Orthosummable orthoalgebras

Abstract

We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomod-ular 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.

Convergence theorems for topological group valued measures on effect algebras

“…In the preceding section we have obtained a number of convergence theorems for measures defined on effect algebras. If the effect algebra L is an orthoalgebra (see Example 2.4), then every orthogonal sequence is contained in a Boolean subalgebra of L (see [23]). As a consequence, in order to proof a convergence theorem for measures defined on orthoalgebras, or orthomodular posets, or lattices, it is possible to use the classical results for measures on Boolean algebras (see [3], [4] or [24]).…”

“…An effect algebra L is an orthoalgebra if and only if a € L and a ^ 1 © a imply a = 0 (see [13] or [16]). Clearly, each complete orthoalgebra (see [39]) (that is, orthosummable orthoalgebra by Habil [23]) is a complete effect algebra, and all a -orthoalgebra (see Feldman and Wilce [15] or Habil [23]) is a <r-complete effect algebra. EXAMPLE 2.5.…”