Amenability and Weak Amenability for Beurling and Lipschitz Algebras

The Cyclic and Simplicial Cohomology of the Bicyclic Semigroup Algebra

“…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}).…”

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

“…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.…”

Function Spaces in Analysis

“…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.…”

Weak and cyclic amenability for non-commutative Banach algebras

“…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.…”