Ultrapowers of Banach algebras

Abstract

In this paper, we consider ultrapowers of Banach algebras as Banach algebras and the product (J,U) on the second dual of Banach algebras. For a Banach algebra A, we show that if there is a continuous derivation from A into itself, then there is a continuous derivation from (A ∗∗ , (J,U)) into it. Moreover, we show that if there is a continuous derivation from A into X∗∗, where X is a Banach A-bimodule, then there is a continuous derivation from A into ultrapower of X i. e., (X)U. Ultra (character) amenability of Banach algebras is investigated and it will be shown that if every continuous derivation from A into (X)U is inner, then A is ultra amenable. Some results related to left (resp. right) multipliers on (A ∗∗ , (J,U)) are also given.