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Unitary element

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Source updated May 17, 2026

In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.

01Definition

Let ${\mathcal {A}}$ be a *-algebra with unit $e$ . An element $a\in {\mathcal {A}}$ is called unitary if $aa^{*}=a^{*}a=e$ . In other words, if $a$ is invertible and $a^{-1}=a^{*}$ holds, then $a$ is unitary.

The set of unitary elements is denoted by ${\mathcal {A}}_{U}$ or $U({\mathcal {A}})$ .

A special case from particular importance is the case where ${\mathcal {A}}$ is a complete normed *-algebra. This algebra satisfies the C*-identity ( $\left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}$ ) and is called a C*-algebra.

02Criteria

Let ${\mathcal {A}}$ be a unital C*-algebra and $a\in {\mathcal {A}}_{N}$ a normal element. Then, $a$ is unitary if the spectrum $\sigma (a)$ consists only of elements of the circle group $\mathbb {T}$ , i.e. $\sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}$ .

03Examples

The unit $e$ is unitary.

Let ${\mathcal {A}}$ be a unital C*-algebra, then:

Every projection, i.e. every element $a\in {\mathcal {A}}$ with $a=a^{*}=a^{2}$ , is unitary. For the spectrum of a projection consists of at most $0$ and $1$ , as follows from the continuous functional calculus.
If $a\in {\mathcal {A}}_{N}$ is a normal element of a C*-algebra ${\mathcal {A}}$ , then for every continuous function $f$ on the spectrum $\sigma (a)$ the continuous functional calculus defines an unitary element $f(a)$ , if $f(\sigma (a))\subseteq \mathbb {T}$ .

04Properties

Let ${\mathcal {A}}$ be a unital *-algebra and $a,b\in {\mathcal {A}}_{U}$ . Then:

The element $ab$ is unitary, since ${\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab$ . In particular, ${\mathcal {A}}_{U}$ forms a multiplicative group.
The element $a$ is normal.
The adjoint element $a^{*}$ is also unitary, since $a=(a^{*})^{*}$ holds for the involution *.
If ${\mathcal {A}}$ is a C*-algebra, $a$ has norm 1, i.e. $\left\|a\right\|=1$ .

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Sources and credits

This article is adapted from the Wikipedia article “Unitary 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.

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Unitary element

01Definition

02Criteria

03Examples

04Properties

Sources and credits

Related topics

Unitary matrix

Unitary operator