{"paper":{"title":"Injective convolution operators on ${\\ell}^{\\infty}(\\Gamma)$ are surjective","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Yemon Choi","submitted_at":"2006-06-15T15:29:13Z","abstract_excerpt":"Let $\\Gamma$ be a discrete group and let $f \\in \\ell^1(\\Gamma)$. We observe that if the natural convolution operator $\\rho_f:\\ell^{\\infty}(\\Gamma)\\to \\ell^{\\inf ty}(\\Gamma)$ is injective, then f is invertible in $\\ell^1(\\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra $\\ell^1(\\Gamma)$. We give simple examples to show that in general one cannot replace $\\ell^{\\infty}$ with $\\ell^p$, $1\\leq p< \\infty$, nor with $L^{\\infty}(G)$ for nondiscrete G. Finally, we consider the problem of extending the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606367","kind":"arxiv","version":2},"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"}