Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
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Abstract
Abstract 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 ᙭( 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 )**.
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