Normal element

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

Definition

Let be a *-Algebra. An element is called normal if it commutes with , i.e. it satisfies the equation .[1]

The set of normal elements is denoted by or .

A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity (), which is called a C*-algebra.

Examples

Criteria

Let be a *-algebra. Then:

  • An element is normal if and only if the *-subalgebra generated by , meaning the smallest *-algebra containing , is commutative.[2]
  • Every element can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements , such that , where denotes the imaginary unit. Exactly then is normal if , i.e. real and imaginary part commutate.[1]

Properties

In *-algebras

Let be a normal element of a *-algebra . Then:

  • The adjoint element is also normal, since holds for the involution *.[4]

In C*-algebras

Let be a normal element of a C*-algebra . Then:

  • It is , since for normal elements using the C*-identity holds.[5]
  • Every normal element is a normaloid element, i.e. the spectral radius equals the norm of , i.e. .[6] This follows from the spectral radius formula by repeated application of the previous property.[7]
  • A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of to .[3]

See also

Notes

References

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