{"paper":{"title":"A new approach to convolution and semi-direct products of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Arash Ghaani Farashahi, Rajabali Kamyabi-Gol","submitted_at":"2012-01-09T17:45:33Z","abstract_excerpt":"Let $H$ and $K$ be locally compact groups and $\\tau:H\\to Aut(K)$ be a continuous homomorphism and also let $G_\\tau=H\\ltimes_\\tau K$ be the semi-direct product of $H$ and $K$ with respect to $\\tau$. We define left and also right $\\tau$-convolution on $L^1(G_\\tau)$ such that $L^1(G_\\tau)$ with respect to each of them is a Banach algebra. Also we define $\\tau$-convolution as a linear combination of the left and right $\\tau$-convolution. We show that the $\\tau$-convolution is commutative if and only if $K$ is abelian and also when $H$ and $K$ are second countable groups, the $\\tau$-convolution coi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1854","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"}