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

The set 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 a complete normed *-algebra, that satisfies the C*-identity (${\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}}$ ), which is called a C*-algebra.

• Every self-adjoint element of a a *-algebra is normal.^[1]
• Every unitary element of a a *-algebra is normal.^[2]
• If ${\displaystyle {\mathcal {A}}}$  is a C*-Algebra and ${\displaystyle a\in {\mathcal {A}}_{N}}$  a normal element, then for every continuous function ${\displaystyle f}$  on the spectrum of ${\displaystyle a}$  the continuous 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 ${\displaystyle a}$ , is commutative.^[2]
• Every element ${\displaystyle a\in {\mathcal {A}}}$  can be uniquely decomposed into a real 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 the imaginary unit. Exactly then ${\displaystyle a}$  is normal if ${\displaystyle a_{1}a_{2}=a_{2}a_{1}}$ , i.e. real and imaginary part commutate.^[1]

In *-algebras edit

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

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

In C*-algebras edit

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

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