Weak and cyclic amenability for Fourier algebras of connected Lie groups

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 Our results complement earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and Forrest--Samei--Spronk (IUMJ 2009).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.Comment: v4: AMS-LaTeX, 26 pages. Final version, to appear in JFA. Includes an authors' correction added at proof stag

Robust recovery of Robinson LpL^p-graphons

In this paper, we study the Robinson graphon completion/recovery problem for the class of $L^p$-graphons. We introduce a function $\Lambda$ on the space of $L^p$-graphons, which measures the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{w(x,y),w(y,z)\}$. We prove that the function $\Lambda$ satisfies the following: (1) $\Lambda$ is compatible with the cut-norm, in the sense that if two graphons are close in the cut-norm, then their $\Lambda$ values are close; and (2) when $p > 5$, every $L^p$-graphon $w$ can be approximated by a Robinson graphon, with error of the approximation bounded in terms of \IN{$\Lambda(w)$}. When $w$ is a noisy version of a Robinson graphon, our method provides a concrete formula for recovering the Robinson graphon approximating $w$ in cut-norm

Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups

A special case of a conjecture raised by Forrest and Runde (2005) [10] asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was already known to hold in the non-abelian compact cases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note). In recent work (Choi and Ghandehari, 2014 [4]) the 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

Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets

We consider the problem of characterizing the Sobolev wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, with the latter being defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive necessary and sufficient criteria for elements of the Sobolev wavefront set, formulated in terms of the decay behaviour of a given generalized continuous wavelet transform. It turns out that the characterization of directed smoothness of finite order can be performed in the two important cases: (1) bandlimited wavelets, and (2) wavelets with finitely many vanishing moments (e.g.~compactly supported wavelets). The main results of this paper are based on a number of fairly technical conditions on the dilation group. In order to demonstrate their wide applicability, we exhibit a large class of generalized shearlet groups in arbitrary dimensions fulfilling all required conditions, and give estimates of the involved constants

Harmonic analysis of Rajchman algebras

Abstract harmonic analysis is mainly concerned with the study of locally compact groups, their unitary representations, and the function spaces associated with them. The Fourier and Fourier-Stieltjes algebras are two of the most important function spaces associated with a locally compact group. The Rajchman algebra associated with a locally compact group is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity. This is a closed, complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. In the Abelian case, the Rajchman algebras can be identified with the algebra of Rajchman measures on the dual group. Such measures have been widely studied in the classical harmonic analysis. In contrast, for non-commutative locally compact groups little is known about these interesting algebras. In this thesis, we investigate certain Banach algebra properties of Rajchman algebras associated with locally compact groups. In particular, we study various amenability properties of Rajchman algebras, and observe their diverse characteristics for different classes of locally compact groups. We prove that amenability of the Rajchman algebra of a group is equivalent to the group being compact and almost Abelian, a property that is shared by the Fourier-Stieltjes algebra. In contrast, we also present examples of large classes of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which Rajchman algebras are not even operator weakly amenable. Moreover, we establish various extension theorems that allow us to generalize the previous result to all non-compact connected SIN-groups. Finally, we investigate the spectral behavior of Rajchman algebras associated with Abelian locally compact groups, and construct point derivations at certain elements of their spectrum using Varopoulos’ decompositions for Rajchman algebras. Having constructed similar decompositions, we obtain analytic discs around certain idempotent characters of Rajchman algebras. These results, and others that we obtain, illustrate the inherent distinction between the Rajchman algebra and the Fourier algebra of many locally compact groups