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