2021/01/04 by Bhat, B. V. Rajarama, Kumar, Manish · 1 citation
#46K50 #47L30 #47L35 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2101.00782
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\inA\ is dense in (resp. equal to) the set of all positive and invertible elements of M. There are large classes of well studied algebras, both in commutative and non-commutative settings, which are known to be logmodular. In this paper, we show that the lattice of projections in a 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 a question of Paulsen and Raghupathi [Trans. Amer. Math. Soc., 363 (2011) 2627-2640]. We also discuss some sufficient criteria under which an algebra having factorization is automatically reflexive and is a nest algebra.