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This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: ...

Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...