Covering Maps and von Neumann Algebras

Abstract

Distinct classes of multiplication operators have been investigated in Chaps. 4 and 5 on the Bergman space, involving their reducing subspaces. Precisely, these multiplication operators arise from finite and thin Blaschke products. The reducing subspaces of a single multiplication operator M ϕ naturally correspond to those projections, which generate a von Neumann algebra \(\mathcal{V}^{{\ast}}(\phi )\). In the above settings, this von Neumann algebra turns out to be abelian, sometimes even trivial, and hence is of type I. However, it is not always the case if the function ϕ varies.