike
MulEquiv.instEquivLike
starMulAut
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
starMulEquiv_apply 📖mathematicalDFunLike.coe
MulEquiv
MulOpposite
MulOpposite.instMul
EquivLike.toFunLike
MulEquiv.instEquivLike
starMulEquiv
MulOpposite.op
Star.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
starRingAut_apply 📖mathematicalDFunLike.coe
RingEquiv
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Distrib.toAdd
EquivLike.toFunLike
RingEquiv.instEquivLike
starRingAut
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
StarRing.toStarAddMonoid
starRingEnd_apply 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
starRingEnd
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
starRingEnd_self_apply 📖mathematicalDFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
starRingEnd
star_star
starRingEquiv_apply 📖mathematicalDFunLike.coe
RingEquiv
MulOpposite
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
MulOpposite.instMul
Distrib.toAdd
MulOpposite.instAdd
EquivLike.toFunLike
RingEquiv.instEquivLike
starRingEquiv
MulOpposite.op
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
StarRing.toStarAddMonoid
star_div 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
CommGroup.toGroup
DivInvMonoid.toDiv
map_div
MulEquivClass.instMonoidHomClass
MulEquiv.instMulEquivClass
star_div₀ 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
CommGroupWithZero.toGroupWithZero
DivInvMonoid.toDiv
GroupWithZero.toDivInvMonoid
MulOpposite.op_injective
division_def
MulOpposite.op_div
mul_comm
StarMul.star_mul
star_inv₀
MulOpposite.op_mul
MulOpposite.op_inv
star_eq_iff_star_eq 📖mathematicalStar.star
InvolutiveStar.toStar
eq_star_iff_eq_star
star_eq_zero 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddEquiv.map_eq_zero_iff
star_id_of_comm 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
starMulOfComm
star_inj 📖mathematicalStar.star
InvolutiveStar.toStar
star_injective
star_injective 📖mathematicalStar.star
InvolutiveStar.toStar
Function.Involutive.injective
InvolutiveStar.star_involutive
star_intCast 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
StarRing.toStarAddMonoid
AddGroupWithOne.toIntCast
AddCommGroupWithOne.toAddGroupWithOne
NonAssocRing.toAddCommGroupWithOne
map_intCast
RingEquivClass.toRingHomClass
RingEquiv.instRingEquivClass
MulOpposite.unop_intCast
star_inv 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
MulOpposite.op_injective
MonoidHom.map_inv
MulOpposite.op_inv
star_invOf 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Invertible.invOf
MulOne.toOne
mul_invOf_self
mul_one
mul_assoc
StarMul.star_mul
star_one
one_mul
star_inv₀ 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
MulOpposite.op_injective
map_inv₀
MulEquivClass.toMonoidWithZeroHomClass
MulEquiv.instMulEquivClass
MulOpposite.op_inv
star_mem_iff 📖mathematicalSetLike.instMembership
Star.star
InvolutiveStar.toStar
StarMemClass.star_mem
star_star
star_mul' 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
CommMagma.toMul
StarMul.star_mul
mul_comm
star_mul_star 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
StarMul.star_mul
star_star
star_natCast 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
map_natCast
RingEquivClass.toRingHomClass
RingEquiv.instRingEquivClass
MulOpposite.unop_natCast
star_ne_zero 📖
star_neg 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddEquiv.map_neg
star_nsmul 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddMonoid.toNatSMul
AddMonoidHom.map_nsmul
star_ofNat 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
StarRing.toStarAddMonoid
star_natCast
star_one 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
MulOne.toOne
MulOpposite.op_injective
MulEquiv.map_one
MulOpposite.op_one
star_pow 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.toNatPow
MulOpposite.op_injective
MonoidHom.map_pow
MulOpposite.op_pow
star_star 📖mathematicalStar.star
InvolutiveStar.toStar
InvolutiveStar.star_involutive
star_star_mul 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
StarMul.star_mul
star_star
star_sub 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SubNegMonoid.toSub
AddEquiv.map_sub
star_zero 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddEquiv.map_zero
star_zpow 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
DivInvMonoid.toZPow
MulOpposite.op_injective
MonoidHom.map_zpow
MulOpposite.op_zpow
star_zpow₀ 📖mathematicalStar.star
InvolutiveStar.toStar
StarMul.toInvolutiveStar
MulZeroClass.toMul
MulZeroOneClass.toMulZeroClass
MonoidWithZero.toMulZeroOneClass
GroupWithZero.toMonoidWithZero
DivInvMonoid.toZPow
GroupWithZero.toDivInvMonoid
MulOpposite.op_injective
map_zpow₀
MulEquivClass.toMonoidWithZeroHomClass
MulEquiv.instMulEquivClass
MulOpposite.op_zpow
star_zsmul 📖mathematicalStar.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SubNegMonoid.toZSMul
AddMonoidHom.map_zsmul

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