Amenability and Weak Amenability for Beurling and Lipschitz Algebras

Bade, William G.; Curtis, Philip C.; Dales, H. G.

doi:10.1093/plms/s3-55_2.359

Cited by 198 publications

(236 citation statements)

References 10 publications

Supporting

Mentioning

234

Contrasting

Order By: Relevance

“…The two sequences {D(p n )(1)} n≥1 and {D(q n )(1)} n≥1 must be bounded and any two bounded sequences define uniquely (up to a coboundary) a simplicial derivation. We now recall that the map HH 1 ( 1 (B)) → HC 0 ( 1 (B)) in the Connes-Tzygan long exact sequence is given by D(·)(·) → D(·) (1). It is now clear that the image of this map is isomorphic to ∞ (Z \ {0}).…”

Section: Four Of the Terms Cancel And Onlymentioning

confidence: 95%

“…Similarly, ∞ (B n+2 ) has the standard bimodule structure arising from its identification with the dual of 1 …”

Section: The Resolutionmentioning

confidence: 99%

“…This is rather a rare phenomenon for Banach algebras, and so we shall only ask that the simplicial cohomology is supported in low dimensions. This idea resulted in another successful move in transferring a De Rham type theory to Banach algebras which was the notion of n-weak amenability introduced by Johnson in [13], which built on the definition of weak amenability in [1]. This successfully introduced a notion of dimension to Banach algebras, which tries to cohomologically capture the dimension of the maximal ideal space by using alternating n-cochains, which reflects De Rham theory.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Cyclic and Simplicial Cohomology of the Bicyclic Semigroup Algebra

Gourdeau

White

2010

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

Abstract. Let A = 1 (B) be the semigroup algebra of B, the bicyclic semigroup. We give a resolution of ∞ (B) which simplifies the computation of the cohomology of 1 (B) dual bimodules. We apply this to the dual module ∞ (B) and show that the simplicial cohomology groups H n (A, A ) vanish for n ≥ 2. Using the Connes-Tzygan exact sequence, these results are used to show that the cyclic cohomology groups HC n (A, A ) vanish when n is odd and are one-dimensional when n is even (n ≥ 2).

show abstract

Section: Four Of the Terms Cancel And Onlymentioning

confidence: 95%

“…Similarly, ∞ (B n+2 ) has the standard bimodule structure arising from its identification with the dual of 1 …”

Section: The Resolutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Cyclic and Simplicial Cohomology of the Bicyclic Semigroup Algebra

Gourdeau

White

2010

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…It is known ( [1]) that a commutative Banach algebra A is weakly amenable if and only if there are no non-zero, continuous derivations from A into A , the dual module of A. Definition 2.3.…”

Section: Preliminariesmentioning

confidence: 99%

Function Spaces in Analysis

Jarosz

2015

Contemporary Mathematics

View full text Add to dashboard Cite

Abstract. Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable examples of Swiss cheese sets in the literature. We explain the various abstract notions of Swiss cheeses, and how they can be manipulated to obtain desirable properties. In particular, we discuss the Feinstein-Heath classicalisation theorem and related results. We conclude with the construction of a new counterexample to a conjecture of S. E. Morris, using a classical Swiss cheese set.

show abstract

“…The notion of weak amenability was introduced in Bade et al [1] as a concept for commutative Banach algebras: A commutative Banach algebra s/ is called weakly amenable if there are no non-zero bounded derivations into any symmetric Banach simodule (symmetric meaning that the left and right module multiplications agree). In the same paper the authors showed that one only has to consider the module si*, the dual space of si equipped with the canonical module structure.…”

Section: Introductionmentioning

confidence: 99%