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1. CJM 2012 (vol 64 pp. 1378)

Raghavan, Dilip; Steprāns, Juris
 On Weakly Tight Families Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by HruÅ¡Ã¡k and GarcÃ­a Ferreira, who applied it to the KatÃ©tov order on almost disjoint families. Keywords:maximal almost disjoint family, cardinal invariantsCategories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

2. CJM 2011 (vol 64 pp. 455)

Sherman, David
 On Cardinal Invariants and Generators for von Neumann Algebras We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, Is every countably-generated von Neumann algebra singly-generated?" and Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$. Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation densityCategory:46L10

3. 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