Normal element

Examples

• Every self-adjoint element of a a *-algebra is normal.
• Every unitary element of a a *-algebra is normal.
• 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 ${\displaystyle f(a)}$.

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.
• 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.

Properties

In *-algebras

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

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

In C*-algebras

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

• It is ${\displaystyle \Vert a^2\Vert ={\Vert a\Vert }^2}$, 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.
• Every normal element is a normaloid element, i.e. the spectral radius ${\displaystyle r(a)}$ equals the norm of ${\displaystyle a}$, i.e. ${\displaystyle r(a)=\Vert a\Vert }$. 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 ${\displaystyle a}$ to ${\displaystyle a}$.