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 $M_n(\mathbb)$, $\Delta(A) = \left, \det (A) \^$ which is the normalized form of the absolute value of the determinant of $A$.

Definition

Let $\mathcal$ be a finite factor with the canonical normalized trace $\tau$ and let $X$ be an invertible operator in $\mathcal$. Then the Fuglede−Kadison determinant of $X$ is defined as
:$\Delta(X) := \exp \tau(\log (X^*X)^),$
(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 $X, Y \in \mathcal$,
* $\Delta (\exp A) = \left, \exp \tau(A) \ = \exp \Re \tau(A)$ for $A \in \mathcal.$
* $\Delta$ is norm-continuous on $GL_1(\mathcal)$, the set of invertible operators in $\mathcal,$
* $\Delta(X)$ 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 $\mathcal$. 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 $\mathcal$, is continuous in the uniform topology.

Algebraic extension

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

Analytic extension

For an operator $A$ in $\mathcal$, 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
:$\lim_ \Delta(H + \varepsilon I) = \Delta(H)$ 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 $\Delta_\tau(\cdot)$.

References

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{{DEFAULTSORT:Fuglede-Kadison determinant
Von Neumann algebras