Abstract view
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level


Published:20060801
Printed: Aug 2006
Zhiguo Hu
Matthias Neufang
Abstract
The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal nonzero
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
Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre

MSC Classifications: 
22D05, 43A20, 43A30, 03E55, 46L10 show english descriptions
General properties and structure of locally compact groups $L^1$algebras on groups, semigroups, etc. Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. Large cardinals General theory of von Neumann algebras
22D05  General properties and structure of locally compact groups 43A20  $L^1$algebras on groups, semigroups, etc. 43A30  Fourier and FourierStieltjes transforms on nonabelian groups and on semigroups, etc. 03E55  Large cardinals 46L10  General theory of von Neumann algebras
