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.Template:Sfn

Definition

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

The set of unitary elements is denoted by ${𝒜}_U$ or Template:Nowrap

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. Template:Nowrap

Examples

• The unit $e$ is unitary.Template:Sfn

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 Template:Nowrap
• 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 Template:Nowrap

Properties

Let $𝒜$ be a unital *-algebra and Template:Nowrap Then:

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

See also

• Unitary matrix
• Unitary operator

References

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• Template:Cite book English translation of Template:Cite book
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