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词条 Fuglede−Kadison determinant
释义

  1. Definition

  2. Properties

  3. Extensions to singular operators

     Algebraic extension  Analytic extension 

  4. Generalizations

  5. References

{{expert needed|mathematics|reason=review the article|date=October 2018}}

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

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

  • 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

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

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

Analytic extension

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

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

  • {{citation

| last1 = Fuglede | first1 = Bent
| last2 = Kadison | first2 = Richard
| journal = Ann. Math. |series=Series 2
| pages = 520−530
| title = Determinant theory in finite factors
| volume = 55
| year = 1952}}.
  • {{citation

| last = de la Harpe| first = Pierre
| journal = Proc. Natl. Acad. Sci. USA
| pages = 15864–15877
| title = Fuglede−Kadison determinant: theme and variations
| volume = 110
| year = 2013}}.{{DEFAULTSORT:Fuglede-Kadison determinant}}

1 : Von Neumann algebras

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