Test sets for tautologies in modular quantum logic

Abstract

As defined by Dunn, Moss, and Wang, an universal test set in an ortholattice $L$ is a subset $T$ such that each term takes value $1$, only, if it does so under all substitutions from $T$. Generalizing their result for ortholattices of subspaces of finite dimensional Hilbert spaces, we show that no infinite modular ortholattice of finite dimension admits a finite universal test set. On the other hand, answering a question of the same authors, we provide a countable universal test set for the ortholattice of projections of any type II${}_1$ von Neumann algebra factor as well as for von Neumann's algebraic construction of a continuous geometry. These universal test sets consist of elements having rational normalized dimension with denominator a power of $2$.