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In mathematics, an element of a *-algebra is called normal if it commutates with its adjoint.cite-ref-footnotedixmier19774-1-0[1]

• Definition

• Examples

• Criteria

• Properties

• In *-algebras

• In C*-algebras

• See also

• Notes

• References

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} .cite-ref-footnotedixmier19774-1-1[1]

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.

• Every self-adjoint element of a a *-algebra is normal.cite-ref-footnotedixmier19774-1-2[1]

• Every unitary element of a a *-algebra is normal.cite-ref-footnotedixmier19775-2-0[2]

• 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)} .cite-ref-footnotedixmier197713-3-0[3]

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.cite-ref-footnotedixmier19775-2-1[2]

• 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.cite-ref-footnotedixmier19774-1-3[1]

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 *.cite-ref-footnotedixmier19773-4-4-0[4]

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

• It is ‖ a 2 ‖ = ‖ a ‖ 2 {\displaystyle \left\|a^{2}\right\|=\left\|a\right\|^{2}} , since for normal elements using the C*-identity ‖ a 2 ‖ 2 = ‖ ( a 2 ) ( a 2 ) ∗ ∗ ‖ = ‖ ( a ∗ ∗ a ) ∗ ∗ ( a ∗ ∗ a ) ‖ = ‖ a ∗ ∗ a ‖ 2 = ( ‖ a ‖ 2 ) 2 {\displaystyle \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.cite-ref-footnotewerner2018518-5-0[5]

• Every normal element is a normaloid element, i.e. the spectral radius r ( a ) {\displaystyle r(a)} equals the norm of a {\displaystyle a} , i.e. r ( a ) = ‖ a ‖ {\displaystyle r(a)=\left\|a\right\|} .cite-ref-footnoteheuser1982390-6-0[6] This follows from the spectral radius formula by repeated application of the previous property.cite-ref-footnotewerner2018284-285-518-7-0[7]

• A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of a {\displaystyle a} to a {\displaystyle a} .cite-ref-footnotedixmier197713-3-1[3]

• Normal matrix

• Normal operator

cite-note-footnotedixmier19774-11. ↑ Dixmier 1977, p. 4.

cite-note-footnotedixmier19775-22. ↑ Dixmier 1977, p. 5.

cite-note-footnotedixmier197713-33. ↑ Dixmier 1977, p. 13.

cite-note-footnotedixmier19773-4-44. ↑ Dixmier 1977, pp. 3–4.

cite-note-footnotewerner2018518-55. ↑ Werner 2018, p. 518.

cite-note-footnoteheuser1982390-66. ↑ Heuser 1982, p. 390.

cite-note-footnotewerner2018284-285-518-77. ↑ Werner 2018, pp. 284–285, 518.

• citerefdixmier1977Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of citerefdixmier1969Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.

• citerefheuser1982Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2.

• citerefwerner2018Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4.