Yemon Choi*, Mahya Ghandehari
* Corresponding author
MSC 2010: Primary 43A30. Secondary 46J10, 47B47
Appeared as J. Funct. Anal. 268 (2015) no. 8, 2440–2463
Preprint version available at arXiv 1405.6403 (final accepted version, not including journal's final formatting and house style)
[ Math Review | Zentralblatt ]
A special case of a conjecture raised by Forrest and Runde (Math. Zeit., 2005) asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note). In recent work (JFA, 2014) the present authors verified this conjecture for the real ax+b group and hence, by structure theory, for any semisimple Lie group.
In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis. En route, we use the known fusion rules for Schrödinger representations to give a concrete realization of the ``dual convolution'' for this group as a kind of twisted, operator-valued convolution. We also give some partial results for solvable groups which give further evidence to support the general conjecture.
After this paper was submitted for publication, we learned of the interesting work of Lee–Ludwig–Samei–Spronk, arXiv 1502.05214, which proves the Lie case of the Forrest–Runde conjecture using a different perspective. In particular, the case of the motion group Euc(2), left open here, is resolved by these authors.
Particular thanks are due to the referee of this article, who in response to the original submission pointed out to us that Section 9 of Stinespring's TAMS 1959 paper discusses "dual convolution" for $L^1(VN(G))$ when $G$ is a unimodular group, and also pointed out that Stinespring's results justify certain technical issues concerning noncommutative Fourier inversion.