Given a unital C*-algebra $1\in\mathcal{A}$.
Then implication holds: $$J\in\mathcal{A}:\quad JJ^*J=J\implies\sigma(J)\geq0$$
How can I check this?
(Operator-algebraically?)
2026-04-03 18:31:33.1775241093
Partial Isometries: Positivity
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This doesn't seem to be true. Take e.g. the $C^*$ algebra $\mathcal{A} = \mathbb{C}$, the complex numbers. Then the element $-1$ satisfies $$(-1)(-1)^*(-1) = -1$$ but its spectrum (when viewed as an operator) is $\{-1\}$.