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

The set of unitary elements is denoted by [math]\displaystyle{ \mathcal{A}_U }[/math] or [math]\displaystyle{ U(\mathcal{A}) }[/math].

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

Criteria

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

Examples

• The unit [math]\displaystyle{ e }[/math] is unitary.

Let [math]\displaystyle{ \mathcal{A} }[/math] be a unital C*-algebra, then:

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

Properties

Let [math]\displaystyle{ \mathcal{A} }[/math] be a unital *-algebra and [math]\displaystyle{ a,b \in \mathcal{A}_U }[/math]. Then:

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

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.