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1. CJM 2010 (vol 62 pp. 845)
Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.
Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebra Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 |
2. 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 |
3. CJM 2001 (vol 53 pp. 944)
ReprÃ©sentations irrÃ©ductibles bornÃ©es des groupes de Lie exponentiels Let $G$ be a solvable exponential Lie group. We characterize all the
continuous topologically irreducible bounded representations $(T,
\calU)$ of $G$ on a Banach space $\calU$ by giving a $G$-orbit in
$\frn^*$ ($\frn$ being the nilradical of $\frg$), a topologically
irreducible representation of $L^1(\RR^n, \o)$, for a certain weight
$\o$ and a certain $n \in \NN$, and a topologically simple extension
norm. If $G$ is not symmetric, \ie, if the weight $\o$ is
exponential, we get a new type of representations which are
fundamentally different from the induced representations.
Soit $G$ un groupe de Lie r\'esoluble exponentiel. Nous
caract\'erisons toutes les repr\'esentations $(T, \calU)$ continues
born\'ees topologiquement irr\'eductibles de $G$ dans un espace de
Banach $\calU$ \`a l'aide d'une $G$-orbite dans $\frn^*$ ($\frn$
\'etant le radical nilpotent de $\frg$), d'une repr\'esentation
topologiquement irr\'eductible de $L^1(\RR^n, \o)$, pour un certain
poids $\o$ et un certain $n \in \NN$, d'une norme d'extension
topologiquement simple. Si $G$ n'est pas sym\'etrique, c. \`a d. si
le poids $\o$ est exponentiel, nous obtenons un nouveau type de
repr\'esentations qui sont fondamentalement diff\'erentes des
repr\'esentations induites.
Keywords:groupe de Lie rÃ©soluble exponentiel, reprÃ©sentation bornÃ©e topologiquement irrÃ©ductible, orbite, norme d'extension, sous-espace invariant, idÃ©al premier, idÃ©al primitif Category:43A20 |
4. CJM 1999 (vol 51 pp. 96)
Partial Characters and Signed Quotient Hypergroups If $G$ is a closed subgroup of a commutative hypergroup $K$, then the
coset space $K/G$ carries a quotient hypergroup structure. In this
paper, we study related convolution structures on $K/G$ coming from
deformations of the quotient hypergroup structure by certain functions
on $K$ which we call partial characters with respect to $G$. They are
usually not probability-preserving, but lead to so-called signed
hypergroups on $K/G$. A first example is provided by the Laguerre
convolution on $\left[ 0,\infty \right[$, which is interpreted as a
signed quotient hypergroup convolution derived from the Heisenberg
group. Moreover, signed hypergroups associated with the Gelfand pair
$\bigl( U(n,1), U(n) \bigr)$ are discussed.
Keywords:quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functions Categories:43A62, 33C25, 43A20, 43A90 |