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Unitary element

From Wikipedia, the free encyclopedia

In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.[1]

Definition

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Let be a *-algebra with unit . An element is called unitary if . In other words, if is invertible and holds, then is unitary.[1]

The set of unitary elements is denoted by or .

A special case from particular importance is the case where is a complete normed *-algebra. This algebra satisfies the C*-identity () and is called a C*-algebra.

Criteria

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  • Let be a unital C*-algebra and a normal element. Then, is unitary if the spectrum consists only of elements of the circle group , i.e. .[2]

Examples

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  • The unit is unitary.[3]

Let be a unital C*-algebra, then:

  • Every projection, i.e. every element with , is unitary. For the spectrum of a projection consists of at most and , as follows from the continuous functional calculus.[4]
  • If is a normal element of a C*-algebra , then for every continuous function on the spectrum the continuous functional calculus defines an unitary element , if .[2]

Properties

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Let be a unital *-algebra and . Then:

  • The element is unitary, since . In particular, forms a multiplicative group.[1]
  • The element is normal.[3]
  • The adjoint element is also unitary, since holds for the involution *.[1]
  • If is a C*-algebra, has norm 1, i.e. .[5]

See also

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Notes

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  1. ^ a b c d Dixmier 1977, p. 5.
  2. ^ a b Kadison & Ringrose 1983, p. 271.
  3. ^ a b Dixmier 1977, pp. 4–5.
  4. ^ Blackadar 2006, pp. 57, 63.
  5. ^ Dixmier 1977, p. 9.

References

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  • Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. pp. 57, 63. ISBN 3-540-28486-9.
  • Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
  • Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.