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• Singular_trace abstract "In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanisheson operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators.Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported.Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes.In heurestic terms, a singular trace corresponds to a way of summingnumbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to the usual sum a1 + a2 + a3 + ... .This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.".
• Singular_trace wikiPageExternalLink Vestigia.pdf.
• Singular_trace wikiPageExternalLink book94bigpdf.pdf.
• Singular_trace wikiPageExternalLink 177778.
• Singular_trace wikiPageExternalLink abstract.
• Singular_trace wikiPageExternalLink AGPS4.pdf.
• Singular_trace wikiPageID "40083877".
• Singular_trace wikiPageLength "27521".
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• Singular_trace wikiPageRevisionID "674999959".
• Singular_trace wikiPageWikiLink Alain_Connes.
• Singular_trace wikiPageWikiLink Calkin_correspondence.
• Singular_trace wikiPageWikiLink Category:Hilbert_space.
• Singular_trace wikiPageWikiLink Category:Von_Neumann_algebras.
• Singular_trace wikiPageWikiLink Cauchy_distribution.
• Singular_trace wikiPageWikiLink Cesàro_mean.
• Singular_trace wikiPageWikiLink Closed_manifold.
• Singular_trace wikiPageWikiLink Commutator_subspace.
• Singular_trace wikiPageWikiLink Compact_operator_on_Hilbert_space.
• Singular_trace wikiPageWikiLink Compact_operator_on_hilbert_space.
• Singular_trace wikiPageWikiLink Correspondence_principle.
• Singular_trace wikiPageWikiLink Dirac_operator.
• Singular_trace wikiPageWikiLink Dixmier_trace.
• Singular_trace wikiPageWikiLink Eigenstate.
• Singular_trace wikiPageWikiLink Eigenvalue.
• Singular_trace wikiPageWikiLink Eigenvalues_and_eigenvectors.
• Singular_trace wikiPageWikiLink Finite-rank_operator.
• Singular_trace wikiPageWikiLink Fully_symmetric_ideal.
• Singular_trace wikiPageWikiLink Harmonic_series_(mathematics).
• Singular_trace wikiPageWikiLink Hilbert_space.
• Singular_trace wikiPageWikiLink Jacques_Dixmier.
• Singular_trace wikiPageWikiLink Laplace-Beltrami_operator.
• Singular_trace wikiPageWikiLink Laplace–Beltrami_operator.
• Singular_trace wikiPageWikiLink Limit_of_a_function.
• Singular_trace wikiPageWikiLink Linear_map.
• Singular_trace wikiPageWikiLink Linear_operators.
• Singular_trace wikiPageWikiLink Lp_space.
• Singular_trace wikiPageWikiLink Matrix_ring.
• Singular_trace wikiPageWikiLink Measure_(mathematics).
• Singular_trace wikiPageWikiLink Measure_theory.
• Singular_trace wikiPageWikiLink Nigel_Kalton.
• Singular_trace wikiPageWikiLink Nilpotent_operator.
• Singular_trace wikiPageWikiLink Noncommutative_geometry.
• Singular_trace wikiPageWikiLink Noncommutative_residue.
• Singular_trace wikiPageWikiLink Orthonormal_basis.
• Singular_trace wikiPageWikiLink Paul_Halmos.
• Singular_trace wikiPageWikiLink Probability.
• Singular_trace wikiPageWikiLink Pseudo-differential_operator.
• Singular_trace wikiPageWikiLink Pseudo-differential_operators.
• Singular_trace wikiPageWikiLink Quantum_state.
• Singular_trace wikiPageWikiLink Quantum_statistical_mechanics.
• Singular_trace wikiPageWikiLink Schatten_class_operator.
• Singular_trace wikiPageWikiLink Self-adjoint.
• Singular_trace wikiPageWikiLink Selfadjoint.
• Singular_trace wikiPageWikiLink Singular_measure.
• Singular_trace wikiPageWikiLink Singular_value.
• Singular_trace wikiPageWikiLink Summation.
• Singular_trace wikiPageWikiLink Symmetric_functional.
• Singular_trace wikiPageWikiLink Trace_(linear_algebra).
• Singular_trace wikiPageWikiLink Trace_class.
• Singular_trace wikiPageWikiLink Victor_Guillemin.
• Singular_trace wikiPageWikiLink Volume_form.
• Singular_trace wikiPageWikiLink Von_Neumann_algebra.
• Singular_trace wikiPageWikiLink Weak_topology.
• Singular_trace wikiPageWikiLink Weak_trace-class_operator.
• Singular_trace wikiPageWikiLink Category:Traces.
• Singular_trace wikiPageWikiLinkText "Singular trace".
• Singular_trace wikiPageWikiLinkText "singular trace".
• Singular_trace wikiPageWikiLinkText "traces".
• Singular_trace hasPhotoCollection Singular_trace.
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• Singular_trace wikiPageUsesTemplate Template:Cite_journal.
• Singular_trace wikiPageUsesTemplate Template:EquationNote.
• Singular_trace wikiPageUsesTemplate Template:EquationRef.
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• Singular_trace subject Category:Hilbert_space.
• Singular_trace subject Category:Von_Neumann_algebras.
• Singular_trace subject Category:Traces.
• Singular_trace type Thing.
• Singular_trace comment "In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanisheson operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank.".
• Singular_trace label "Singular trace".
• Singular_trace seeAlso Dixmier_trace.
• Singular_trace sameAs m.0x1yq_p.
• Singular_trace sameAs Q17103311.
• Singular_trace sameAs Q17103311.
• Singular_trace wasDerivedFrom Singular_trace?oldid=674999959.
• Singular_trace isPrimaryTopicOf Singular_trace.