Normal conditional expectations of finite index and sets of modular generators

Abstract

Normal conditional expectations E: M --> N in M of finite index on von Neumann algebras M with discrete center are investigated to find an estimate for the minimal number of generators of M as a Hilbert N-module. Analyzing the case of M being finite type I with discrete center we obtain that these von Neumann algebras M are always finitely generated Hilbert N-modules with a minimal generator set consisting of at most [K(E)]^2 generators, where [.] denotes the integer part of a real number and K(E) = {K: K.E-id_M >= 0}. This result contrasts remarkable examples by P. Jolissaint and S. Popa showing the existence of normal conditional expectations of finite index on certain type II_1 von Neumann algebras with center l_\infty which are not algebraically of finite index in the sense of Y. Watatani. We show that estimates of the minimal number of module generators by a function of [K(E)] cannot exist for certain type II_1 von Neumann algebras with non-trivial center.