Normal element

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

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]

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]

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.