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,?,f_N be a finite set of N arbitrary distinct functions on some set. Then either the sum set f_i+f_j or the product set has at least N^1+c elements, where c>0 is an absolute constant. We use Freiman's 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,