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Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an element of the group von Neumann algebra of $G$. Is it possible to have $\beta\in\mathbb CG$?

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    $\begingroup$ You're asking whether $1+\alpha$ is invertible in the group algebra. Conjecturally, as you know, this holds only if $1+\alpha$ is a multiple of a group element, so, in your setting, iff $\alpha$ is a real scalar in $]-1,1[$. $\endgroup$
    – YCor
    Oct 21, 2018 at 10:41
  • $\begingroup$ @YCor For an element $g$ in the support of $\alpha$, $g^n$ belongs to the support of $\alpha^n$, so hopefully, it seems, it is not possible to have $\beta\in\mathbb CG$. $\endgroup$ Oct 21, 2018 at 10:50

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