In this article, we consider pair frames in Banach spaces and introduce Banach pair frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as special kinds of (Banach) pair frames. Some frame-like inequalities for (Banach) pair frames are presented. The elements that participant in the construction of (Banach) pair frames are characterized. It is shown that a Banach space X has a Banach pair frame with respect to a Banach scalar sequence space `, when it is precisely isomorphic to a complemented subspace of `. It is shown that if we are allowed to choose the scalar sequence space, pair frames and Banach pair frames with respect to the chosen scalar sequence space denote the same concept.

It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.

For semisimple, commutative Banach algebras A and B, some Banach algebra properties of the Cartesin product A×B are characterized in terms of A and B. A couple of results are also proved for non-commutative Banach algebras.

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.

This paper is devoted to an investigation of frames and Riesz bases for general Banach sequence spaces. We establish various relationships between Bessel (respectively, frames) and Riesz sequences (respectively, Riesz bases), and then some of their applications are presented. Some recent results for Banach frames and atomic decompositions are sharpened with simple proofs. Banach spaces consisting of Bessel or Riesz sequences are introduced and it is shown that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators.