AbstractThe 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 L1(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 ᙭(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)**.
On the topological centre problem for weighted convolution algebras and semigroup compactifications
