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.
","abstract_html":"<p>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 &isin; 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.</p><p>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.</p><p>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&macr;(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.</p>","abstract_has_math":false,"creators":["Eroshkin, Oleg"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Eric Grinberg"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T05:22:25Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholars.unh.edu/dissertation/494","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Eric Grinberg"]},{"key":"dc:creator","label":"Author","values":["Eroshkin, Oleg"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholars.unh.edu/dissertation/494"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Invariant Frechet algebras on bounded symmetric domains"]}]}],"canonical_facts":{"dc:contributor":["Eric Grinberg"],"dc:creator":["Eroshkin, Oleg"],"dc:description.abstract":["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.
"],"dc:identifier":["https://scholars.unh.edu/dissertation/494"],"dc:subject":["Mathematics"],"dc:title":["Invariant Frechet algebras on bounded symmetric domains"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T05:22:25Z"}