Active lattices determine AW*-algebras |
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Authors: | Chris Heunen Manuel L Reyes |
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Institution: | 1. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK;2. Department of Mathematics, Bowdoin College, 8600 College Station, Brunswick, ME 04011, USA |
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Abstract: | We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure. |
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Keywords: | AW*-algebra Active lattice Piecewise complete Boolean algebra Projection Symmetry Unitary action Category |
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