Smooth Fourier Multipliers on Group Von Neumann Algebras

We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove Lp estimates for noncommutative Riesz transforms and characterize L∞ → BMO boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón-Zygmund theory. Introduction and main results Convergence of Fourier series and norm estimates for Fourier multipliers are central in harmonic analysis. As far as Calderón-Zygmund methods are involved very few results have been successfully transferred to other noneuclidean topological groups. In this paper we study smooth Fourier multipliers frequency supported by an arbitrary discrete group. For instance, the frequency group associated to the n-dimensional torus is the integer lattice Z and —by de Leeuw’s compactification theorem [10]— we may impose the discrete topology on the frequency group of R and still obtain the same family of Lp-multipliers. What can we say about Lp-boundedness of Fourier multipliers for arbitrary discrete groups? What do we mean by smoothness of the multiplier in that case? Basic unexplored examples include duals of Cantor cubes, other discrete abelian groups, finite groups of large cardinality, the discretized Heisenberg group, free groups... For nonabelian discrete groups, the compact dual is a quantum group whose underlying space is a group von Neumann algebra. These algebras are widely accepted and very well understood in noncommutative geometry [6] and operator algebra [38]. In harmonic analysis this approach was first considered in the ground-breaking results of Haagerup [21] and Cowling/Haagerup [9] on the approximation property of group von Neumann algebras. Up to isolated contributions [23, 49], the Lp-theory for Fourier multipliers on these algebras is very much unexplored. Let G be a discrete group with left regular representation λ : G→ B(`2(G)) given by λ(g)δh = δgh, where the δg’s form the unit vector basis of `2(G). Write L(G) for its group von Neumann algebra, the weak operator closure of the linear span of λ(G). Given f ∈ L(G), we consider the standard normalized trace τG(f) = 〈δe, fδe〉 where e denotes the identity of G. Any such f has a Fourier series ∑ g∈G f̂(g)λ(g) with f̂(g) = τG(fλ(g −1)) so that τG(f) = f̂(e).