Arens regularity of module actions

• Autorzy: M. Eshaghi Gordji , M. Filali
• Języki publikacji: EN

Abstrakty

EN

We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if 𝓐 has a brai (blai), then the right (left) module action of 𝓐 on 𝓐 * is Arens regular if and only if 𝓐 is reflexive. We find that Arens regularity is implied by the factorization of 𝓐 * or 𝓐 ** when 𝓐 is a left or a right ideal in 𝓐 **. The Arens regularity and strong irregularity of 𝓐 are related to those of the module actions of 𝓐 on the nth dual $𝓐^{(n)}$ of 𝓐. Banach algebras 𝓐 for which Z(𝓐 **) = 𝓐 but $𝓐 ⊊ Z^{t}(𝓐 **)$ are found (here Z(𝓐 **) and $Z^{t}(𝓐 **)$ are the topological centres of 𝓐 ** with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that 𝓐 ⊊ Z(𝓐 **) ⊊ 𝓐 **. Finally, the triangular Banach algebras 𝓣 are used to find Banach algebras having the following properties: (i) 𝓣*𝓣 = 𝓣 𝓣* but $Z(𝓣**) ≠ Z^{t}(𝓣**)$; (ii) $Z(𝓣**) = Z^{t}(𝓣**)$ and 𝓣*𝓣 = 𝓣* but 𝓣 𝓣* ≠ 𝓣*; (iii) Z(𝓣**) = 𝓣 but 𝓣 is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and Ülger.

Kategorie tematyczne

• 46H25: Normed modules and Banach modules, topological modules (if not placed in \SbjNo 13-XX or \SbjNo 16-XX)
• 46H20: Structure, classification of topological algebras
• 43A20: L 1 -algebras on groups, semigroups, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 181
• Numer: 3
• Strony: 237-254

Daty

wydano

2007

Twórcy

autor

M. Eshaghi Gordji

• Department of Mathematics, University of Semnan, Semnan, Iran
• Department of Mathematics, Shahid Beheshti University, Tehran, Iran

autor

M. Filali

• Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland

Identyfikatory

DOI

10.4064/sm181-3-3