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Normal element

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In mathematics, an element of a *-algebra is called normal if it commutates with its adjoint.[1]

Definition

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Let be a *-Algebra. An element is called normal if it commutes with , i.e. it satisfies the equation .[1]

The set of normal elements is denoted by or .

A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity (), which is called a C*-algebra.

Examples

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Criteria

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Let be a *-algebra. Then:

  • An element is normal if and only if the *-subalgebra generated by , meaning the smallest *-algebra containing , is commutative.[2]
  • Every element can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements , such that , where denotes the imaginary unit. Exactly then is normal if , i.e. real and imaginary part commutate.[1]

Properties

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In *-algebras

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Let be a normal element of a *-algebra . Then:

  • The adjoint element is also normal, since holds for the involution *.[4]

In C*-algebras

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Let be a normal element of a C*-algebra . Then:

  • It is , since for normal elements using the C*-identity holds.[5]
  • Every normal element is a normaloid element, i.e. the spectral radius equals the norm of , i.e. .[6] This follows from the spectral radius formula by repeated application of the previous property.[7]
  • A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of to .[3]

See also

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Notes

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  1. ^ a b c d Dixmier 1977, p. 4.
  2. ^ a b Dixmier 1977, p. 5.
  3. ^ a b Dixmier 1977, p. 13.
  4. ^ Dixmier 1977, pp. 3–4.
  5. ^ Werner 2018, p. 518.
  6. ^ Heuser 1982, p. 390.
  7. ^ Werner 2018, pp. 284–285, 518.

References

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  • 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.
  • Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2.
  • Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4.