Lattices of Logmodular Algebras

Abstract

A subalgebra \mathcal{A} of a C^{*} -algebra \mathcal{M} is logmodular (resp. has factorization) if the set \{a^{*}a;\ \text{$a\in\mathcal{M}$ is invertible with $a,a^{-1}\in\mathcal{A}$}\} is dense in (resp. equal to) the set of all positive and invertible elements of \mathcal{M} . In this paper, we show that the lattice of projections in a (separable) von Neumann algebra \mathcal{M} whose ranges are invariant under a logmodular algebra in \mathcal{M} , is a commutative subspace lattice. Further, if \mathcal{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.