Rigidity of CR maps between Shilov boundaries of bounded symmetric domains

Our goal is to establish what seems to be the first rigidity result for CR embeddings between Shilov boundaries of bounded symmetric domains of higher rank. The result states that any such CR embedding is the standard linear embedding up to CR automorphisms. Our basic assumption extends precisely the well-known optimal bound for the rank one case. There are no other restrictions on the ranks, in particular, the difficult case when the target rank is larger than the source rank is also allowed.