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 commutes with $a^{\ast }$, i.e. it satisfies the equation $a{a}^{\ast }=a^{\ast }a$.

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

A special case of particular importance is the case where $𝒜$ is a complete normed *-algebra, that satisfies the C*-identity ($\Vert a^{\ast }a\Vert ={\Vert a\Vert }^2\mathrm{\forall }a\in 𝒜$), 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 $a^{\ast }$ is also normal, since $a=(a^{\ast }{)}^{\ast }$ 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 ${\Vert a^2\Vert }^2=\Vert (a^2)(a^2{)}^{\ast }\Vert =\Vert (a^{\ast }a{)}^{\ast }(a^{\ast }a)\Vert ={\Vert a^{\ast }a\Vert }^2={\left({\Vert a\Vert }^2\right)}^2$ 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. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in français). Gauthier-Villars. 1969.
• Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2.
• Werner, Dirk (2018). Funktionalanalysis (in Deutsch) (8 ed.). Springer. ISBN 978-3-662-55407-4.