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.[1] Because of that the notations , or for the set of self-adjoint elements are also sometimes used, even in the more recent literature.
Let , then is self-adjoint, since . A similarly calculation yields that is also self-adjoint.[6]
Let be the product of two self-adjoint elements . Then is self-adjoint if and commutate, since always holds.[1]
If is a C*-algebra, then a normal element is self-adjoint if and only if its spectrum is real, i.e. .[5]
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 .[1]
The set of self-adjoint elements is a reallinear subspace of . From the previous property, it follows that is the direct sum of two real linear subspaces, i.e. .[7]
The *-algebra is called a hermitian *-algebra if every self-adjoint element has a real spectrum .[8]
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.[9]
According to the continuous functional calculus, there exist uniquely determined positive elements , such that with . For the norm, holds.[10] The elements and are also referred to as the positive and negative parts. In addition, holds for the absolute value defined for every element .[11]
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.[12]
Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. ISBN3-540-28486-9.
Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN0-7204-0762-1. 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. ISBN0-12-393301-3.
Palmer, Theodore W. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN0-521-36638-0.