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

Definition

, i.e. it satisfies the equation

The set or

is a complete normed *-algebra), which is called a C*-algebra.

Examples

• Every self-adjoint element of a a *-algebra is

• Every unitary element of a a *-algebra is

• a normal element, then for every continuous function on the spectrum the continuous functional calculus defines another normal element

Criteria

be a *-algebra. Then:

• is normal if and only if the *-subalgebra, is

• can be uniquely decomposed into a real and imaginary part denotes the imaginary unit, i.e. real and imaginary part

Properties

In *-algebras

be a normal element of a *-algebra Then:

• holds for the involution

In C*-algebras

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

• Every normal element is a normaloid element, i.e. the spectral radius, i.e. This follows from the spectral radius formula by repeated application of the previous property.

• to

See also

• Normal matrix

• Normal operator

Notes

References

• English translation of