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 $\beta\in\mathbb CG$ such that coefficients of $\beta^*\alpha\beta$ are all positive real numbers?
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3$\begingroup$ When $G$ is commutative (for example $\mathbf Z$), you have the Fourier transform that translates your question to an elementary question on functions on the circle. You will easily find counterexamples in that case. $\endgroup$– Mikael de la SalleCommented May 28, 2019 at 13:23
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$\begingroup$ @MikaeldelaSalle: Could you please give me one of such counterexamples? $\endgroup$– MSMalekanCommented May 28, 2019 at 13:38
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2$\begingroup$ Seen as a function on the circle, $\alpha(z) = |z-1|^2$. Or any positive trigonometric polynomial vanishing at $z=1$. $\endgroup$– Mikael de la SalleCommented May 28, 2019 at 13:40
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$\begingroup$ Because the sum of the coefficients (=Fourier coefficients here) of $\alpha |\beta|^2$ is the value of $\alpha |\beta|^2$ at $z=1$, so is $0$. $\endgroup$– Mikael de la SalleCommented May 28, 2019 at 13:56
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