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

Definition

Let ${ℳ}$ be a finite factor with the canonical normalized trace $\tau$ and let $X$ be an invertible operator in ${ℳ}$. Then the Fuglede−Kadison determinant of $X$ is defined as

$\mathrm{\Delta }(X):=\exp \tau (\log (X^{*}X{)}^{1/2}),$

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

Properties

• $\mathrm{\Delta }(XY)=\mathrm{\Delta }(X)\mathrm{\Delta }(Y)$ for invertible operators $X,Y\in {ℳ}$,
• $\mathrm{\Delta }(\exp A)=|\exp \tau (A)|=\exp \mathrm{\Re }\tau (A)$ for $A\in {ℳ}.$
• $\mathrm{\Delta }$ is norm-continuous on $G{L}_1({ℳ})$, the set of invertible operators in ${ℳ},$
• $\mathrm{\Delta }(X)$ 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 ${ℳ}$. All of them must assign a value of 0 to operators with non-trivial nullspace. No extension of the determinant $\mathrm{\Delta }$ from the invertible operators to all operators in ${ℳ}$, is continuous in the uniform topology.

Algebraic extension

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

Analytic extension

For an operator $A$ in ${ℳ}$, the analytic extension of $\mathrm{\Delta }$ uses the spectral decomposition of $|A|=\int \lambda d{E}_{\lambda }$ to define $\mathrm{\Delta }(A):=\exp (\int \log \lambda d\tau (E_{\lambda }))$ with the understanding that $\mathrm{\Delta }(A)=0$ if $\int \log \lambda d\tau (E_{\lambda })=-\mathrm{\infty }$. This extension satisfies the continuity property

$\underset{\epsilon \to 0}{\lim }\mathrm{\Delta }(H+\epsilon I)=\mathrm{\Delta }(H)$ 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 ${\mathrm{\Delta }}_{\tau }(\cdot )$.

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.