arens product
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Author(s):  
Mina Ettefagh

Let A be a Banach algebra such that its (2n)−th dual for some(n ≥ 1) with first Arens product is m−weakly amenable for some (m > 2n).We introduce some conditions by which if m is odd [even], then A is weakly [2-weakly] amenable.



2019 ◽  
Vol 71 (03) ◽  
pp. 717-747
Author(s):  
Ross Stokke

AbstractMotivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.



Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1493-1496
Author(s):  
A. Zivari-Kazempour

Let A' and A'' be the dual and bidual spaces of a locally convex algebra A with dual and weak* topology, respectively. In this paper, we show that A has a bounded right (left) approximate identity if and only if A'' has a right (left) unit with respect to the first (second) Arens product.



2014 ◽  
Vol 66 (1) ◽  
pp. 295-321 ◽  
Author(s):  
R. Stokke
Keyword(s):  


2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Abbas Sahleh ◽  
Abbas Zivari-Kazempour

Let be a Banach algebra and its second dual equipped with the first Arens product. We consider three -bimodule structures on the fourth dual of . This paper discusses the situation that makes these structures coincide.



2009 ◽  
Vol 61 (2) ◽  
pp. 382-394
Author(s):  
Tianxuan Miao

Abstract. Let 𝒜 be a Banach algebra with a bounded right approximate identity and let ℬ be a closed ideal of 𝒜. We study the relationship between the right identities of the double duals ℬ** and 𝒜** under the Arens product. We show that every right identity of ℬ** can be extended to a right identity of 𝒜** in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra A(G) of a locally compact group G, an element ϕ ∈ A(G)** is in A(G) if and only if A(G)ϕ ⊆ A(G) and Eϕ = ϕ for all right identities E of A(G)**. We also prove some results about the topological centers of ℬ** and 𝒜**.



2008 ◽  
Vol 50 (3) ◽  
pp. 539-555 ◽  
Author(s):  
MATTHEW DAWS

AbstractThe Arens products are the standard way of extending the product from a Banach algebrato its bidual″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that ifis Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.



2008 ◽  
Vol 45 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Mina Ettefagh

Let A be a Banach algebra and ( A ″, □) be its second dual with first Arens product. The third dual of A can be regarded as dual of A ″ ( A ‴ = ( A ″)′) or as the second dual of A ′ ( A ‴ = ( A ′)″), so there are two ( A ″, □)-bimodule structures on A ‴ that are not always equal. This paper determines the conditions that make these structures equal. As a consequence, there are some relations between weak amenability of A and ( A ″, □).





2002 ◽  
Vol 65 (2) ◽  
pp. 191-197 ◽  
Author(s):  
F. Ghahramani ◽  
J. Laali

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .



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