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.
An element is normal if and only if the *-
subalgebra generated by , meaning the smallest *-algebra containing , is commutative.[2]
Every element can be uniquely decomposed into a
real and imaginary part, which means there exist self-adjoint elements , such that , where denotes the
imaginary unit. Exactly then is normal if , i.e. real and imaginary part commutate.[1]
Properties
In *-algebras
Let be a normal element of a *-algebra . Then:
The adjoint element is also normal, since holds for the
involution*.[4]
In C*-algebras
Let be a normal element of a C*-algebra . Then:
It is , since for normal elements using the C*-identity holds.[5]
Every normal element is a normaloid element, i.e. the
spectral radius equals the norm of , i.e. .[6] This follows from the
spectral radius formula by repeated application of the previous property.[7]
A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of to .[3]
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.
Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd.
ISBN0-471-10069-2.
Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer.
ISBN978-3-662-55407-4.