Normal element explained

In mathematics, an element of a

• -algebra

is called normal if it commutates with its

Definition

Let

The set of normal elements is denoted by

l{A} _{N} or

A special case of particular importance is the case where

l{A} is a complete normed *-algebra, that satisfies the C*-identity (\left\| a ^{*a} \right\| = \left\| a \right\| ^{2} \forall a \in l{A} ), 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

l{A} is a C*-Algebra and a \in l{A} _{N} a normal element, then for every continuous function f on the spectrum of a the continuous functional calculus defines another normal element

Criteria

Let

l{A} be a *-algebra. Then:

• An element

a \in l{A} is normal if and only if the *-subalgebra generated by a , meaning the smallest *-algebra containing a , is

• Every element

a \in l{A} can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements a _{1,a} _{2} \in l{A} _{sa} , such that a = a _{1} + i a _{2} , where i denotes the imaginary unit. Exactly then a is normal if a _{1} a _{2} = a _{2} a _{1} , i.e. real and imaginary part

Properties

In *-algebras

Let

a \in l{A} _{N} be a normal element of a *-algebra Then:

• The adjoint element

In C*-algebras

Let

a \in l{A} _{N} be a normal element of a C*-algebra Then:

• It is

\left\| a ^{2} \right\| = \left\| a \right\| ^{2} , since for normal elements using the C*-identity \left\| a ^{2} \right\| ^{2} = \left\| (a ^{2)} (a ^{2)} ^{*} \right\| = \left\| (a ^{*a)} ^{*} (a ^{*a)} \right\| = \left\| a ^{*a} \right\| ^{2} = \left( \left\| a \right\| ^{2} \right) ^{2} r(a) equals the norm of a , 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

a to

See also

• Normal matrix
• Normal operator

References

• Book: Dixmier, Jacques . C*-algebras . North-Holland . Amsterdam/New York/Oxford . 1977 . 0-7204-0762-1 . Jellett . Francis . English translation of Book: Dixmier, Jacques . 0 . Les C*-algèbres et leurs représentations . fr . Gauthier-Villars . 1969 .
• Book: Heuser, Harro . Functional analysis . John Wiley & Sons Ltd. . 1982. 0-471-10069-2 . Horvath . John.
• Book: Werner, Dirk . Funktionalanalysis . 8 . de . Springer . 2018 . 978-3-662-55407-4 .