Lattices of Logmodular Algebras
Article Type
Research Article
Publication Title
Publications of the Research Institute for Mathematical Sciences
Abstract
A subalgebra A of a C∗-algebra M is logmodular (resp. has factorization) if the set {a∗ a; a ∈ M is invertible with a, a−1 ∈ A} 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.
First Page
507
Last Page
537
DOI
10.4171/prims/60-3-3
Publication Date
1-1-2024
Recommended Citation
Bhat, B. V.Rajarama and Kumar, Manish, "Lattices of Logmodular Algebras" (2024). Journal Articles. 4876.
https://digitalcommons.isical.ac.in/journal-articles/4876
Comments
Open Access; Green Open Access