A type of GNS-construction for Banach algebras

We show that every Banach algebra A admits a representation on a certain Banach space E. In particular, any Banach algebra A contained in autoperiodic functionals on A such that separate the points of A could be imbedded in B(E) for some reflexive Banach space E

Solubility of groups can be characterized by configuration

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. We show that group properties of being soluble or FC can be characterized by configuration sets. Then we investigate some condition on configuration pairs, which leads to isomorphism. We introduce a somewhat different notion of configuration equivalence, namely strong configuration equivalence, and prove that strong configuration equivalence coincides with isomorphism.Comment: 19 page

On the Arens regularity of the Herz Algebra

Let $G$ be a locally compact group, $A_p (G)$ be the Herz algebra of $G$ associated with $1 <p< \infty$. We show that $A_p (G)$ is Arens regular if and only if $G$ is a discrete group and for each countable subgroup $H$ of $G$, $A_p (H)$ is Arens regular. In the case $G$ is a countable discrete group we investigate the relations between Arens regularity of $A_p (G)$ and the iterated limit condition. We consider the problem of Arens regularity of $l^1 (G)$ as a subspace of $A_p (G)$. A few related results when the unit ball of $(l^1 (G),.,A_p(G))$ is bounded under $\|.\|_1$-norm are also determined

Configuration Equivalence is not Equivalent to Isomorphism

Giving a condition for the the amenability of groups, Rosenblatt and Willis, first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this open question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will study some types of subgroups in configuration equivalent groups. In particular, we will prove this conjecture, due to Rosenblatt and Willis, that configuration equivalent groups, both include the free non-Abelian group of same rank or not. Finally, we prove that two-sided equivalent groups have same class numbers

Configuration of nilpotent groups and isomorphism

The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.Comment: to appear in Journal of Algebra and its Application

Amenability of groups and semigroups characterized by Configuration

In 2005, Abdollahi and Rejali, studied the relations between paradoxical decompositions and configurations for semigroups. In the present paper, we introduce another concept of amenability on semigroups and groups which includes amenability of semigroups and inner-amenability of groups. We have the previous known results to semigroups and groups satisfying this concept.Comment: 12 pages, 0 figure

Weakly almost periodic Banach algebras on semi-groups

Let WAP(A) be the space of all weakly almost periodic functionals on a Banach algebra A. The Banach algebra A for which the natural embedding of A into WAP(A)* is bounded below is called a WAP-algebra. We show that the second dual of a Banach algebra A is a WAP-algebra, under each Arens products, if and only if A** is a dual Banach algebra. This is equivalent to the Arens regularity of A. For a locally compact foundation semigroup S, we show that the absolutely continuous semigroup measure algebra M_a(S) is a WAP-algebra if and only if the measure algebra M_b(S) is so

First cohomology on weighted semigroup algebras

The aim of this work is to generalize Johnson's techniques in order to apply them to establish a bijective correspondence between $S$-derivations and continuous derivations on $M_a(S,\omega),$ where $S$ is a locally compact foundation semigroup with identity $e$, and $\omega$ is a weight function on $S$, and apply it to find a necessary condition for amenability of weighted group algebras.Comment: 12 page

Topological Invariant Means on dual Space of Multiplier algebra and weakly compact Multiplier on Herz-algebra

In this paper we investigate the compact and weakly compact multipliers of the Herz-algebras $A_p(G)$. Let $B_p(G)$ be the space of pointwise multipliers of $A_p(G)$. We show that there is a topological invariant mean on $B^*_p (G)$. Furthermore, we show that if $B^*_p(G)$ is separable, then $G$ is a discrete group

Group Representations on Reflexive Spaces

For weighted group convoltion measure algebra we construct a representation on reflexsive space.Comment: 13. arXiv admin note: text overlap with arXiv:1501.0642