scholarly journals Module Actions on Iterated Duals of Banach Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Abbas Sahleh ◽  
Abbas Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Arens Product ◽  
Second Dual

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.

scholarly journals Amenability and topological centres of the second duals of Banach algebras

2002 ◽  
Vol 65 (2) ◽  
pp. 191-197 ◽  
Author(s):  
F. Ghahramani ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dual Algebra ◽  
Arens Product ◽  
Weak Amenability ◽  
Arens Regularity ◽  
Second Dual

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 .

On amenability of the Banach algebras l∞(S, [Ufr ])

2002 ◽  
Vol 132 (2) ◽  
pp. 319-322
Author(s):  
FÉLIX CABELLO SÁNCHEZ ◽  
RICARDO GARCÍA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Short Note ◽  
Basic Information ◽  
Arens Product ◽  
Pointwise Multiplication ◽  
Weak Amenability ◽  
Usual Norm ◽  
Bounded Functions ◽  
Second Dual

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

Isomorphisms and multipliers on second dual algebras of Banach algebras

1992 ◽  
Vol 111 (1) ◽  
pp. 161-168 ◽  
Author(s):  
Fereidoun Ghahramani ◽  
Anthony To-Ming Lau
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Weakly Compact ◽  
Dual Algebra ◽  
Arens Product ◽  
Second Dual

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

Weak Amenability of a Class of Banach Algebras

2001 ◽  
Vol 44 (4) ◽  
pp. 504-508 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Dual Algebra ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that, if a Banach algebra is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of implies the (2m+ 1)-weak amenability of for all m ≥ 1.

The Second Conjugates of Certain Banach Algebras

1975 ◽  
Vol 27 (5) ◽  
pp. 1029-1035 ◽  
Author(s):  
Pak-Ken Wong
Keyword(s):  
Recent Result ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Arens Product ◽  
Conjugate Space ◽  
Natural Extensions ◽  
Semisimple Banach Algebra

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

scholarly journals ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

2008 ◽  
Vol 50 (3) ◽  
pp. 539-555 ◽  
Author(s):  
MATTHEW DAWS
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Main Tool ◽  
Bilinear Map ◽  
General Will ◽  
Arens Product ◽  
C Algebras ◽  
Local Reflexivity

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.

The third dual of a Banach algebra

2008 ◽  
Vol 45 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Mina Ettefagh
Keyword(s):  
Banach Algebra ◽  
Arens Product ◽  
The Third ◽  
Weak Amenability ◽  
Second Dual

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 ″, □).

scholarly journals Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

2019 ◽  
Vol 71 (03) ◽  
pp. 717-747
Author(s):  
Ross Stokke
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Bounded Operators ◽  
Locally Compact ◽  
Type Representation ◽  
Arens Product ◽  
Isometric Representation ◽  
Product Algebra

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.

scholarly journals m−WEAK AMENABILITY OF (2n)−TH DUALS OF BANACH ALGEBRAS

2019 ◽  
pp. 1
Author(s):  
Mina Ettefagh
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Arens Product ◽  
Weak Amenability

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.

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