Question

what is spectrum of an element

Answer 1

The set of numbers  for which  is non-invertible (the algebra is assumed to be complex,  is a given element of it and is the identity of the algebra). The spectrum is a non-empty compact set (the Gel'fand–Mazur theorem). In the case of a commutative algebra, the spectrum coincides with the set of values on this element of all the characters of the algebra (cf. Character of a algebra)

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