A sum–product theorem in semi-simple commutative Banach algebras

The following analogue of the Erdös–Szemerédi sum-product theorem is shown. Let A=f1,⋯,fN be a finite set of N arbitrary distinct functions on some set. Then either the sum set fi+fj or the product set fi fj has at least N1+c elements, where c>0 is an absolute constant. We use Freimanʹs lemma and Balog–Szemerédi–Gowers Theorem on graphs and combinatorics. As a corollary, we obtain an Erdös–Szemerédi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂R is a finite set, |A| large enough, then|A+A|+|A.A|>|A|1+c,where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=|H|, v=1N∑x∈Hδx. Then, for all constant c, there is a k=k(c) such that maxz∈Rv(k)(z)