Minimal generating reflexive lattices of projections in finite von Neumann algebras

Abstract

We show that the reflexive lattice generated by a double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to the two-dimensional sphere S2 (plus two distinct points corresponding to zero and I). Furthermore, such a reflexive lattice is in general minimally generating for the von Neumann algebra it generates. As an application, we show that if a reflexive lattice \({\mathcal F}\) generates the algebra \({M_n(\mathbb C)}\) of all n × n complex matrices, for some n ≥ 3, then \({\mathcal F\setminus\{0,I\}}\) is connected if and only if it is homeomorphic to S2.