Normal Element

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

Definition

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

The set of normal elements is denoted by $\mathcal_N$ or

A special case of particular importance is the case where $\mathcal$ is a complete normed *-algebra, that satisfies the C*-identity ($\left\, a^*a \right\, = \left\, a \right\, ^2 \ \forall a \in \mathcal$), 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 $\mathcal$ is a C*-Algebra and $a \in \mathcal_N$ a normal element, then for every continuous function $f$ on the spectrum of $a$ the continuous functional calculus defines another normal element

Criteria

Let $\mathcal$ be a *-algebra. Then:

* An element $a \in \mathcal$ is normal if and only if the *- subalgebra generated by $a$, meaning the smallest *-algebra containing $a$, is
* Every element $a \in \mathcal$ can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements $a_1,a_2 \in \mathcal_$, such that $a = a_1 + \mathrm a_2$, where $\mathrm$ 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 $a \in \mathcal_N$ be a normal element of a *-algebra Then:

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

In C*-algebras

Let $a \in \mathcal_N$ 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 $\left\, a^2 \right\, ^2 = \left\, (a^2) (a^2)^* \right\, = \left\, (a^*a)^* (a^*a) \right\, = \left\, a^*a \right\, ^2 = \left( \left\, a \right\, ^2 \right)^2$
* Every normal element is a normaloid element, i.e. the spectral radius $r(a)$ 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

Notes

References

* English translation of
*
*

{{SpectralTheory

Abstract algebra
C*-algebras