A spectral gap property for random walks under unitary representations

Let $G$ be a locally compact group and $μ$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(π, \cal H)$ of $G,$ we study spectral properties of the operator $π(μ)$ acting on $\cal H.$ Assume that $μ$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $π\otimes \barπ.$ We show that $π(μ)$ has a spectral gap, that is, for the spectral radius $r_{\rm spec}(π(μ))$ of $π(μ),$ we have $r_{\rm spec}(π(μ))<1.$ This provides a common generalization of several previously known results. Another consequence is that, if $G$ has Kazhdan's Property (T), then $r_{\rm spec}(π(μ))<1$ for every unitary representation $π$ of $G$ without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.

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