Yemon Choi*, Mahya Ghandehari
* Corresponding author
MSC 2010: Primary 43A30, Secondary 46J10, 47B47
Published as J. Funct. Anal. 266 (2014), no. 11, 6501–6530
Preprint version available at arXiv 1304.3710 (final accepted version, not incorporating copy-editor's changes)
[ Math Review | Zentralblatt ]
Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real ax+b group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and Forrest–Samei–Spronk (IUMJ 2009). As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable.
When the article was submitted for publication, both authors worked at the University of Saskatchewan. By the time it was accepted, we had both moved on to positions elsewhere.
The work presented in this article was the result of discussions in late 2012. I had been trying to understand Johnson's 1994 article for several years, with a view to examining the ``higher-dimensional weak amenability'' of A(SO(3)). Although this is still unresolved (at time of writing), I had made some notes on how once could recover Johnson's estimates for derivations on Pol(SO(3)) using Schur orthogonality relations; this approach was to some extent motivated by calculations for Beurling algebras on T. On showing these notes to my coauthor, she quickly saw that the orthogonality relations for coefficient functions on the ax+b group could be applied in a very similar way to construct derivations on its Fourier algebra.
Once the ax+b case had been hammered out, we looked at other AR groups where the method might apply. Unfortunately, our estimates didn't appear strong enough in many of those cases. On the other hand, the reduced Heisenberg group is not quite AR, but it is close enough that the same arguments with coefficient functions and orthogonality can be applied. Having noticed that the ax+b case then yields results for SL(2,R) via Herz restriction, it was then natural to see how many other Lie groups would follow cheaply in the same way.