scholarly journals Weak amenability of the second dual of a Banach algebra

2007 ◽  
Vol 182 (3) ◽  
pp. 205-205 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
M. Filali
Keyword(s):  
Banach Algebra ◽  
Weak Amenability ◽  
Second Dual

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 .

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.

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].

scholarly journals LIFTING DERIVATIONS AND n-WEAK AMENABILITY OF THE SECOND DUAL OF A BANACH ALGEBRA

2010 ◽  
Vol 83 (1) ◽  
pp. 122-129 ◽  
Author(s):  
S. BAROOTKOOB ◽  
H. R. EBRAHIMI VISHKI
Keyword(s):  
Banach Algebra ◽  
Positive Answer ◽  
Unified Approach ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that for n≥2, n-weak amenability of the second dual 𝒜** of a Banach algebra 𝒜 implies that of 𝒜. We also provide a positive answer for the case n=1, which sharpens some older results. Our method of proof also provides a unified approach to give short proofs for some known results in the case where n=1.

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 Weak amenability for the second dual of Banach modules

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Fatemeh Anousheh ◽  
Davood Ebrahimi Bagha ◽  
Abasalt Bodaghi
Keyword(s):  
Banach Algebra ◽  
Mild Conditions ◽  
Weak Amenability ◽  
Banach Modules ◽  
Second Dual

AbstractLet A be a Banach algebra, E be a Banach A-bimodule and Δ E → A be a bounded Banach A-bimodule homomorphism. It is shown that under some mild conditions, the weakΔ''-amenability of E'' (as an A''-bimodule) necessitates weak Δ-amenability of E (as an A-bimodule). Some examples of weak-amenable Banach modules are provided as well.

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].

On James' Quasi-Reflexive Banach Space as a Banach Algebra

1980 ◽  
Vol 32 (5) ◽  
pp. 1080-1101 ◽  
Author(s):  
Alfred D. Andrew ◽  
William L. Green
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Reflexive Banach Space ◽  
Equivalent Norm ◽  
New Techniques ◽  
Codimension One ◽  
Gelfand Transform ◽  
Schauder Bases ◽  
Second Dual ◽  
Arens Multiplication

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

scholarly journals The second dual of a Banach algebra

1979 ◽  
Vol 84 (3-4) ◽  
pp. 309-325 ◽  
Author(s):  
J. Duncan ◽  
S. A. R. Hosseiniun
Keyword(s):  
Banach Algebra ◽  
Open Problems ◽  
Current State ◽  
Second Dual ◽  
The Subject

SynopsisWe give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

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