Let $A$ be a $C^*$-algebra and $E$ be a left Hilbert $A$-module. In this paper we define a product on $E$ that making it into a Banach algebra and show that under the certain conditions $E$  is Arens regular. We also study the relationship between derivations of $A$ and $E$.

in this paper, we first define spaces of single difference sequences defined by a sequence of orlicz functions without convexity and investigate their properties. then we extend this idea to spaces of double sequences and present a new matrix theoretic approach construction of such double sequence spaces.

in this paper, we associate canonically a precyclic mod- ule to a regular multiplier hopf algebra endowed with a group-like projection and a modular pair in involution satisfying certain con- dition

let a be a banach algebra and x be a banach a-bimodule. in this paper, we dene a new product on a  x and generalize the module extension banach algebras. we obtain characterizations of arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new banach algebra.

For given triangular real Lie algebra $\mathfrak{g}$ we construct a certain completion $C^\infty(\mathfrak{g})$ of its universal enveloping algebra. It is Frechet-Arens-Michael algebra, which consists elements polynomial growth and satisfies to the following property: every homomorphism from Banach all whose are has an extension continuous $C^\infty(\mathfrak{g})$. Elements this can be called f...

For a simple Lie algebra g we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of g-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example g = sl2(C) the regular solution can be expressed via products of Airy functions and their derivatives; this matrix valued function...

A survey is given of the work on strong regularity for uniform algebras over the last thirty years, and some new results are proved, including the following. Let A be a uniform algebra on a compact space X and let E be the set of all those points x ∈ X such that A is not strongly regular at x. If E has no non-empty, perfect subsets then A is normal, and X is the character space of A. If X is ei...