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 \( {\displaystyle \Delta (A)} \).
For a matrix A in \( M_{n}({\mathbb {C}}) \), \( {\displaystyle \Delta (A)=\left|\det(A)\right|^{1/n}} \) which is the normalized form of the absolute value of the determinant of A.
Definition
Let M {\displaystyle {\mathcal {M}}} {\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
\( {\displaystyle \Delta (X):=\exp \tau (\log(X^{*}X)^{1/2}),} \)
(cf. Relation between determinant and trace via eigenvalues). The number \( \Delta (X) \) is well-defined by continuous functional calculus.
Properties
\( {\displaystyle \Delta (XY)=\Delta (X)\Delta (Y)} \) for invertible operators \( {\displaystyle X,Y\in {\mathcal {M}}} \) ,
\( {\displaystyle \Delta (\exp A)=\left|\exp \tau (A)\right|=\exp \Re \tau (A)} \) for \( {\displaystyle A\in {\mathcal {M}}.} \)
\( \Delta \) is norm-continuous on \( {\displaystyle GL_{1}({\mathcal {M}})} \), the set of invertible operators in \( {\displaystyle {\mathcal {M}},} \)
\( \Delta (X) \) 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 \( {\displaystyle |A|=\int \lambda \;dE_{\lambda }} \) 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
lim \( {\displaystyle \lim _{\varepsilon \rightarrow 0}\Delta (H+\varepsilon I)=\Delta (H)} \) 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 ( \( \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: 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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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