Rigidity results for automorphisms of Hardy--Toeplitz C∗-algebras

We prove a number of results on the automorphisms and isomorphisms between Hardy--Toeplitz algebras T(D) associated to bounded symmetric domains D: that the stable isomorphism class of T(D) determines D (even when it is reducible), that for reducible domains D=D1×⋯×Ds the automorphisms of the Shilov boundary ˇS(D) induced by those of T(D) permute the Shilov boundaries ˇS(Di), and that by contrast to arbitrary solvable algebras, automorphisms of T(D) that are trivial on their character spaces ˇS(D) are trivial on the entire spectrum ˆT(D).