Lattices of Logmodular Algebras

Abstract

A subalgebra $\mathcal{A}$ of a $C^{\ast }$-algebra $\mathcal{M}$ is logmodular (resp. has factorization) if the set $\{a^{\ast }a;\text{a∈M is invertible with a,a−1∈A}\}$ 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.