{"paper":{"title":"A spectral gap property for random walks under unitary representations","license":"","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DS","authors_text":"Bachir Bekka, Yves Guivarc'h","submitted_at":"2005-08-11T10:05:34Z","abstract_excerpt":"Let $G$ be a locally compact group and $\\mu$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(\\pi, \\cal H)$ of $G,$ we study spectral properties of the operator $\\pi(\\mu)$ acting on $\\cal H.$ Assume that $\\mu$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $\\pi\\otimes \\bar\\pi.$ We show that $\\pi(\\mu)$ has a spectral gap, that is, for the spectral radius $r_{\\rm spec}(\\pi(\\mu))$ of $\\pi(\\mu),$ we have $r_{\\rm spec}(\\pi(\\mu))<1.$ This provides a common "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508195","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}