Fuglede−Kadison determinant

Template:Expert needed 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 $M_n(ℂ)$, $\mathrm{\Delta }(A)={|det(A)|}^{1/n}$ 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

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

(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 which case it is denoted by ${\mathrm{\Delta }}_{\tau }(\cdot )$.

References

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