Topological Orthoalgebras
نویسنده
چکیده
We define topological orthoalgebras (TOAs), and study their properties. While every topological orthomodular lattice is a TOA, the lattice of projections of a Hilbert space (in its norm or strong operator topology) is an example of a lattice-ordered TOA that is not a topological lattice. On the other hand, we show that every compact Boolean TOA is a topological Boolean algebra. We also show that a compact TOA in which 0 is an isolated point is atomic and of finite height. We identify and study a particularly tractable class of TOAs, which we call stably ordered: those in which the upper-set generated by an open set is open. This includes all topological OMLs, and also the projection lattices of Hilbert spaces. Finally, we obtain a version of the Foulis-Randall representation theory for stably-ordered topological orthoalgebras.
منابع مشابه
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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 th...
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