to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

Projective limits of Banach algebras have been studied sporadically by many authors since 1952, when they were first introduced by Arens [2] and Michael [11]. Projective limits of complex C∗-algebras were first mentioned by Arens [2]. They have since been studied under various names by Wenjen [20], Sya Do-Shin [18], Brooks [4], Inoue [10], Schmüdgen [17], Fritzsche [6–7], Fragoulopoulou [5], Ph...

We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception...

We investigate the Arens products on the biduals of certain algebras of operators on nonreflexive Banach spaces. To be precise, we study the α-nuclear operators, where α is a tensor norm. This includes the approximable and nuclear operators, and we use these, together with the 2-nuclear operators, as motivating examples. The structure of the two topological centres of the bidual are studied, an...

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 $E$.