Fuglede−Kadison determinant

{{expert needed|mathematics|reason=review the article|date=October 2018}}

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 $\Delta_\tau(\cdot)$ .

References

{{citation

| last1 = Fuglede | first1 = Bent

| last2 = Kadison | first2 = Richard

| journal = Ann. Math. |series=Series 2

| pages = 520–530

| title = Determinant theory in finite factors

| volume = 55

| year = 1952 | issue = 3

| doi=10.2307/1969645| jstor = 1969645

}}.

{{citation

| last = de la Harpe| first = Pierre

| journal = Proc. Natl. Acad. Sci. USA

| pages = 15864–15877

| title = Fuglede−Kadison determinant: theme and variations

| volume = 110

| year = 2013

| issue = 40

| doi=10.1073/pnas.1202059110| pmid = 24082099

| pmc = 3791716

| doi-access = free

}}.

Category:Von Neumann algebras