Decomposability of von Neumann algebras and the Mazur property of higher level
Canadian Journal of Mathematics , Volume 58 - Issue 4 p. 768- 795
The decomposability number of a von Neumann algebra M (denoted by dec(M)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in M. In this paper, we explore the close connection between dec(M) and the cardinal level of the Mazur property for the predual M* of M, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L 1 (G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say M, the cardinal number dec(M) and a certain cardinal level of the Mazur property of M& are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: The compact covering number κ(G) of G and the least cardinality χ(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.