Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

Abstract

For any amenable locally compact group G, space of multipliers MA(G) of Fourier algebra A(G) coincides with space B(G) of functions on G that are linear combinations of continuous positive definite functions. We prove that MA(G)\B(G) * 0 for many non-amenable connected groups. More specifically we prove that MOA(G)\B(G) * 0 for classical complex Lie groups, and general Lorentz groups SOO(n, 1), n > 2. MOA(G) is a certain subspace of MA(G), which we call space of completely bounded multipliers of A(G). Unlike MA(G), space MOA(G) has nice stability properties with respect to direct products of groups. It is known that Fourier algebra of free group on N generators (N 2 2) admits an unbounded approximate unit ((Pn), which is bounded in multiplier norm. We extend this result to any closed subgroup of general Lorentz group SOO(n, 1). Moreover we show that for these groups ((Pn) can be chosen to be bounded with respect to MOA(G)-norm. By a duality argument we obtain that reduced C*-algebra of every discrete subgroup of SOO(n, 1) has the completely bounded approximation property. In particular this property holds for C* (F2), reduced C*-algebra of free group on two generators. We also prove