Parallelogram law and comparability axioms in orthomodular lattices
by
B. N. Waphare
Department of Mathematics, University of Pune, Pune-411 007(India)

Berberian in his book Baer *-rings [1972] provides monographic and elegant treatment of Baer *-rings in which the natural merger of the parts of von Neumann algebra that indicate functional analytic stream and lattice theoretic stream is conspicuous. Berberian's volumnious and detailed exposition mentions several open problems for Baer *-rings. We concetrate on the following two open problems.
Open problem 1: If A is a Baer *-ring with Partial Comparability (PC), does it follow that A has Generalized Comparability (GC)?
Open problem 2: If A is a Baer *-ring with (GC) and if e, f are finite projections in A, is e \/ f finite? In other words, is it true that in a Baer *-ring with (GC) the finite projections form a sublattice of the lattice of all projections of A?

We have studied the comparability axioms in orthomodular lattices with suitable equivalence relation, and obtained equivalence between (GC) and (PC) in a general *-ring imposing some restrictions on the lattice of projections. In the similar way using parallelogram law we have obtained the sublatticeness of finite projections in a *-ring.

Date received: March 12, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caiv-07.