{"paper":{"title":"On the commutant of $C(X)$ in $C^*$-crossed products by $\\mathbb{Z}$ and their representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.OA","authors_text":"Christian Svensson, Jun Tomiyama","submitted_at":"2008-07-18T10:18:39Z","abstract_excerpt":"For the $C^*$-crossed product $C^*(\\Sigma)$ associated with an arbitrary topological dynamical system $\\Sigma = (X, \\sigma)$, we provide a detailed analysis of the commutant, in $C^* (\\Sigma)$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\\tilde{\\pi}$ of $C^* (\\Sigma)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^*"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.2940","kind":"arxiv","version":3},"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"}