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 {\displaystyle {\mathcal {A}}} be a *-algebra with unit e {\displaystyle e} . An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is called unitary if a a ∗ = a ∗ a = e {\displaystyle aa^{*}=a^{*}a=e} . In other words, if a {\displaystyle a} is invertible and a − 1 = a ∗ {\displaystyle a^{-1}=a^{*}} holds, then a {\displaystyle a} is unitary. The set of unitary elements is denoted by A U {\displaystyle {\mathcal {A}}_{U}} or U ( A ) {\displaystyle U({\mathcal {A}})} . A special case from particular importance is the case where A {\displaystyle {\mathcal {A}}} is a complete normed *-algebra. This algebra satisfies the C*-identity ( ‖ a ∗ a ‖ = ‖ a ‖ 2 ∀ a ∈ A {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}} ) and is called a C*-algebra.
Criteria Let A {\displaystyle {\mathcal {A}}} be a unital C*-algebra and a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} a normal element. Then, a {\displaystyle a} is unitary if the spectrum σ ( a ) {\displaystyle \sigma (a)} consists only of elements of the circle group T {\displaystyle \mathbb {T} } , i.e. σ ( a ) ⊆ T = { λ ∈ C ∣ | λ | = 1 } {\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}} .
Examples The unit e {\displaystyle e} is unitary. Let A {\displaystyle {\mathcal {A}}} be a unital C*-algebra, then:
Every projection, i.e. every element a ∈ A {\displaystyle a\in {\mathcal {A}}} with a = a ∗ = a 2 {\displaystyle a=a^{*}=a^{2}} , is unitary. For the spectrum of a projection consists of at most 0 {\displaystyle 0} and 1 {\displaystyle 1} , as follows from the continuous functional calculus. If a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} is a normal element of a C*-algebra A {\displaystyle {\mathcal {A}}} , then for every continuous function f {\displaystyle f} on the spectrum σ ( a ) {\displaystyle \sigma (a)} the continuous functional calculus defines an unitary element f ( a ) {\displaystyle f(a)} , if f ( σ ( a ) ) ⊆ T {\displaystyle f(\sigma (a))\subseteq \mathbb {T} } .
Properties Let A {\displaystyle {\mathcal {A}}} be a unital *-algebra and a , b ∈ A U {\displaystyle a,b\in {\mathcal {A}}_{U}} . Then:
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