Normal element

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

Definition

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-Algebra. An element [math]\displaystyle{ a \in \mathcal{A} }[/math] is called normal if it commutates with [math]\displaystyle{ a^* }[/math], i.e. it satisfies the equation [math]\displaystyle{ aa^* = a^*a }[/math].[1]

The set of normal elements is denoted by [math]\displaystyle{ \mathcal{A}_N }[/math] or [math]\displaystyle{ N(\mathcal{A}) }[/math].

A special case from particular importance is the case where [math]\displaystyle{ \mathcal{A} }[/math] is a complete normed *-algebra, that satisfies the C*-identity ([math]\displaystyle{ \left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A} }[/math]), which is called a C*-algebra.

Examples

Criteria

Let [math]\displaystyle{ \mathcal{A} }[/math] be a *-algebra. Then:

  • An element [math]\displaystyle{ a \in \mathcal{A} }[/math] is normal if and only if the *-subalgebra generated by [math]\displaystyle{ a }[/math], meaning the smallest *-algebra containing [math]\displaystyle{ a }[/math], is commutative.[2]
  • Every element [math]\displaystyle{ a \in \mathcal{A} }[/math] can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements [math]\displaystyle{ a_1,a_2 \in \mathcal{A}_{sa} }[/math], such that [math]\displaystyle{ a = a_1 + \mathrm{i} a_2 }[/math], where [math]\displaystyle{ \mathrm{i} }[/math] denotes the imaginary unit. Exactly then [math]\displaystyle{ a }[/math] is normal if [math]\displaystyle{ a_1 a_2 = a_2 a_1 }[/math], i.e. real and imaginary part commutate.[1]

Properties

In *-algebras

Let [math]\displaystyle{ a \in \mathcal{A}_N }[/math] be a normal element of a *-algebra [math]\displaystyle{ \mathcal{A} }[/math]. Then:

  • The adjoint element [math]\displaystyle{ a^* }[/math] is also normal, since [math]\displaystyle{ a = (a^*)^* }[/math] holds for the involution *.[4]

In C*-algebras

Let [math]\displaystyle{ a \in \mathcal{A}_N }[/math] be a normal element of a C*-algebra [math]\displaystyle{ \mathcal{A} }[/math]. Then:

  • It is [math]\displaystyle{ \left\| a^2 \right\| = \left\| a \right\|^2 }[/math], since for normal elements using the C*-identity [math]\displaystyle{ \left\| a^2 \right\|^2 = \left\| (a^2) (a^2)^* \right\| = \left\| (a^*a)^* (a^*a) \right\| = \left\| a^*a \right\|^2 = \left( \left\| a \right\|^2 \right)^2 }[/math] holds.[5]
  • Every normal element is a normaloid element, i.e. the spectral radius [math]\displaystyle{ r(a) }[/math] equals the norm of [math]\displaystyle{ a }[/math], i.e. [math]\displaystyle{ r(a)= \left\| a \right\| }[/math].[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 [math]\displaystyle{ a }[/math] to [math]\displaystyle{ a }[/math].[3]

See also

Notes

  1. 1.0 1.1 1.2 1.3 Dixmier 1977, p. 4.
  2. 2.0 2.1 Dixmier 1977, p. 5.
  3. 3.0 3.1 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

  • Dixmier, Jacques (1977). C*-algebras. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1.  English translation of Dixmier, Jacques (1969) (in fr). Les C*-algèbres et leurs représentations. Gauthier-Villars. 
  • Heuser, Harro (1982). Functional analysis. John Wiley & Sons Ltd.. ISBN 0-471-10069-2. 
  • Werner, Dirk (2018) (in de). Funktionalanalysis (8 ed.). Springer. ISBN 978-3-662-55407-4.