We show that the Lipschitz structure of a separable quasi-Banach space does not determine, in general, its linear structure. Using the notion of the Arens-Eells p-space over a metric space for 0 < p ≤ 1 we construct examples of separable quasi-Banach spaces which are Lipschitz isomorphic but not linearly isomorphic.

In this paper, we introduce the new concept of biprojectivity of a Banach algebra modulo an ideal, as a generalization of this notion in the classical case. By using it , we obtain some necessary and sufficient conditions for contractibility of Banach algebras modulo an ideal. In particular we characterize the contractibility of quotient Banach algebras. Also we study the relationship between t...

In this paper we investigate some hereditary properties of amenability modulo an ideal of Banach algebras. We show that if $(e_alpha)_alpha$ is a bounded approximate identity modulo I of a Banach algebra A and X is a neo-unital modulo I, then $(e_alpha)_alpha$ is a bounded approximate identity for X. Moreover we show that amenability modulo an ideal of a Banach algebra A can be only considered ...

We investigate the notion of Connes-amenability, introduced by Runde in [14], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σWC-virtual diagonal, as introduced in [10], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing tha...

In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain ...

in this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a banach algebra $a$ into a semisimple banach algebra $b$ is continuous.

Let G be a locally compact group. Consider the Banach algebra L1(G) ∗∗, equipped with the first Arens multiplication, as well as the algebra LUC(G)∗, the dual of the space of bounded left uniformly continuous functions on G, whose product extends the convolution in the measure algebra M(G). We present (for the most interesting case of a non-compact group) completely different – in particular, d...

Let  and  be Banach algebras, ,  and . We define an -product on  which is a strongly splitting extension of  by . We show that these products form a large class of Banach algebras which contains all module extensions and triangular Banach algebras. Then we consider spectrum, Arens regularity, amenability and weak amenability of these products.

let  and  be banach algebras, ,  and . we define an -product on  which is a strongly splitting extension of  by . we show that these products form a large class of banach algebras which contains all module extensions and triangular banach algebras. then we consider spectrum, arens regularity, amenability and weak amenability of these products.

We prove that every nondegenerate Banach space representation of the Drinfeld–Jimbo algebra Uq(g) a semisimple complex Lie g is finite dimensional when |q|≠1. As corollary, we find an explicit form Arens–Michael envelope Uq(g), which similar to U(g) obtained by Joseph Taylor in 1970s. In case g=sl2, also consider theory corresponding analytic form, U˜(sl2)ℏ (with eℏ=q) and show it simpler than ...