Amenability and topological centres of the second duals of Banach algebras

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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Weak Amenability of a Class of Banach Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-2001-050-7 ◽

2001 ◽

Vol 44 (4) ◽

pp. 504-508 ◽

Cited By ~ 4

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102005443 ◽

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].

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Isomorphisms and multipliers on second dual algebras of Banach algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075241 ◽

1992 ◽

Vol 111 (1) ◽

pp. 161-168 ◽

Cited By ~ 9

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.

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The third dual of a Banach algebra

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2007.1038 ◽

2008 ◽

Vol 45 (1) ◽

pp. 1-11 ◽

Cited By ~ 1

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 ″, □).

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Module Actions on Iterated Duals of Banach Algebras

ISRN Mathematical Analysis ◽

10.5402/2011/439649 ◽

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.

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ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719001114 ◽

2019 ◽

Vol 102 (1) ◽

pp. 138-150

Author(s):

RUKI MATSUI ◽

YUJI TAKAHASHI

Keyword(s):

Abelian Group ◽

Banach Algebra ◽

Group Algebra ◽

Dual Algebra ◽

Arens Regularity ◽

Second Dual ◽

Discrete Abelian Group ◽

Stieltjes Transforms

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

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

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1901001e ◽

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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Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501694 ◽

2018 ◽

Vol 17 (09) ◽

pp. 1850169 ◽

Cited By ~ 3

Author(s):

Hossein Javanshiri ◽

Mehdi Nemati

Keyword(s):

Banach Algebra ◽

Cohomology Group ◽

Banach Algebras ◽

Multiplier Algebra ◽

Topological Center ◽

Maximal Ideals ◽

Open Questions ◽

Arens Regularity ◽

Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

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Amenability of Banach algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100067840 ◽

1989 ◽

Vol 105 (2) ◽

pp. 351-355 ◽

Cited By ~ 8

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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φ-Multipliers on Banach Algebras and Topological Modules

Abstract and Applied Analysis ◽

10.1155/2015/951021 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6

Author(s):

Marjan Adib

Keyword(s):

Compact Group ◽

Banach Algebra ◽

Banach Algebras ◽

Topological Module ◽

Topological Modules ◽

Arens Regularity

We prove some results concerning Arens regularity and amenability of the Banach algebraMφAof allφ-multipliers on a given Banach algebraA. We also considerφ-multipliers in the general topological module setting and investigate some of their properties. We discuss theφ-strict andφ-uniform topologies onMφA. A characterization ofφ-multipliers onL1G-moduleLpG, whereGis a compact group, is given.

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