Normal element

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

Definition

Let ${𝒜}$ be a *-Algebra. An element $a\in {𝒜}$ is called normal if it commutates with $a^{*}$, i.e. it satisfies the equation $a{a}^{*}=a^{*}a$.

The set of normal elements is denoted by ${{𝒜}}_N$ or $N({𝒜})$.

A special case from particular importance is the case where ${𝒜}$ is a complete normed *-algebra, that satisfies the C*-identity ($\Vert a^{*}a\Vert ={\Vert a\Vert }^2\mathrm{\forall }a\in {𝒜}$), which is called a C*-algebra.

Examples

• Every self-adjoint element of a a *-algebra is normal.
• Every unitary element of a a *-algebra is normal.
• If ${𝒜}$ is a C*-Algebra and $a\in {{𝒜}}_N$ a normal element, then for every continuous function $f$ on the spectrum of $a$ the continuous functional calculus defines another normal element $f(a)$.

Criteria

Let ${𝒜}$ be a *-algebra. Then:

• An element $a\in {𝒜}$ is normal if and only if the *-subalgebra generated by $a$, meaning the smallest *-algebra containing $a$, is commutative.
• Every element $a\in {𝒜}$ can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements $a_1,a_2\in {{𝒜}}_{sa}$, such that $a=a_1+\mathrm{i}{a}_2$, where $\mathrm{i}$ denotes the imaginary unit. Exactly then $a$ is normal if $a_1{a}_2=a_2{a}_1$, i.e. real and imaginary part commutate.

Properties

In *-algebras

Let $a\in {{𝒜}}_N$ be a normal element of a *-algebra ${𝒜}$. Then:

• The adjoint element $a^{*}$ is also normal, since $a=(a^{*}{)}^{*}$ holds for the involution *.

In C*-algebras

Let $a\in {{𝒜}}_N$ be a normal element of a C*-algebra ${𝒜}$. Then:

• It is $\Vert a^2\Vert ={\Vert a\Vert }^2$, since for normal elements using the C*-identity ${\Vert a^2\Vert }^2=\Vert (a^2)(a^2{)}^{*}\Vert =\Vert (a^{*}a{)}^{*}(a^{*}a)\Vert ={\Vert a^{*}a\Vert }^2={\left({\Vert a\Vert }^2\right)}^2$ holds.
• Every normal element is a normaloid element, i.e. the spectral radius $r(a)$ equals the norm of $a$, i.e. $r(a)=\Vert a\Vert$. 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 $a$.

See also

• Normal matrix
• Normal operator

References

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