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1. CJM 2006 (vol 58 pp. 768)

Hu, Zhiguo; Neufang, Matthias
 Decomposability of von Neumann Algebras and the Mazur Property of Higher Level 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 \$\lone\$, the Fourier algebra \$A(G)\$, the measure algebra \$M(G)\$, the algebra \$\luc^*\$, 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 \$\kg\$ of \$G\$ and the least cardinality \$\bg\$ 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 \$\ag^{**}\$. Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centreCategories:22D05, 43A20, 43A30, 03E55, 46L10

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