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.

Definition

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

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

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

Criteria

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

Examples

• The unit $e$ is unitary.

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

• Every projection, i.e. every element $a\in {𝒜}$ 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 $a\in {{𝒜}}_N$ is a normal element of a C*-algebra ${𝒜}$, then for every continuous function $f$ on the spectrum $\sigma (a)$ the continuous functional calculus defines an unitary element $f(a)$, if $f(\sigma (a))\subseteq \mathbb{𝕋}$.

Properties

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

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

See also

• Unitary matrix
• Unitary operator

References

• Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. pp. 57, 63. ISBN 3-540-28486-9. 
• Dixmier, Jacques (1977). C*-algebras. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1.  English translation of Dixmier, Jacques (1969) (in fr). Les C*-algèbres et leurs représentations. Gauthier-Villars. 
• Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory.. New York/London: Academic Press. ISBN 0-12-393301-3.