Let S be a semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second dual of ` (S) is the Banach algebraM(βS...

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 ...

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...

In this paper we first introduce a new multiplication on triangular Banach algebras. Then we study their Arens regularity as well as their topological centers . In this paper we first introduce a new multiplication on triangular Banach algebras. Then we study their Arens regularity as well as their topological centers . In this paper we first introduce a new multiplication on triangular Banach ...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p then it also extends to the simple Cole extension with respect to p. We...

Let A be a Banach algebra. Then A the second dual of A is a Banach algebra with first (second) Arens product. We study the Arens products of A(= (A∗∗)∗∗). We fined some conditions on A to be a left ideal in A. We fined the biggest two sided ideal I of A, in which I is a left (right) ideal of A∗∗.

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on  a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of  level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p, then it also extends to the simple Cole extension with respect to p. W...

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...