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# Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

Published:2006-08-01
Printed: Aug 2006
• Zhiguo Hu
• Matthias Neufang
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## 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 $\lone$, 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 centre
 MSC Classifications: 22D05 - General properties and structure of locally compact groups 43A20 - $L^1$-algebras on groups, semigroups, etc. 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 03E55 - Large cardinals 46L10 - General theory of von Neumann algebras