2014/11/17 by Deguang Han, Han, Deguang, David R. Larson +5 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1411.4559
Contemporary Mathematics, 21 pages. arXiv admin note: substantial text overlap with arXiv:1110.5833
arxiv created 2014/11/17 · openalex publication_date 2014/11/17 · arxiv updated 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homomorphism dilation acting on a Banach space. Here the dilation space often needs to be a Banach space even if the underlying space is a Hilbert space, and the projections are idempotents that are not necessarily self-adjoint. These results lead to some new connections between frame theory and operator algebras, and some of them can be considered as part of the investigation about "noncommutative" frame theory.