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 ${\displaystyle a=a^{*}}$.^[1]

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.^[2]

A special case from 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.^[1] 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.

• Each positive element of a C*-algebra is self-adjoint.^[3]
• For each element ${\displaystyle a}$ of a *-algebra, the elements ${\displaystyle aa^{*}}$ and ${\displaystyle a^{*}a}$ are self-adjoint, since * is an involutive antiautomorphism.^[4]
• For each element ${\displaystyle a}$ of a *-algebra, the real and imaginary parts ${\textstyle \operatorname {Re} (a)={\frac {1}{2}}(a+a^{*})}$ and ${\textstyle \operatorname {Im} (a)={\frac {1}{2\mathrm {i} }}(a-a^{*})}$ are self-adjoint, where ${\displaystyle \mathrm {i} }$ denotes the imaginary unit.^[1]
• If ${\displaystyle a\in {\mathcal {A}}_{N}}$ is a normal element of a C*-algebra ${\displaystyle {\mathcal {A}}}$, then for every real-valued function ${\displaystyle f}$, which is continuous on the spectrum of ${\displaystyle a}$, the continuous functional calculus defines a self-adjoint element ${\displaystyle f(a)}$.^[5]

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

• Let ${\displaystyle a\in {\mathcal {A}}}$, then ${\displaystyle a^{*}a}$ is self-adjoint, since ${\displaystyle (a^{*}a)^{*}=a^{*}(a^{*})^{*}=a^{*}a}$. A similarly calculation yields that ${\displaystyle aa^{*}}$ is also self-adjoint.^[6]
• Let ${\displaystyle a=a_{1}a_{2}}$ be the product of two self-adjoint elements ${\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}}$. Then ${\displaystyle a}$ is self-adjoint if ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ commutate, since ${\displaystyle (a_{1}a_{2})^{*}=a_{2}^{*}a_{1}^{*}=a_{2}a_{1}}$ always holds.^[1]
• If ${\displaystyle {\mathcal {A}}}$ is a C*-algebra, then a normal element ${\displaystyle a\in {\mathcal {A}}_{N}}$ is self-adjoint if and only if its spectrum is real, i.e. ${\displaystyle \sigma (a)\subseteq \mathbb {R} }$.^[5]

• Self-adjoint matrix
• Self-adjoint operator