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.

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 The Relationship between Two Involutive SemigroupsSandSTIs Defined by a Left MultiplierT

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
S. M. Mohammadi ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Mild Conditions ◽  
Underlying Space ◽  
Second Dual ◽  
Left Multiplier ◽  
The Relationship

LetSbe a semigroup with a left multiplierTonS. There exists a new semigroupST, which depends onSandT, which has the same underlying space asS. We study the question of involutions onSTand a Banach algebraAT. We find a condition ofTunder whichSTand the second dualAT****admit an involution. We will show thatATisC*-algebra if and only ifT:AT→Ais an isometry, under mild conditions. Also,AisC*-algebra if and only if so isAT, under other minor conditions.

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 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 Weak module amenability for the second dual of a Banach algebra

2021 ◽  
Vol 25 (2) ◽  
pp. 297-306
Author(s):  
Shabani Soltanmoradi ◽  
Davood Ebrahimi Bagha ◽  
Pourbahri Rahpeyma
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Module Amenability ◽  
Banach Modules ◽  
Weak Module ◽  
Module Derivation ◽  
Second Dual

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

scholarly journals Sectional representation of Banach modules and their multipliers

2003 ◽  
Vol 2003 (13) ◽  
pp. 817-825
Author(s):  
Terje Hõim ◽  
D. A. Robbins
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Banach Module ◽  
The Past ◽  
Banach Modules ◽  
Quotient Modules

LetXbe a Banach module over the commutative Banach algebraAwith maximal ideal spaceΔ. We show that there is a norm-decreasing representation ofXas a space of bounded sections in a Banach bundleπ:ℰ→Δ, whose fibers are quotient modules ofX. There is also a representation ofM(X), the space of multipliersT:A→X, as a space of sections in the same bundle, but this representation may not be continuous. These sectional representations subsume results of various authors over the past three decades.

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].

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