{"paper":{"title":"Geometric essence of \"compact\" operators on Hilbert $C^*$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Evgenij Troitsky","submitted_at":"2018-10-05T17:03:16Z","abstract_excerpt":"We introduce a uniform structure on any Hilbert $C^*$-module $\\mathcal N$ and prove the following theorem: suppose, $F:{\\mathcal M}\\to {\\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\\mathcal A$ and $\\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $\\theta_{x,y}$, $\\theta_{x,y}(z):=x\\langle y,z\\rangle$, $x\\in {\\mathcal N}$, $y,z\\in {\\mathcal M}$ (i.e. $F$ is ${\\mathcal A}$-compact, or \"compact\") if and only if $F$ maps the unit ball of ${\\mathcal M}$ to a totally bounded set with respect to this uniform structure (i."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.02792","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"}