For a Banach algebra A, we introduce c: c(A), the set of all  2 A such that   : A! A is a completely continuous operator, where   is de ned by   (a) = a
 for all a 2 A. We call A, a completely continuous Banach algebra if c: c(A) = A . We give some examples of completely continuous Banach algebras and a su cient condition for an open problem raised for the  rst time by J. E Gal e, T. J. Ransford and M. C. White: Is there exist an in nite dimensional amenable Banach algebra whose underlying Banach space is re exive? We prove that a re exive, amenable, completely continuous Banach algebra with the approximation property is trivial.

For a Banach algebra A, we introduce c: c(A), the set of all  2 A such that   : A! A is a completely continuous operator, where   is de ned by   (a) = a
 for all a 2 A. We call A, a completely continuous Banach algebra if c: c(A) = A . We give some examples of completely continuous Banach algebras and a su cient condition for an open problem raised for the  rst time by J. E Gal e, T. J. Ransford and M. C. White: Is there exist an in nite dimensional amenable Banach algebra whose underlying Banach space is re exive? We prove that a re exive, amenable, completely continuous Banach algebra with the approximation property is trivial.

Let A be a Banach algebra and X be a Banach A-bimodule. Then S = AÅX, the l1-direct sum of A and X becomes a module extension Banach algebra when equipped with the algebra product (a, x).(a’, x’) = (aa’, ax’+xa’). In this paper, we investigate biflatness and biprojectivity for these Banach algebras. We also discuss on automatic continuity of derivations on S = A Å A.

The article studies the concept of a biprojective and pseudo amenable Banach algebra, where is a continuous homomorphism on and. We show if is contractible, then is biprojective. The converse holds, whenever is either unital or commutative and there exists such that.

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra A and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space Â.

We introduce the notions of $\mathcal{T}_{M}$-amenability and $\phi$-$\mathcal{T}_{M}$-amenability. Then, we characterize $\phi$-$\mathcal{T}_{M}$-amenability  in terms of $WAP$-diagonals and $\phi$-invariant means. Some concrete cases are also discussed.

An element of Banach algebra is EP, if it commutes with its Moore-Penrose inverse. We present a number of new characterizations of EP elements in Banach algebra.