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 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].

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.

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.

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.

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 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 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.

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 Functional Identities and Zero Lie Product Determined Banach Algebras

2020 ◽  
Vol 71 (2) ◽  
pp. 649-665
Author(s):  
Matej Brešar
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
The Third ◽  
Linear Maps ◽  
Functional Identities

Abstract Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity and the third one concerns bijective commutativity preserving linear maps.

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