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

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1. CJM Online first

Lee, Hun Hee; Youn, Sang-gyun
New deformations of convolution algebras and Fourier algebras on locally compact groups
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information of the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

Keywords:Fourier algebra, convolution algebra, operator algebra, Beurling algebra
Categories:43A20, 43A30, 47L30, 47L25

2. 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

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

4. 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
Category:43A20

5. 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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