Holomorphic retractions and boundary berezin transforms

In an earlier paper, the first two authors have shown that the convolution of a function f continuous on the closure of a Cartan domain and a K- invariant finite measure a. on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face F depends only on the restriction of f to F and is equal to the convolution, in F, of the latter restriction with some measure (if on F uniquely determined by μ In this article, we give an explicit formula for μ_F in terms of F, showing in particular that for measures μ corresponding to the Berezin transforms the measures p again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.