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

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 ${\displaystyle aa^{*}=a^{*}a}$.

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.

Let ${\displaystyle {\mathcal {A}}}$ be a *-algebra. Then:

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

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