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## Abstract

Let $G$ be a locally compact $\sigma$-compact group. Motivated by an earlier notion
for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra $A^n(G)$ of $G$. We characterise the completely bounded multidimensional multipliers associated with
$A^n(G)$ 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 $G^{n+1}$ 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.

Original language | English |
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Pages (from-to) | 459-484 |

Number of pages | 26 |

Journal | Operators and Matrices |

Volume | 4 |

Issue number | 4 |

Publication status | Published - 2010 |

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