The Universal Factorization Property for Commutative Subspace Lattices

Abstract

We present a necessary and sufficient condition for a commutative subspace lattice (CSL) to have the property that every sublattice has the universal factorization property (UFP). Then we derive a sufficient condition for a CSL to have the UFP which applies to all known examples of CSL’s with the UFP. (The preceding results are, in fact, obtained for CSL- subalgebras of factor von Neumann algebras.) Finally, we give an example of a CSL which has the UFP, even though it is not a direct sum of a complemented lattice and a CSL with the property that every linearly ordered sublattice is countable.