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 $$\mathcal{A}$$ 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.

Criteria

 * 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 \Complex \mid

Examples

 * 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 $f(\sigma(a)) \subseteq \mathbb{T}$.

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


 * The element $$ab$$ is unitary, since $((ab)^*)^{-1} = (b^*a^*)^{-1} = (a^*)^{-1} (b^*)^{-1} = ab$. 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\