Embedding of the Free Abelian Topological Group A(X x X) into A(X)

In my talk I will discuss the following question: For which separable metrizable spaces X does the free abelian topological group A(X x X) isomorphically embed into A(X)?
While for many natural spaces X such an embedding exist, I will show that there is a compact metric space X (Cook continuum) such that A(X x X) does not embed into A(X) as a topological subgroup.
The results which I am going to present were obtained jointly with A. Leiderman and S. Morris.