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
which is the normalized form of the absolute value of the determinant of
be a finite factor with the canonical normalized trace
be an invertible operator in
is defined as (cf.
Relation between determinant and trace via eigenvalues).
is well-defined by continuous functional calculus.
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
, is continuous in the uniform topology.
The algebraic extension of
assigns a value of 0 to a singular operator in
, the analytic extension of
uses the spectral decomposition of
∫ log λ
∫ log λ
This extension satisfies the continuity property 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