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 .
Definition Let 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 for invertible operators , for is norm-continuous on , the set of invertible operators in does not exceed the spectral radius of .
Extensions to singular operators There 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 extension The algebraic extension of assigns a value of 0 to a singular operator in .
Analytic extension For 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
Generalizations Although 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 which case it is denoted by .
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