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In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on 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 summing numbers ''a''1, ''a''2, ''a''3, ... that is completely orthogonal or 'singular' with respect to the usual sum ''a''1 + ''a''2 + ''a''3 + ... . 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. == Origin == By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras,〔 , 〕 thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal up to some trivial counterexamples.〔 〕 Over the course of 15 years Dixmier, aided by a suggestion of Nachman Aronszajn and inequalities proved by Joseph Hersch, developed an example of a non-trivial yet non-normal trace on weak trace-class operators,〔 〕 disproving his earlier view. Singular traces based on Dixmier's construction are called Dixmier traces. Independently and by different methods, German mathematician Albrecht Pietsch (de) investigated traces on ideals of operators on Banach spaces.〔 〕 In 1987 Nigel Kalton answered a question of Piestch by showing that the operator trace is not the unique trace on quasi-normed proper subideals of the trace-class operators on a Hilbert space.〔 〕 József Varga independently studied a similar question.〔 〕 To solve the question of uniqueness of the trace on the full ideal of trace-class operators, Kalton developed a spectral condition for the commutator subspace of trace class operators following on from results of Gary Weiss.〔 A consequence of the results of Weiss and the spectral condition of Kalton was the existence of non-trivial singular traces on trace class operators .〔〔 Also independently, and from a different direction, Mariusz Wodzicki investigated the noncommutative residue, a trace on classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative of the dimension of the manifold.〔 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「singular trace」の詳細全文を読む スポンサード リンク
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