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This thread has been refreshed!
Problem
Given a C*-algebra $\mathcal{A}$.
Consider an element $A\in\mathcal{A}$.
Introduce an abstract polynomial calculus: $$A=A^*:\quad p(A):=\sum_ka_kA^k\quad(p\in\mathbb{C}[X])$$ This induces the concret polynomial calculus: $$A=A^*:\quad p(A):=\sum_ka_kA^k\quad(p\in\mathcal{P}[\sigma(A)])$$ Why is the concret polynomial calculus well defined: $$p(x)\equiv q(x)\quad x\in\sigma(A)\implies p(A)=q(A)$$
(Besides, what can happen for nonselfadjoint ones?)