词条 | Fuglede−Kadison determinant |
释义 |
In mathematics, the Fuglede−Kadison determinant of an invertible operator in a finite factor is a positive real number associated with it. It defines a multiplicative homomorphism from the set of invertible operators to the set of positive real numbers. The Fuglede−Kadison determinant of an operator is often denoted by . For a matrix in , which is the normalized form of the absolute value of the determinant of . DefinitionLet be a finite factor with the canonical normalized trace and let be an invertible operator in . Then the Fuglede−Kadison determinant of is defined as (cf. Relation between determinant and trace via eigenvalues). The number is well-defined by continuous functional calculus. Properties
Extensions to singular operatorsThere are many possible extensions of the Fuglede−Kadison determinant to singular operators in . All of them must assign a value of 0 to operators with non-trivial nullspace. No extension of the determinant from the invertible operators to all operators in , is continuous in the uniform topology. Algebraic extensionThe algebraic extension of assigns a value of 0 to a singular operator in . Analytic extensionFor an operator in , the analytic extension of uses the spectral decomposition of to define with the understanding that if . This extension satisfies the continuity property for GeneralizationsAlthough originally the Fuglede−Kadison determinant was defined for operators in finite factors, it carries over to the case of operators in von Neumann algebras with a tracial state () in the case of which it is denoted by . References
| last1 = Fuglede | first1 = Bent | last2 = Kadison | first2 = Richard | journal = Ann. Math. |series=Series 2 | pages = 520−530 | title = Determinant theory in finite factors | volume = 55 | year = 1952}}.
| last = de la Harpe| first = Pierre | journal = Proc. Natl. Acad. Sci. USA | pages = 15864–15877 | title = Fuglede−Kadison determinant: theme and variations | volume = 110 | year = 2013}}.{{DEFAULTSORT:Fuglede-Kadison determinant}} 1 : Von Neumann algebras |
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