Normal conditional expectations of finite index and sets of modular generators

1998/09/26 by Michael Frank, Frank, Michael
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.math/9809152

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

openalex publication_date 1998/09/26 · arxiv created 2000/02/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Related

• Intersections of finite families of finite index subfactors
• Conditional expectations onto maximal abelian *-subalgebras
• Orbits of conditional expectations
• Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras
• Maximality and finiteness of type 1 subdiagonal algebras