2010/07/19 by Kazem Haghnejad Azar, Azar, Kazem Haghnejad
Mathematics · #Advanced Topology and Set Theory #F.2.2 #FOS: Mathematics #Functional Analysis (math.FA) #I.2.7 #Rings, Modules, and Algebras #math.FA
paper · pdf · doi:10.48550/arxiv.1007.3110
arxiv created 2010/07/19 · openalex publication_date 2010/07/19 · arxiv updated 2010/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we will study some Arens regularity properties of module actions. Let B be a Banach A-bimodule and let Z^ℓB**(A**) and Z^ℓA**(B**) be the topological centers of the left module action π_ℓ:~A× B→ B and the right module action πr:~B× A→ B, respectively. In this paper, we will extend some problems from topological center of second dual of Banach algebra A, Z1(A**), into spaces Z^ℓB**(A**) and Z^ℓA**(B**). We investigate some relationships between Z1(A**) and topological centers of module actions. For an unital Banach A-module B we show that Z^ℓA**(B**)Z1(A**)=Z^ℓA**(B**) and as results in group algebras, for locally compact group G, we have Z^ℓ_L1(G)**(M(G)**)M(G)=Z^ℓ_L1(G)**(M(G)**) and Z^ℓM(G)**(L1(G)**)M(G)=Z^ℓM(G)**(L1(G)**). For Banach A-bimodule B, if we assume that B^*B**⊆ A^*, then ~B**Z1(A**)⊆ Z^ℓA**(B**) and moreover if B is an unital as Banach A-module, then we conclude that B**Z1(A**)=Z^ℓA**(B**). Let Z^ℓA**(B**)A⊆ B and suppose that B is WSC, so we conclude that Z^ℓA**(B**)=B. If B*A≠ B^* and B** has a left unit A**-module, then Z^ℓB**(A**)≠ A**. We will also establish some relationships of Arens regularity of Banach algebras A, B and Arens regularity of projective tensor product A⊗B.