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
- Every self-adjoint element of a a *-algebra is normal.[1]
- Every unitary element of a a *-algebra is normal.[2]
- If is a C*-Algebra and a normal element, then for every continuous function on the spectrum of the continuous functional calculus defines another normal element .[3]
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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