Normal conditional expectations of finite index and sets of modular generators

Details Normal conditional expectations of finite index and sets of modular generators Normal conditional expectations of finite index and sets of modular generators

1998-09-26

arxiv.org/abs/math/9809152v2

Mathematics

Operator Algebras

7 pages, minor changes and additions, complemented list of literature

Scientific paper

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.

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