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Normal element

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Source updated May 17, 2026

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

01Definition

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 $aa^{*}=a^{*}a$ .

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.

02Examples

Every self-adjoint element of a a *-algebra is normal.
Every unitary element of a a *-algebra is normal.
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 $f(a)$ .

03Criteria

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.
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.

04Properties

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 *.

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.
Every normal element is a normaloid element, i.e. the spectral radius $r(a)$ equals the norm of $a$ , i.e. $r(a)=\left\|a\right\|$ . 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 $a$ to $a$ .

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Sources and credits

This article is adapted from the Wikipedia article “Normal element”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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