| Let D be a bounded domain in the complex vector space Cn . We say that D is symmetric iff, given any two points p, q ∈ D, there is a biholomorphism &phis;, which interchanges p and q. These domains were classified abstractly by Elie Cartan in his general study of symmetric spaces, and were canonically realized in Cn by Harish-Chandra. They include polydisks and Siegel domains.;Let D be a bounded symmetric domain in Cn , and G be the largest connected group of biholomorphic automorphisms of D. The algebra C( D) of all continuous (not necessarily bounded) complex-valued functions on D with compact-open topology is a Frechet algebra. A closed subalgebra of C(D) is called an invariant algebra if it is closed under compositions with elements of G.;We prove that if D is irreducible, then there are only three invariant algebras with identity with maximal ideal space D : C(D), the set of all holomorphic functions H(D) and the set of all antiholomorphic functions H¯(D). This result partially generalizes the Rudin's classification of invariant algebras on unit ball in Cn . For the general symmetric bounded domain D we prove that the only invariant algebras are tensor products of invariant algebras on irreducible factors of D. |
