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 (3): 520–530, doi:10.2307/1969645, JSTOR 1969645.

• de la Harpe, Pierre (2013), "Fuglede−Kadison determinant: theme and variations", Proc. Natl. Acad. Sci. USA, 110 (40): 15864–15877, doi:10.1073/pnas.1202059110, PMC 3791716, PMID 24082099.