In mathematics, the FugledeâÂÂKadison 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 FugledeâÂÂKadison 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 .
Let be a finite factor with the canonical normalized trace and let be an invertible operator in . Then the FugledeâÂÂKadison determinant of is defined as
(cf. Relation between determinant and trace via eigenvalues). The number is well-defined by continuous functional calculus.
There are many possible extensions of the FugledeâÂÂKadison 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.
The algebraic extension of assigns a value of 0 to a singular operator in .
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
Although originally the FugledeâÂÂKadison 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 .