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 the case of which it is denoted by . References . . Von Neumann algebras