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 [math]\displaystyle{ A }[/math] is often denoted by [math]\displaystyle{ \Delta(A) }[/math].

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

Definition

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

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

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

Properties

• [math]\displaystyle{ \Delta(XY) = \Delta(X) \Delta(Y) }[/math] for invertible operators [math]\displaystyle{ X, Y \in \mathcal{M} }[/math],
• [math]\displaystyle{ \Delta (\exp A) = \left| \exp \tau(A) \right| = \exp \Re \tau(A) }[/math] for [math]\displaystyle{ A \in \mathcal{M}. }[/math]
• [math]\displaystyle{ \Delta }[/math] is norm-continuous on [math]\displaystyle{ GL_1(\mathcal{M}) }[/math], the set of invertible operators in [math]\displaystyle{ \mathcal{M}, }[/math]
• [math]\displaystyle{ \Delta(X) }[/math] does not exceed the spectral radius of [math]\displaystyle{ X }[/math].

Extensions to singular operators

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

Algebraic extension

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

Analytic extension

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

[math]\displaystyle{ \lim_{\varepsilon \rightarrow 0} \Delta(H + \varepsilon I) = \Delta(H) }[/math] for [math]\displaystyle{ H \ge 0. }[/math]

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 ([math]\displaystyle{ \tau }[/math]) in the case of which it is denoted by [math]\displaystyle{ \Delta_\tau(\cdot) }[/math].

References

• Fuglede, Bent; Kadison, Richard (1952), "Determinant theory in finite factors", Ann. Math., Series 2 55 (3): 520–530, doi:10.2307/1969645 .

• 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, PMID 24082099 .