dorsal/arxiv
View SchemaCompact Orthoalgebras
| Authors | Alexander Wilce |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0405180 |
| URL | https://arxiv.org/abs/quant-ph/0405180 |
Abstract
We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic, and has a compact center. We prove also that any compact TOA with isolated 0 is of finite height. We then focus on stably ordered TOAs: those in which the upper-set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras -- in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated 0 is determined by that of of its space of atoms.
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"abstract": "We initiate a study of topological orthoalgebras (TOAs), concentrating on the\ncompact case. Examples of TOAs include topological orthomodular lattices, and\nalso the projection lattice of a Hilbert space. As the latter example\nillustrates, a lattice-ordered TOA need not be a topological lattice. However,\nwe show that a compact Boolean TOA is a topological Boolean algebra. Using\nthis, we prove that any compact regular TOA is atomistic, and has a compact\ncenter. We prove also that any compact TOA with isolated 0 is of finite height.\nWe then focus on stably ordered TOAs: those in which the upper-set generated by\nan open set is open. These include both topological orthomodular lattices and\ninterval orthoalgebras -- in particular, projection lattices. We show that the\ntopology of a compact stably-ordered TOA with isolated 0 is determined by that\nof of its space of atoms.",
"arxiv_id": "quant-ph/0405180",
"authors": [
"Alexander Wilce"
],
"categories": [
"quant-ph"
],
"title": "Compact Orthoalgebras",
"url": "https://arxiv.org/abs/quant-ph/0405180"
},
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