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 be a *-algebra with unit . An element is called unitary if . In other words, if is invertible and holds, then is unitary.[1]
The set of unitary elements is denoted by or .
A special case from particular importance is the case where is a complete normed *-algebra. This algebra satisfies the C*-identity () and is called a C*-algebra.
Criteria
- Let be a unital C*-algebra and a normal element. Then, is unitary if the spectrum consists only of elements of the circle group , i.e. .[2]
Examples
- The unit is unitary.[3]
Let be a unital C*-algebra, then:
- Every projection, i.e. every element with , is unitary. For the spectrum of a projection consists of at most and , as follows from the continuous functional calculus.[4]
- If is a normal element of a C*-algebra , then for every continuous function on the spectrum the continuous functional calculus defines an unitary element , if .[2]
Properties
Let be a unital *-algebra and . Then:
- The element is unitary, since . In particular, forms a multiplicative group.[1]
- The element is normal.[3]
- The adjoint element is also unitary, since holds for the involution *.[1]
- If is a C*-algebra, has norm 1, i.e. .[5]
See also
- Unitary matrix
- Unitary operator
Notes
- ^ a b c d Dixmier 1977, p. 5.
- ^ a b Kadison 1983, p. 271. sfn error: no target: CITEREFKadison1983 (help)
- ^ 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.
- v
- t
- e
Spectral theory and *-algebras
- Involution/*-algebra
- Banach algebra
- B*-algebra
- C*-algebra
- Noncommutative topology
- Projection-valued measure
- Spectrum
- Spectrum of a C*-algebra
- Spectral radius
- Operator space
- Isospectral
- Normal operator
- Hermitian/Self-adjoint operator
- Unitary operator
- Unit
- Decomposition of a spectrum
- Continuous
- Point
- Residual
- Approximate point
- Compression
- Direct integral
- Discrete
- Spectral abscissa
- 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