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 ${\displaystyle A}$ is often denoted by ${\displaystyle \Delta (A)}$.

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

Definition

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

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

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

Properties

• ${\displaystyle \Delta (XY)=\Delta (X)\Delta (Y)}$ for invertible operators ${\displaystyle X,Y\in {\mathcal {M}}}$,
• ${\displaystyle \Delta (\exp A)=\left|\exp \tau (A)\right|=\exp \Re \tau (A)}$ for ${\displaystyle A\in {\mathcal {M}}.}$
• ${\displaystyle \Delta }$ is norm-continuous on ${\displaystyle GL_{1}({\mathcal {M}})}$, the set of invertible operators in ${\displaystyle {\mathcal {M}},}$
• ${\displaystyle \Delta (X)}$ does not exceed the spectral radius of ${\displaystyle X}$.

Extensions to singular operators

There are many possible extensions of the Fuglede−Kadison determinant to singular operators in ${\displaystyle {\mathcal {M}}}$. All of them must assign a value of 0 to operators with non-trivial nullspace. No extension of the determinant ${\displaystyle \Delta }$ from the invertible operators to all operators in ${\displaystyle {\mathcal {M}}}$, is continuous in the uniform topology.

Algebraic extension

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

Analytic extension

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

${\displaystyle \lim _{\varepsilon \rightarrow 0}\Delta (H+\varepsilon I)=\Delta (H)}$ for ${\displaystyle H\geq 0.}$

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 (${\displaystyle \tau }$) in the case of which it is denoted by ${\displaystyle \Delta _{\tau }(\cdot )}$.

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.