A Characterization of the Leinert property

Let $G$ be a discrete group and denote by $\lambda _G$ its left regular representation on $\ell _2(G)$. Denote further by ${\mathbf {F}}_n$ the free group on $n$ generators $\{g_1,g_2,\ldots ,g_n\}$ and $\lambda$ its left regular representation. In this paper we show that a subset $S=\{ t_1, t_2, \ldots , t_n \}$ of $G$ has the Leinert property if and only if for some real positive coefficients $\alpha _1,\alpha _2,\ldots ,\alpha _n$ the identity

holds. Using the same method we obtain some metric estimates about abstract unitaries $U_1,U_2,\ldots , U_n$ satisfying the similar identity $\biggl \|\sum _{i=1}^n U_i \otimes \overline {U_i}\biggr \|_{\min }$
$=2\sqrt {n-1}.$