Multipliers of multidimensional Fourier algebras

Abstract

Let G be a locally compact σ -compact group. Motivated by an earlier notion for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra An(G) of G . We characterise the completely bounded multidimensional multipliers associated with An(G) in several equivalent ways. In particular, we establish a completely isometric embedding of the space of all n -dimensional completely bounded multipliers into the space of all Schur multipliers on Gn+1 with respect to the (left) Haar measure. We show that in the case G is amenable the space of completely bounded multidimensional multipliers coincides with the multidimensional Fourier-Stieltjes algebra of G introduced by Ylinen. We extend some well-known results for abelian groups to the multidimensional setting. Mathematics subject classification (2010): Primary 47B49; Secondary 47L60, 43A22.