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 $a{a}^{\ast }=a^{\ast }a=e$. In other words, if $a$ is invertible and $a^{-1}=a^{\ast }$ 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 ($\Vert a^{\ast }a\Vert ={\Vert a\Vert }^2\mathrm{\forall }a\in 𝒜$) 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 $𝕋$, i.e. $\sigma (a)\subseteq 𝕋=\{\lambda \in ℂ\mid |\lambda |=1\}$.

Examples

• The unit $e$ is unitary.

Let $𝒜$ be a unital C*-algebra, then:

• Every projection, i.e. every element $a\in 𝒜$ with $a=a^{\ast }=a^2$, 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 $f(\sigma (a))\subseteq 𝕋$.

Properties

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

• The element $ab$ is unitary, since $((ab{)}^{\ast }{)}^{-1}=(b^{\ast }{a}^{\ast }{)}^{-1}=(a^{\ast }{)}^{-1}(b^{\ast }{)}^{-1}=ab$. In particular, ${𝒜}_U$ forms a multiplicative group.
• The element $a$ is normal.
• The adjoint element $a^{\ast }$ is also unitary, since $a=(a^{\ast }{)}^{\ast }$ 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. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in français). Gauthier-Villars. 1969.
• 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.