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 (respectively, right) multipliers on (A**, \({{\bigcirc }_{{(J,\mathcal{U})}}}\)) are also given.
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Ebadian, A., Jabbari, A. Ultrapowers of Banach Algebras. Vestnik St.Petersb. Univ.Math. 55, 336–346 (2022). https://doi.org/10.1134/S1063454122030074
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DOI: https://doi.org/10.1134/S1063454122030074