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1. CJM 2011 (vol 64 pp. 455)
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 density Category:46L10 |
2. CJM 2011 (vol 63 pp. 551)
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras |
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras In the paper, we introduce a new concept,
topological orbit dimension of an $n$-tuple of elements in a unital
C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's
topological free
entropy dimension of every finite family of self-adjoint generators of a
nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the
Voiculescu's topological free entropy dimension is additive in the full free
product of some unital C$^{\ast}$-algebras. We show that the unital full free
product of Blackadar and Kirchberg's unital MF
algebras is also an MF algebra. As an application, we obtain that
$\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.
Keywords: topological free entropy dimension, unital C$^{*}$-algebra Categories:46L10, 46L54 |
3. CJM 2006 (vol 58 pp. 768)
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 centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 |