By Kanupriya (IIT Delhi)

Abstract 

In this talk, we study several aspects of the multidimensional Fourier algebra A n (G)
associated with a locally compact group G. We begin with spectral synthesis and estab-
lish multidimensional versions of the subgroup lemma, injection theorem, inverse pro-
jection theorem, and Malliavin’s theorem, along with a parallel synthesis result between
A n (G) and A n+1 (G). We then discuss approximation identities, operator amenability,
approximation properties (AP n ), and weak amenability of A n (G). Finally, we investi-gate
invariant means on V N n (G), including their existence and properties for discrete and
non-discrete groups, their behavior under open subgroups, and invariant means on the
dual of A n (G). These results provide a unified study of spectral, approximation,
amenability, and invariant mean properties in the setting of multidimensional Fourier
algebras.