Abstract:
In this paper, we consider ultrapowers of Banach algebras as Banach algebras and the product $\bigcirc_{(J,\mathcal{U })}$ 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 $(A^{**},\bigcirc_{(J,\mathcal{U})})$ 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., $(X)_\mathcal{U}$ . Ultra (character) amenability of Banach algebras is investigated and it will be shown that if every continuous derivation from $A$ into $(X)_\mathcal{U}$ is inner, then $A$ is ultra amenable. Some results related to left (resp. right) multipliers on $(A^{**}, \bigcirc_{(J,\mathcal{U})})$ are also given.
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