characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Question

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that $A=2P-I$. Is this true?

Answer 1

Let ${\mathcal A}=B(H)$, where $H$ is a Hilbert space. Let $0\ne Q \in B(H)$ be a quasinilpotent operator, i.e., $\sigma(Q)=\{ 0\}$, and let $A=I+Q$. Then $\sigma(A)=\{ 1\}\subseteq \{ -1,1\}$. If there were a projection $P\in B(H)$ such that $A=2P-I$, then we would have $Q=2P-2I$, where $I$ is the identity operator, and consequently $$ Q^2=4P^2-8P+4I=-2(2P-2I). $$ It would follow that $$ Q^2+2Q=0$$ which is impossible for a non-zero quasinilpotent operator $Q$ because $Q+2I$ is invertible.