Normal element

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

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].

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

• Every self-adjoint element of a a *-algebra is normal.
• Every unitary element of a a *-algebra is normal.
• If [math]\displaystyle{ \mathcal{A} }[/math] is a C*-Algebra and [math]\displaystyle{ a \in \mathcal{A}_N }[/math] a normal element, then for every continuous function [math]\displaystyle{ f }[/math] on the spectrum of [math]\displaystyle{ a }[/math] the continuous functional calculus defines another normal element [math]\displaystyle{ f(a) }[/math].

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.
• 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.

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 *.

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.
• 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]. This follows from the spectral radius formula by repeated application of the previous property.
• 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].

See also

• Normal matrix
• Normal operator

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.