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.

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\ 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 $a$.