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Weak amenability of the second dual of a Banach algebra

100%

Gordji M., Filali M.

Studia Mathematica

2007

tom 182

nr 3

205-213

It is known that a Banach algebra 𝒜 inherits amenability from its second Banach dual 𝒜**. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L¹(G), the Fourier algebra A(G) when G is amenable, the Banach algebras 𝒜 which are left ideals in 𝒜**, the dual Banach algebras, and the Banach algebras 𝒜 which are Arens regular and have every derivation from 𝒜 into 𝒜* weakly compact. In this paper, we extend this class of algebras to the Banach algebras for which the second adjoint of each derivation D:𝒜 → 𝒜* satisfies D''(𝒜**)⊆ WAP(𝒜), the Banach algebras 𝒜 which are right ideals in 𝒜** and satisfy 𝒜**𝒜 = 𝒜**, and to the Figà-Talamanca-Herz algebra $A_p(G)$ for G amenable. We also provide a short proof of the interesting recent criterion on when the second adjoint of a derivation is again a derivation.

Arens regularity of module actions

100%

Gordji M., Filali M.

Studia Mathematica

2007

tom 181

nr 3

237-254

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.

Ograniczanie wyników

2 Studia Mathematica

2 Filali M.

2 Gordji M.

2 2007

Wyniki wyszukiwania

Weak amenability of the second dual of a Banach algebra

Arens regularity of module actions