In mathematics, an element of a *-algebra is called normal if it commutates with its adjoint.
Definition Let A {\displaystyle {\mathcal {A}}} be a *-Algebra. An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is called normal if it commutes with a ∗ {\displaystyle a^{*}} , i.e. it satisfies the equation a a ∗ = a ∗ a {\displaystyle aa^{*}=a^{*}a} . The set of normal elements is denoted by A N {\displaystyle {\mathcal {A}}_{N}} or N ( A ) {\displaystyle N({\mathcal {A}})} . A special case of particular importance is the case where A {\displaystyle {\mathcal {A}}} is a complete normed *-algebra, that satisfies the C*-identity ( ‖ a ∗ a ‖ = ‖ a ‖ 2 ∀ a ∈ A {\displaystyle \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 A {\displaystyle {\mathcal {A}}} is a C*-Algebra and a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} a normal element, then for every continuous function f {\displaystyle f} on the spectrum of a {\displaystyle a} the continuous functional calculus defines another normal element f ( a ) {\displaystyle f(a)} .
Criteria Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then:
An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is normal if and only if the *-subalgebra generated by a {\displaystyle a} , meaning the smallest *-algebra containing a {\displaystyle a} , is commutative. Every element a ∈ A {\displaystyle 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 ∈ A s a {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}} , such that a = a 1 + i a 2 {\displaystyle a=a_{1}+\mathrm {i} a_{2}} , where i {\displaystyle \mathrm {i} } denotes the imaginary unit. Exactly then a {\displaystyle a} is normal if a 1 a 2 = a 2 a 1 {\displaystyle a_{1}a_{2}=a_{2}a_{1}} , i.e. real and imaginary part commutate.
Properties
In *-algebras Let a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} be a normal element of a *-algebra A {\displaystyle {\mathcal {A}}} . Then:
The adjoint element a ∗ {\displaystyle a^{*}} is also normal, since a = ( a ∗ ) ∗ {\displaystyle a=(a^{*})^{*}} holds for the involution *.
In C*-algebras Let a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} be a normal element of a C*-algebra A {\displaystyle {\mathcal {A}}} . Then:
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