{"paper":{"title":"Amenable groups and bounded $\\Delta$-weak approximate identities","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohammad Fozouni","submitted_at":"2014-04-08T19:50:16Z","abstract_excerpt":"Let $A$ be a Banach algebra with a non-empty character space. We say that a bounded net $\\{e_{\\alpha}\\}$ in $A$ is a bounded $\\Delta$-weak approximate identity for $A$ if, for each $a\\in A$ and compact subset $K$ of $\\Delta(A)$, $||\\widehat{e_{\\alpha}a}-\\widehat{a}||_{K}=\\sup_{\\phi\\in K}|\\phi(e_{\\alpha}a)-\\phi(a)|\\rightarrow 0$. For each $1<p<\\infty$, we prove that the Figa-Talamanca Herz algebra, $A_{p}(G)$ has a bounded $\\Delta$-weak approximate identity if and only if $G$ is an amenable group. Also we give a sufficient condition for amenability of group $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2262","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"}