Encyclopedia

Self-adjoint element

Element of *-algebra where x* equals x

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated May 19, 2026

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

01Definition

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

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

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

A special case of particular importance is the case where ${\mathcal {A}}$ is a complete normed *-algebra, that satisfies the C*-identity ( $\left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}$ ), 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 ${\mathcal {A}}_{h}$ , ${\mathcal {A}}_{H}$ or $H({\mathcal {A}})$ for the set of self-adjoint elements are also sometimes used, even in the more recent literature.

02Examples

Each positive element of a C*-algebra is self-adjoint.
For each element $a$ of a *-algebra, the elements $aa^{*}$ and $a^{*}a$ are self-adjoint, since * is an involutive antiautomorphism.
For each element $a$ 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 $\mathrm {i}$ denotes the imaginary unit.
If $a\in {\mathcal {A}}_{N}$ is a normal element of a C*-algebra ${\mathcal {A}}$ , then for every real-valued function $f$ , which is continuous on the spectrum of $a$ , the continuous functional calculus defines a self-adjoint element $f(a)$ .

03Criteria

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

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

04Properties

In *-algebras

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

Each element $a\in {\mathcal {A}}$ can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements $a_{1},a_{2}\in {\mathcal {A}}_{sa}$ , so that $a=a_{1}+\mathrm {i} a_{2}$ holds. Where ${\textstyle a_{1}={\frac {1}{2}}(a+a^{*})$ and ${\textstyle a_{2}={\frac {1}{2\mathrm {i} }}(a-a^{*})$ .
The set of self-adjoint elements ${\mathcal {A}}_{sa}$ is a real linear subspace of ${\mathcal {A}}$ . From the previous property, it follows that ${\mathcal {A}}$ is the direct sum of two real linear subspaces, i.e. ${\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}$ .
If $a\in {\mathcal {A}}_{sa}$ is self-adjoint, then $a$ is normal.
The *-algebra ${\mathcal {A}}$ is called a hermitian *-algebra if every self-adjoint element $a\in {\mathcal {A}}_{sa}$ has a real spectrum $\sigma (a)\subseteq \mathbb {R}$ .

In C*-algebras

Let ${\mathcal {A}}$ be a C*-algebra and $a\in {\mathcal {A}}_{sa}$ . Then:

For the spectrum $\left\|a\right\|\in \sigma (a)$ or $-\left\|a\right\|\in \sigma (a)$ holds, since $\sigma (a)$ is real and $r(a)=\left\|a\right\|$ holds for the spectral radius, because $a$ is normal.
According to the continuous functional calculus, there exist uniquely determined positive elements $a_{+},a_{-}\in {\mathcal {A}}_{+}$ , such that $a=a_{+}-a_{-}$ with $a_{+}a_{-}=a_{-}a_{+}=0$ . For the norm, $\left\|a\right\|=\max(\left\|a_{+}\right\|,\left\|a_{-}\right\|)$ holds. The elements $a_{+}$ and $a_{-}$ are also referred to as the positive and negative parts. In addition, $|a|=a_{+}+a_{-}$ holds for the absolute value defined for every element ${\textstyle |a|=(a^{*}a)^{\frac {1}{2}}$ .
For every $a\in {\mathcal {A}}_{+}$ and odd $n\in \mathbb {N}$ , there exists a uniquely determined $b\in {\mathcal {A}}_{+}$ that satisfies $b^{n}=a$ , i.e. a unique $n$ -th root, as can be shown with the continuous functional calculus.

Watch videos about Self-adjoint elementExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Self-adjoint element”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.