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 ${\mathcal {A}}$ 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.^[1]

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. $\sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid |\lambda |=1\}$ .^[2]

Examples

The unit $e$ is unitary.^[3]

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.^[4]
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 $f(\sigma (a))\subseteq \mathbb {T}$ .^[2]

Properties

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

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

Notes

^ ^a ^b ^c ^d Dixmier 1977, p. 5.
^ ^a ^b Kadison & Ringrose 1983, p. 271.
^ ^a ^b Dixmier 1977, pp. 4–5.
^ Blackadar 2006, pp. 57, 63.
^ Dixmier 1977, p. 9.

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.

[FOOTNOTEDixmier19775-1] Dixmier 1977, p. 5.

[FOOTNOTEKadisonRingrose1983271-2] Kadison & Ringrose 1983, p. 271.

[FOOTNOTEDixmier19774–5-3] Dixmier 1977, pp. 4–5.

[FOOTNOTEBlackadar200657,_63-4] Blackadar 2006, pp. 57, 63.

[FOOTNOTEDixmier19779-5] Dixmier 1977, p. 9.

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v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem