Normal element

In mathematics, an element of a *-algebra is called normal if it commutates with its

Definition

Let be a *-Algebra. An element is called normal if it commutes with, i.e. it satisfies the equation
The set of normal elements is denoted by or
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.

Examples

Every self-adjoint element of a a *-algebra is
Every unitary element of a a *-algebra is
If is a C*-Algebra and a normal element, then for every continuous function on the spectrum of the continuous functional calculus defines another normal element

Criteria

Let be a *-algebra. Then:

An element is normal if and only if the *-subalgebra generated by, meaning the smallest *-algebra containing, is
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

Properties

In *-algebras

Let be a normal element of a *-algebra Then:

The adjoint element is also normal, since holds for the involution

In C*-algebras

Let be a normal element of a C*-algebra Then:

It is, since for normal elements using the C*-identity
Every normal element is a normaloid element, i.e. the spectral radius equals the norm of, i.e. This follows from the spectral radius formula by repeated application of the previous property.
A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of to