The bidual of a Banach algebra associated with a bilinear map

Abstract

We associate a Banach algebra \(\mathbf A _\Phi \) to each bounded bilinear map \(\Phi : \mathrm X\times \mathrm Y\rightarrow \mathrm {Z}\) on Banach spaces, and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras. For example, the construction of \(\mathbf A _\Phi \) enables us to provide many simple examples of Banach algebras with different topological centers, which are neither Arens regular nor either left or right strongly Arens irregular. It also gives examples of Banach algebras which are not n-weakly amenable for each natural number n. We also find out that the dual of \(\mathbf A _\Phi \) does not enjoy the factorization property of any level.

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Correspondence to Sedigheh Barootkoob.

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Barootkoob, S., Khosravi, A. & Ebrahimi Vishki, H.R. The bidual of a Banach algebra associated with a bilinear map. Boll Unione Mat Ital 12, 569–574 (2019). https://doi.org/10.1007/s40574-019-00193-w

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Keywords

  • Bidual
  • Arens product
  • Bilinear map
  • Topological center
  • Weak amenability

Mathematics Subject Classification

  • Primary 46H20
  • Secondary 46H25