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 S 2 (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 S 2.
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Hou, C., Yuan, W. Minimal generating reflexive lattices of projections in finite von Neumann algebras. Math. Ann. 353, 499–517 (2012). https://doi.org/10.1007/s00208-011-0695-7
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DOI: https://doi.org/10.1007/s00208-011-0695-7