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

Definition

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.

Examples

• 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]

Criteria

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]

Properties

In *-algebras

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

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]