Yemon Choi

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of l1-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.

Surveys in contemporary mathematics

Preface 1. Rank and determinant functions for matrices over semi-rings A. E. Guterman 2. Algebraic geometry over Lie algebras I. V. Kazachkov 3. Destabilization of closed braids A. V. Malyutin 4. n-dimensional local fields and adeles on n-dimensional schemes D. V. Osipov 5. Cohomology of face rings, and torus actions T. E. Panov 6. Three lectures on the Borsuk partition problem A. M. Raigorodskii 7. Embedding and knotting of manifolds in Euclidean spaces A. B. Skopenkov 8. On Maxwellian and Boltzmann distributions V. V. Ten.

APPROXIMATE AMENABILITY OF SCHATTEN CLASSES, LIPSCHITZ ALGEBRAS AND SECOND DUALS OF FOURIER ALGEBRAS

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for approximate amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that bounded approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.

A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A C -ALGEBRA

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.

Simplicial cohomology of band semigroup algebras

We establish the simplicial triviality of the convolution algebra

Simplicial homology and hochschild cohomology of banach semilattice algebras

\ell^1(S)

Quotients of Fourier algebras, and representations which are not completely bounded

, where

Weak and cyclic amenability for Fourier algebras of connected Lie groups

S

ZL-amenability constants of finite groups with two character degrees

is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on

On commutative, operator amenable subalgebras of finite von Neumann algebras

\ell^1(S)