Normal element

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

Definition

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

The set of normal elements is denoted by ${\mathcal {A}}_{N}$ or $N({\mathcal {A}})$ .

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

Examples

Every self-adjoint element of a a *-algebra is normal.^[1]
Every unitary element of a a *-algebra is normal.^[2]
If ${\mathcal {A}}$ is a C*-Algebra and $a\in {\mathcal {A}}_{N}$ a normal element, then for every continuous function $f$ on the spectrum of $a$ the continuous functional calculus defines another normal element $f(a)$ .^[3]

Criteria

Let ${\mathcal {A}}$ be a *-algebra. Then:

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

Properties

In *-algebras

Let $a\in {\mathcal {A}}_{N}$ be a normal element of a *-algebra ${\mathcal {A}}$ . Then:

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

In C*-algebras

Let $a\in {\mathcal {A}}_{N}$ be a normal element of a C*-algebra ${\mathcal {A}}$ . Then:

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

Notes

Dixmier 1977, p. 4.
Dixmier 1977, p. 5.
Dixmier 1977, p. 13.
Dixmier 1977, pp. 3–4.
Werner 2018, p. 518.
Heuser 1982, p. 390.
Werner 2018, pp. 284–285, 518.

References

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.