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Normal element

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

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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} .2

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

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.6
  • 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.7

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 *.8

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:

  • 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.9
  • 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\|} .10 This follows from the spectral radius formula by repeated application of the previous property.11
  • 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} .12

See also

Notes

  • Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
  • Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2.
  • Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4.

References

  1. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  2. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  3. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  4. Dixmier 1977, p. 5. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  5. Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  6. Dixmier 1977, p. 5. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  7. Dixmier 1977, p. 4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  8. Dixmier 1977, pp. 3–4. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩

  9. Werner 2018, p. 518. - Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4. ↩

  10. Heuser 1982, p. 390. - Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2. ↩

  11. Werner 2018, pp. 284–285, 518. - Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4. ↩

  12. Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. ↩