Normal element

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

Definition

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

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

A special case of particular importance is the case where ${\mathcal {A}}$ is a complete normed *-algebra, that satisfies the C*-identity ($\left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}$), 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 ${\mathcal {A}}$ is a C*-Algebra and $a\in {\mathcal {A}}_{N}$ 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 ${\mathcal {A}}$ be a *-algebra. Then:

• An element $a\in {\mathcal {A}}$ is normal if and only if the *-subalgebra generated by $a$, meaning the smallest *-algebra containing $a$, is commutative.
• Every element $a\in {\mathcal {A}}$ can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements $a_{1},a_{2}\in {\mathcal {A}}_{sa}$, 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 $a\in {\mathcal {A}}_{N}$ be a normal element of a *-algebra ${\mathcal {A}}$. Then:

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

In C*-algebras

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

• It is $\left\|a^{2}\right\|=\left\|a\right\|^{2}$, since for normal elements using the C*-identity $\left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}$ holds.
• Every normal element is a normaloid element, i.e. the spectral radius $r(a)$ equals the norm of $a$, i.e. $r(a)=\left\|a\right\|$. 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 French). 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 German) (8 ed.). Springer. ISBN 978-3-662-55407-4.