The Fourier algebra of a locally compact group was first defined by Eymard in 1964. Eymard showed that this algebra is in fact the space of all coefficient functions of the left regular representation equipped with pointwise operations. The Fourier algebra is a semi-simple commutative Banach algebra, and thus it admits no non-zero continuous derivation into itself. In this talk we study weak amenability, which is a weaker form of differentiability, for Fourier algebras. A commutative Banach algebra is called weakly amenable if it admits no non-zero continuous derivations into its dual space. The notion of weak amenability was first defined and studied for certain important examples by Bade, Curtis and Dales.
In 1994, Johnson constructed a non-zero continuous derivation from the Fourier algebra of the rotation group in 3 dimensions into its dual. Subsequently, using the structure theory of Lie groups and Lie algebras, this result was extended to any non-Abelian, compact, connected group. Using techniques of non-commutative harmonic analysis, we prove that semi-simple connected Lie groups and 1-connected non-Abelian nilpotent Lie groups are not weak amenable by reducing the problem to two special cases: the $ax+b$ group and the 3-dimensional Heisenberg group. These are the first examples of classes of locally compact groups with non-weak amenable Fourier algebras which do not contain closed copies of the rotation group in 3 dimensions.
- Functional Analysis Seminar