Normal conditional expectations of finite index and sets of module generators

Normal conditional expectations E : M → N ⊆ 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 projective N -modules with a minimal generator set consisting of at most [K(E)] generators, where [.] denotes the integer part of a real number and K(E) = inf{K : K · E − idM ≥ 0}. This result contrasts with remarkable examples of P. Jolissaint and S. Popa showing the existence of normal conditional expectations of finite index on certain type II1 von Neumann algebras with center l∞ 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 II1 von Neumann algebras with non-trivial center. We provide a more detailed analysis of the type II1 situation. Primary Classification: 46L10. Secondary Classification: 46H25,16D40.