Multipliers of multidimensional Fourier algebras, Operators and Matrices

Abstract. Let G be a locally compact σ-compact group. Mo-tivated 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. 1