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\geq 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

Fuglede, Bent; Kadison, Richard (1952), "Determinant theory in finite factors", Ann. Math., Series 2, 55: 520–530, doi:10.2307/1969645.

de la Harpe, Pierre (2013), "Fuglede−Kadison determinant: theme and variations", Proc. Natl. Acad. Sci. USA, 110: 15864–15877, doi:10.1073/pnas.1202059110.

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