Continuing the research on the Banach-Saks and Schur properties started in (cf. [10]) we investigate analogous properties in the module context. As an environment serves the class of Hilbert C∗-modules. Some properties of weak module topologies on Hilbert C∗-modules are described. Natural module analogues of the classical weak Banach-Saks and the classical Schur properties are defined and studi...

Completely continuous operators Let A be a Banach algebra with |A| % {1}, where A := {a ∈ A : |ab| = |a| |b| for all b ∈ A} is the set of all multiplicative units in A (equivalently, A = {a ∈ A : |a| |a| = 1}). Then for any Banach modules M,N over A, an A-linear map L : M → N is continuous iff supm6=0 |L(m)| |m| < ∞. Let BA(M,N) be the space of such maps. With the norm |L| := supm6=0 |L(m)| |m|...

For two normed algebras $A$ and $B$ with the character space   $bigtriangleup(B)neq emptyset$  and a left $B-$module $X,$  a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$  as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $bigtriangleup(B)=lbrace varphirbrace$ then $CMH_B(A, X)bigcup lbrace 0rbrace$ is...

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 a recent paper, S.-E. Takahasi defined the notion of a BSE Banach module over a commutative Banach algebra A with bounded approximate identity. We show that the multiplier space &f(X) of X can be represented as a space of sections in a bundle of Banach spaces, and we use bundle techniques to obtain shorter proofs of various of Takahasi’s results on P-algebra modules and to answer several que...

We utilize the notion of module extension to reduce the problem of stability of derivations to that of ring homomorphisms studied by R. Badora in the context of Banach bimodules over Banach algebras. AMS subject classifications: Primary 39B82; Secondary 39B52, 46H25