Element of algebra where x* equals x
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. ).
Definition
Let be a *-algebra. An element is called self-adjoint if .
The set of self-adjoint elements is referred to as .
A subset that is closed under the involution *, i.e. , is called self-adjoint.
A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity (), which is called a C*-algebra.
Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian. Because of that the notations , or for the set of self-adjoint elements are also sometimes used, even in the more recent literature.
Examples
- Each positive element of a C*-algebra is self-adjoint.
- For each element of a *-algebra, the elements and are self-adjoint, since * is an involutive antiautomorphism.
- For each element of a *-algebra, the real and imaginary parts and are self-adjoint, where denotes the imaginary unit.
- If is a normal element of a C*-algebra , then for every real-valued function , which is continuous on the spectrum of , the continuous functional calculus defines a self-adjoint element .
Criteria
Let be a *-algebra. Then:
- Let , then is self-adjoint, since . A similarly calculation yields that is also self-adjoint.
- Let be the product of two self-adjoint elements . Then is self-adjoint if and commutate, since always holds.
- If is a C*-algebra, then a normal element is self-adjoint if and only if its spectrum is real, i.e. .
Properties
In *-algebras
Let be a *-algebra. Then:
- Each element can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements , so that holds. Where and .
- The set of self-adjoint elements is a real linear subspace of . From the previous property, it follows that is the direct sum of two real linear subspaces, i.e. .
- If is self-adjoint, then is normal.
- The *-algebra is called a hermitian *-algebra if every self-adjoint element has a real spectrum .
In C*-algebras
Let be a C*-algebra and . Then:
- For the spectrum or holds, since is real and holds for the spectral radius, because is normal.
- According to the continuous functional calculus, there exist uniquely determined positive elements , such that with . For the norm, holds. The elements and are also referred to as the positive and negative parts. In addition, holds for the absolute value defined for every element .
- For every and odd , there exists a uniquely determined that satisfies , i.e. a unique -th root, as can be shown with the continuous functional calculus.
See also
Notes
References
- Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 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.
- Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0.