Unitary element

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.^[1]

Definition

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

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

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

Criteria

• Let ${\displaystyle {\mathcal {A}}}$ be a unital C*-algebra and ${\displaystyle a\in {\mathcal {A}}_{N}}$ a normal element. Then, ${\displaystyle a}$ is unitary if the spectrum ${\displaystyle \sigma (a)}$ consists only of elements of the circle group ${\displaystyle \mathbb {T} }$, i.e. ${\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}}$.^[2]

Examples

• The unit ${\displaystyle e}$ is unitary.^[3]

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

• Every projection, i.e. every element ${\displaystyle a\in {\mathcal {A}}}$ with ${\displaystyle a=a^{*}=a^{2}}$, is unitary. For the spectrum of a projection consists of at most ${\displaystyle 0}$ and ${\displaystyle 1}$, as follows from the continuous functional calculus.^[4]
• If ${\displaystyle a\in {\mathcal {A}}_{N}}$ is a normal element of a C*-algebra ${\displaystyle {\mathcal {A}}}$, then for every continuous function ${\displaystyle f}$ on the spectrum ${\displaystyle \sigma (a)}$ the continuous functional calculus defines an unitary element ${\displaystyle f(a)}$, if ${\displaystyle f(\sigma (a))\subseteq \mathbb {T} }$.^[2]

Properties

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

• The element ${\displaystyle ab}$ is unitary, since ${\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab}$. In particular, ${\displaystyle {\mathcal {A}}_{U}}$ forms a multiplicative group.^[1]
• The element ${\displaystyle a}$ is normal.^[3]
• The adjoint element ${\displaystyle a^{*}}$ is also unitary, since ${\displaystyle a=(a^{*})^{*}}$ holds for the involution *.^[1]
• If ${\displaystyle {\mathcal {A}}}$ is a C*-algebra, ${\displaystyle a}$ has norm 1, i.e. ${\displaystyle \left\|a\right\|=1}$.^[5]

See also

• Unitary matrix
• Unitary operator