Lattices of Logmodular Algebras

A subalgebra A of a C* -algebra M is logmodular (resp. has factorization) if the set \{a*a;\ a is invertible with a,a⁻¹\} is dense in (resp. equal to) the set of all positive and invertible elements of M . In this paper, we show that the lattice of projections in a (separable) von Neumann algebra M whose ranges are invariant under a logmodular algebra in M , is a commutative subspace lattice. Further, if M is a factor then this lattice is a nest. As a special case, it follows that all reflexive (in particular, completely distributive CSL) logmodular subalgebras of type I factors are nest algebras, thus answering in the affirmative a question by Paulsen and Raghupathi (Trans. Amer. Math. Soc. 363 (2011) 2627–2640). We also give a complete characterization of logmodular subalgebras in finite-dimensional von Neumann algebras.