Inmathematics, anelement of a*-algebra is callednormal if itcommutates with itsadjoint.^[1]

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

Theset of normal elements is denoted by${\displaystyle {\mathcal{A}}_N}$ or${\displaystyle N(\mathcal{A})}$.

A special case of particular importance is the case where${\displaystyle \mathcal{A}}$ is acomplete normed *-algebra, that satisfies the C*-identity (${\displaystyle \Vert a^{\ast }a\Vert ={\Vert a\Vert }^2\mathrm{\forall }a\in \mathcal{A}}$), which is called aC*-algebra.

• Everyself-adjoint element of a a *-algebra isnormal.^[1]
• Everyunitary element of a a *-algebra isnormal.^[2]
• If${\displaystyle \mathcal{A}}$ is a C*-Algebra and${\displaystyle a\in {\mathcal{A}}_N}$ a normal element, then for everycontinuous function${\displaystyle f}$ on thespectrum of${\displaystyle a}$ thecontinuous functional calculus defines another normal element${\displaystyle f(a)}$.^[3]

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

• An element${\displaystyle a\in \mathcal{A}}$ is normal if and only if the *-subalgebra generated by${\displaystyle a}$, meaning the smallest *-algebra containing${\displaystyle a}$, iscommutative.^[2]
• Every element${\displaystyle a\in \mathcal{A}}$ can be uniquely decomposed into areal and imaginary part, which means there exist self-adjoint elements${\displaystyle a_1,a_2\in {\mathcal{A}}_{sa}}$, such that${\displaystyle a=a_1+\mathrm{i}{a}_2}$, where${\displaystyle \mathrm{i}}$ denotes theimaginary unit. Exactly then${\displaystyle a}$ is normal if${\displaystyle a_1{a}_2=a_2{a}_1}$, i.e. real and imaginary partcommutate.^[1]

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

• The adjoint element${\displaystyle a^{\ast }}$ is also normal, since${\displaystyle a=(a^{\ast }{)}^{\ast }}$ holds for theinvolution*.^[4]

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

• It is${\displaystyle \Vert a^2\Vert ={\Vert a\Vert }^2}$, since for normal elements using the C*-identity${\displaystyle {\Vert a^2\Vert }^2=\Vert (a^2)(a^2{)}^{\ast }\Vert =\Vert (a^{\ast }a{)}^{\ast }(a^{\ast }a)\Vert ={\Vert a^{\ast }a\Vert }^2={\left({\Vert a\Vert }^2\right)}^2}$holds.^[5]
• Every normal element is a normaloid element, i.e. thespectral radius${\displaystyle r(a)}$ equals the norm of${\displaystyle a}$, i.e.${\displaystyle r(a)=\Vert a\Vert }$.^[6] This follows from thespectral 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${\displaystyle a}$ to${\displaystyle a}$.^[3]

• Normal matrix
• Normal operator

• Dixmier, Jacques (1977).C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland.ISBN 0-7204-0762-1. English translation ofLes 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.

• Abstract algebra
• C*-algebras

• CS1 French-language sources (fr)
• CS1 German-language sources (de)