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

See also

• Normal matrix
• Normal operator

Notes

References

• English translation of