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Fuglede−Kadison determinant

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

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

Definition

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

Δ ( X ) := exp ⁡ τ ( log ⁡ ( X ∗ X ) 1 / 2 ) , {\displaystyle \Delta (X):=\exp \tau (\log(X^{*}X)^{1/2}),}

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

Properties

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

Extensions to singular operators

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

Algebraic extension

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

Analytic extension

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

lim ε → 0 Δ ( H + ε I ) = Δ ( H ) {\displaystyle \lim _{\varepsilon \rightarrow 0}\Delta (H+\varepsilon I)=\Delta (H)} for H ≥ 0. {\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 ( τ {\displaystyle \tau } ) in the case of which it is denoted by Δ τ ( ⋅ ) {\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.
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