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2007 | 181 | 3 | 237-254
Tytuł artykułu

Arens regularity of module actions

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Abstrakty
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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.
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Twórcy
  • Department of Mathematics, University of Semnan, Semnan, Iran
  • Department of Mathematics, Shahid Beheshti University, Tehran, Iran
autor
  • Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
Bibliografia
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-3-3
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