On isomorphisms between ideals of Fourier algebras of finite abelian groups

Abstract

We study linear isomorphisms between the ideals of Fourier algebras A(G₁) and A(G₂) where $\widehat{{G_1}}$ and $\widehat{{G_2}}$ are infinite compact abelian torsion groups, assuming that there exists a infinite subgroup of G₁ non-isomorphic to any subgroup of G₂. Then ∥T_n∥ → ∞ for any sequences of ideals I ⁽ⁿ⁾₂ ⊂ A(G₂), and any increasing sequence of ideals I ⁽ⁿ⁾₁ ⊂ A(G₁) such that $\overline { \cup I_1^{(n)}} = A({G_1})$, and a sequence of isomorphisms T_n: I ⁽ⁿ⁾₁ → I ⁽ⁿ⁾₂ . Moreover, in two cases: (1) when G₂ is an infinite p-group and (2) G₁ contains a p-group and G₂ does not contain any infinite p-group, we estimate the norm as an iterated logarithm of the dimension. This result can be reformulated in terms of the equivalence of systems of functions consisting of the group characters. The reformulation answers a question of A. Pelczynski. Our proof is based on the deep results from additive combinatorics (Green and Sanders Theorem) and number theory (results on the number of solutions of S-unit equations by Evertse, van der Poorten and Sclickewei).