let $mathcal{a}$ be a banach algebra with bai and $e$ be an introverted subspace of $mathcal{a'}$.in this paper we study the quotient arens regularity of $mathcal{a}$ with respect to $e$ and prove that the group algebra $l^1(g)$ for a locally compact group $g$, is quotient arens regular with respect to certain introverted subspace $e$ of $l^infty(g)$.some related result are given as well.

Let $mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $mathcal{A}^prime$. In this paper we study the quotient Arens regularity of $mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^infty(G)$. Some related result are given as well.

A Banach algebra is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols A⁎=WAP(A). To identify the opposite behaviour, Granirer called a extremely non-Arens regular (enAr, for short) quotient A⁎/WAP(A) contains closed subspace that has A⁎ as quotient. In this paper we propose simplification and quantification of concept. We say r-enAr, with r≥1, there an iso...

In this paper we define the notion of weak Arens regular Banach algebras and extend the concept of quasi-multipliers to this certain class of Banach algebras. Among other the relationship between Arens regularity of the algebra A∗∗ of a weak Arens regular Banach algebra A and the space QMr(A∗) of all bilinear and separately continuous right quasimultipliers of A∗ is investigated. Further, we st...

‎We present a characterization of Arens regular semigroup algebras‎ ‎$ell^1(S)$‎, ‎for a large class of semigroups‎. ‎Mainly‎, ‎we show that‎ ‎if the set of idempotents of an inverse semigroup $S$ is finite‎, ‎then $ell^1(S)$ is Arens regular if and only if $S$ is finite‎.

Let A be a Banach algebra with the second dual A∗∗. If A has a bounded approximate identity (= BAI), then A∗∗ is unital if and only if A∗∗ has a weak∗ bounded approximate identity(= W ∗BAI). If A is Arens regular and A has a BAI, then A∗ factors on both sides. In this paper we introduce new concepts LW ∗W and RW ∗W property and we show that under certain conditions if A has LW ∗W and RW ∗W prop...

‎we present a characterization of arens regular semigroup algebras‎ ‎$ell^1(s)$‎, ‎for a large class of semigroups‎. ‎mainly‎, ‎we show that‎ ‎if the set of idempotents of an inverse semigroup $s$ is finite‎, ‎then $ell^1(s)$ is arens regular if and only if $s$ is finite‎.

The Arens products are the standard way of extending the product from a Banach algebra A to its bidual A′′. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if A is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known ...

Robin Harte Dedicated to Goldie Hawn, on her birthday Abstract We attempt to deconstruct the Arens-Royden Theorem. Suppose A is a Banach algebra (by default complex, with identity 1): we shall write 0.1 A−1 = {a ∈ A : 1 ∈ Aa∩aA} for the open subgroup of invertible elements, and A−1 0 for the connected component of the identity in A −1: it turns out ([8], [11] Theorem 7.11.4) that 0.2 A−1 0 = Ex...