If $A$ is a non-unital separable $C^*$ algebra,does there exist a strictly positive idempotent element in $A$ ?
2026-03-26 10:07:09.1774519629
strictly positive element
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A strictly positive element $e$ in $A$ satisfies $\overline{ eAe}=A$. If $e$ is an idempotent, then $e$ would be a unit for $A$, which you are assuming non-unital.