The Second Conjugates of Certain 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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Symmetric amenability and the nonexistence of Lie and Jordan derivations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075010 ◽

1996 ◽

Vol 120 (3) ◽

pp. 455-473 ◽

Cited By ~ 68

Author(s):

B. E. Johnson

Keyword(s):

Positive Result ◽

Banach Algebra ◽

Recent Work ◽

Banach Algebras ◽

Jordan Derivation ◽

Symmetric Tensors ◽

C Algebras ◽

Semisimple Algebras ◽

First Time ◽

Semisimple Banach Algebra

A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

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2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

Journal of the Australian Mathematical Society ◽

10.1017/s144678872000021x ◽

2020 ◽

pp. 1-12

Author(s):

WENBO HUANG ◽

JIANKUI LI

Keyword(s):

Banach Algebra ◽

Banach Algebras ◽

Subspace Lattice ◽

Local Derivation ◽

Essential Ideal ◽

Semisimple Banach Algebra

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$ . We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$ , then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$ -subspace lattice algebras, is a derivation.

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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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ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

Glasgow Mathematical Journal ◽

10.1017/s0017089508004400 ◽

2008 ◽

Vol 50 (3) ◽

pp. 539-555 ◽

Cited By ~ 2

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.

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A note on annihilator Banach algebras

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700027258 ◽

1988 ◽

Vol 38 (1) ◽

pp. 77-81

Author(s):

Pak-Ken Wong

Keyword(s):

Banach Algebra ◽

Banach Algebras ◽

Dense Subalgebra ◽

Semisimple Banach Algebra

LetAbe a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebraBwith | · | such that ‖ · ‖ majorises | · | onA. The purpose of this paper is to investigate the annihilator property between the algebrasAandB.

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Continuity of Lie derivations on Banach algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500019933 ◽

1998 ◽

Vol 41 (3) ◽

pp. 625-630 ◽

Cited By ~ 14

Author(s):

M. I. Berenguer ◽

A. R. Villena

Keyword(s):

Banach Algebra ◽

Banach Algebras ◽

Lie Derivation ◽

Separating Subspace ◽

Lie Derivations ◽

Semisimple Banach Algebra

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.

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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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Finite dimensionality in scole of Banach algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171284000570 ◽

1984 ◽

Vol 7 (3) ◽

pp. 519-522 ◽

Cited By ~ 2

Author(s):

Sin-Ei Takahasi

Keyword(s):

Banach Algebra ◽

Banach Algebras ◽

Finite Dimensional ◽

Finite Dimensionality ◽

Semisimple Banach Algebra

It is shown that if the soclesoc(A)of a semisimple Banach algebraAis norm-closed, thensoc(A)is already finite dimensional. The proof makes use of the Al-Moajil theorem. However it is remarked that our main theorem is an extension of the Al-Moajil's.

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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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