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.^[1]

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.^[1]

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\}} .^[2]

Examples

• The unit e {\displaystyle e} is unitary.^[3]

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.^[4]
• 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} } .^[2]

Properties

Let A {\displaystyle {\mathcal {A}}} be a unital *-algebra and a , b ∈ A U {\displaystyle a,b\in {\mathcal {A}}_{U}} . Then:

• The element a b {\displaystyle ab} is unitary, since ( ( a b ) ∗ ) − 1 = ( b ∗ a ∗ ) − 1 = ( a ∗ ) − 1 ( b ∗ ) − 1 = a b {\textstyle ((ab)^{*})^{-1}=(b^{*}a^{*})^{-1}=(a^{*})^{-1}(b^{*})^{-1}=ab} . In particular, A U {\displaystyle {\mathcal {A}}_{U}} forms a multiplicative group.^[1]
• The element a {\displaystyle a} is normal.^[3]
• The adjoint element a ∗ {\displaystyle a^{*}} is also unitary, since a = ( a ∗ ) ∗ {\displaystyle a=(a^{*})^{*}} holds for the involution *.^[1]
• If A {\displaystyle {\mathcal {A}}} is a C*-algebra, a {\displaystyle a} has norm 1, i.e. ‖ a ‖ = 1 {\displaystyle \left\|a\right\|=1} .^[5]

See also

• Unitary matrix
• Unitary operator

Notes

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 French). 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.