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 $$A$$ is often denoted by $$\Delta(A)$$.

For a matrix $$A$$ in $$M_n(\mathbb{C})$$, $$\Delta(A) = \left| \det (A) \right|^{1/n}$$ which is the normalized form of the absolute value of the determinant of $$A$$.

Definition
Let $$\mathcal{M}$$ be a finite factor with the canonical normalized trace $$\tau$$ and let $$X$$ be an invertible operator in $$\mathcal{M}$$. Then the Fuglede−Kadison determinant of $$X$$ is defined as
 * $$\Delta(X) := \exp \tau(\log (X^*X)^{1/2}),$$

(cf. Relation between determinant and trace via eigenvalues). The number $$\Delta(X)$$ is well-defined by continuous functional calculus.

Properties

 * $$\Delta(XY) = \Delta(X) \Delta(Y)$$ for invertible operators $$X, Y \in \mathcal{M}$$,
 * $$\Delta (\exp A) = \left| \exp \tau(A) \right| = \exp \Re \tau(A)$$ for $$A \in \mathcal{M}.$$
 * $$\Delta$$ is norm-continuous on $$GL_1(\mathcal{M})$$, the set of invertible operators in $$\mathcal{M},$$
 * $$\Delta(X)$$ does not exceed the spectral radius of $$X$$.

Extensions to singular operators
There are many possible extensions of the Fuglede−Kadison determinant to singular operators in $$\mathcal{M}$$. All of them must assign a value of 0 to operators with non-trivial nullspace. No extension of the determinant $$\Delta$$ from the invertible operators to all operators in $$\mathcal{M}$$, is continuous in the uniform topology.

Algebraic extension
The algebraic extension of $$\Delta$$ assigns a value of 0 to a singular operator in $$\mathcal{M}$$.

Analytic extension
For an operator $$A$$ in $$\mathcal{M}$$, the analytic extension of $$\Delta$$ uses the spectral decomposition of $$|A| = \int \lambda \; dE_\lambda$$ to define $$\Delta(A) := \exp \left( \int \log \lambda \; d\tau(E_\lambda) \right)$$ with the understanding that $$\Delta(A) = 0$$ if $$\int \log \lambda \; d\tau(E_\lambda) = -\infty$$. This extension satisfies the continuity property
 * $$\lim_{\varepsilon \rightarrow 0} \Delta(H + \varepsilon I) = \Delta(H)$$ for $$H \ge 0.$$

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 ($$\tau$$) in the case of which it is denoted by $$\Delta_\tau(\cdot)$$.