Spectral Synthesis in the Multidimensional Fourier Algebra and the Varopolous Algebra for Compact Groups

Let $G$ be a compact group and let $A^n(G)$ denote the multidimensional Fourier algebra introduced by Todorov and Turowska. In this note, we first define the multidimensional version of the Varopolous algebra and show that the multidimensional Fourier algebra can be embedded into the multidimensional Varopolous algebra. Using this embedding, we also prove a result on parallel synthesis, subsuming the earlier results of Varopolous, Spronk-Turowska and Parthsarathy-Prakash.