Let A be a Banach algebra and I closed ideal of A. We say that is amenable relative to I, if A/I an algebra. study the amenability algebras investigate triangular associated locally compact groups. generalize some previous known results by applying concept algebras, especially, we present generalization Johnson?s theorem in amenability.

We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and Żuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.

In this paper, we study approximate amenability of matrix algebras. We show that every derivation from Mn(A) into Mn(E) is the sum of an inner derivation and a derivation induced by a derivation from A into E(m), where A is a Banach algebra and E is a Banach A-bimodule. By using this, we provide many results in approximate and permanent weak amenability of these algebras.

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...

Recently, Zhang and Song [Q. Zhang, Y. Song, Fixed point theory forgeneralized $varphi$-weak contractions,Appl. Math. Lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. In this paper, we prove a common fixed point theorem fora family of compatible maps. In fact, a new generalization of Zhangand Song's theorem is given.