Compact orthoalgebras

Wilce, Alexander

doi:10.1090/S0002-9939-05-07884-6

Compact orthoalgebras
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Proc. Amer. Math. Soc. 133 (2005), 2911-2920 Request permission

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 its space of atoms.

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