Fuglede−Kadison determinant

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
. 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

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.