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Search: MSC category 43A20 ( $L^1$-algebras on groups, semigroups, etc. )

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1. CJM 2010 (vol 62 pp. 845)

Samei, Ebrahim; Spronk, Nico; Stokke, Ross
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)

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 centre
Categories:22D05, 43A20, 43A30, 03E55, 46L10

3. CJM 2001 (vol 53 pp. 944)

Ludwig, J.; Molitor-Braun, C.
Représentations irréductibles bornées 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 résoluble exponentiel, représentation bornée topologiquement irréductible, orbite, norme d'extension, sous-espace invariant, idéal premier, idéal primitif

4. CJM 1999 (vol 51 pp. 96)

Rösler, Margit; Voit, Michael
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

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