It is shown that hermitian symmetric spaces of non-compact type (bounded symmetric domains) give rise to an interesting non-commutative geometry in the sense of Alain Connes. More precisely, we study Toeplitz operators and Toeplitz C*-algebras on the Hardy and Bergman space over symmetric domains and determine the full spectrum (irreducible representations) of these C*-algebras. As a very recent work (with G. Misra) we extend this theory to the boundary orbits under the semi-simple Lie group. Interesting examples (unit ball, spin factor) arise already for domains of small rank. The necessary Jordan algebraic background is reviewed at the beginning of the talk.
