In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. ${\displaystyle a=a^{*}}$).

Let ${\displaystyle {\mathcal {A}}}$ be a *-algebra. An element ${\displaystyle a\in {\mathcal {A}}}$ is called self-adjoint if ${\displaystyle a=a^{*}}$.

The set of self-adjoint elements is referred to as ${\displaystyle {\mathcal {A}}_{sa}}$.

A subset ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {A}}}$ that is closed under the involution *, i.e. ${\displaystyle {\mathcal {B}}={\mathcal {B}}^{*}}$, is called self-adjoint.

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.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian. Because of that the notations ${\displaystyle {\mathcal {A}}_{h}}$, ${\displaystyle {\mathcal {A}}_{H}}$ or ${\displaystyle H({\mathcal {A}})}$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

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

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