Normal element

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

In mathematics, an element of a *-algebra is called normal if it commutates with its A special case of particular importance is the case where \mathcal{A} is a complete normed *-algebra, that satisfies the C*-identity ( \left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A} ), which is called a C*-algebra. Every element 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 \in \mathcal{A}_{sa} , such that a = a_1 + \mathrm{i} a_2 , where \mathrm{i} denotes the imaginary unit. Exactly then a is normal if a_1 a_2 = a_2 a_1 , i.e. real and imaginary part The adjoint element a^* is also normal, since a = (a^*)^* holds for the involution It is \left\| a^2 \right\| = \left\| a \right\|^2 , since for normal elements using the C*-identity \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 Every normal element is a normaloid element, i.e. the spectral radius r(a) equals the norm of a , i.e. This follows from the spectral radius formula by repeated application of the previous property. A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of a to

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