Normal element

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title: "Normal element" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["abstract-algebra", "c*-algebras"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Normal_element" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

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

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
Every unitary element of a a *-algebra is
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

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
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

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
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

Notes

References

English translation of

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