BANACH ALGEBRAS RELATED TO THE ELEMENTS OF THE UNIT BALL OF A BANACH ALGEBRA

Suppose A is a Banach algebra and c is in A with ||epsilon|| <= 1. In this note we aim to study the algebraic properties of the Banach algebra A(epsilon), where the product on A(epsilon) is given by a circle dot b = a epsilon b, for a, b is an element of A. In particular we study the Arens regularity, amenability and derivations on A(epsilon). Also we prove that if A has an involution then A(epsilon) has the same involution just when epsilon = 1 or -1.