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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 is often denoted by .

For a matrix in , which is the normalized form of the absolute value of the determinant of .

Definition

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Let be a finite factor with the canonical normalized trace and let be an invertible operator in . Then the Fuglede−Kadison determinant of is defined as

(cf. Relation between determinant and trace via eigenvalues). The number is well-defined by continuous functional calculus.

Properties

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  • for invertible operators ,
  • for
  • is norm-continuous on , the set of invertible operators in
  • does not exceed the spectral radius of .

Extensions to singular operators

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

Algebraic extension

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The algebraic extension of assigns a value of 0 to a singular operator in .

Analytic extension

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For an operator in , the analytic extension of uses the spectral decomposition of to define with the understanding that if . This extension satisfies the continuity property

for

Generalizations

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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 () in the case of which it is denoted by .

References

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