Introduction Fock Representation Shilov Boundary Shilov boundary for "holomorphic functions" on a quantum matrix ball Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Department of Mathematics Chalmers University of Technology June 22, 2017 Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary The classical notion of a Shilov boundary I Was introduced by Georgii Shilov. I It generalizes the maximus modulus priciple: I If f (z) 2 C(D) is an holomorphic function on the unit disc D; then max jf (z)j = max jf (z)j; z2D z2T where T is the unit circle. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Definition A boundary of X relative to A is a closed subset B ⊆ X; such that for every f (z) 2 A max jf (z)j = max jf (z)j: z2X z2B Definition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary. It can be shown that the Shilov boundary exists and that it is unique. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Definition A boundary of X relative to A is a closed subset B ⊆ X; such that for every f (z) 2 A max jf (z)j = max jf (z)j: z2X z2B Definition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary. It can be shown that the Shilov boundary exists and that it is unique. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Definition A boundary of X relative to A is a closed subset B ⊆ X; such that for every f (z) 2 A max jf (z)j = max jf (z)j: z2X z2B Definition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary. It can be shown that the Shilov boundary exists and that it is unique. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Definition A boundary of X relative to A is a closed subset B ⊆ X; such that for every f (z) 2 A max jf (z)j = max jf (z)j: z2X z2B Definition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary. It can be shown that the Shilov boundary exists and that it is unique. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Examples of Shilov Boundary Example In C(D) let A(D) ⊆ C(D) be the holomorphic functions. The Shilov boundary of D relative to A(D) is then T: Example Let Dn be the closed unit ball in Mn(C); then the sub-algebra A(Dn) ⊆ C(Dn) of holomorphic functions has as its Shilov boundary Un; the Lie group of unitary n × n matrices. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Examples of Shilov Boundary Example In C(D) let A(D) ⊆ C(D) be the holomorphic functions. The Shilov boundary of D relative to A(D) is then T: Example Let Dn be the closed unit ball in Mn(C); then the sub-algebra A(Dn) ⊆ C(Dn) of holomorphic functions has as its Shilov boundary Un; the Lie group of unitary n × n matrices. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary A non-commutative version of the Shilov boundary was proposed by Arveson [’70, Acta Math.]: ∗ C(X) unital C -algebras B (noncom. topological spaces) K ⊂ X, closed closed ideals in B ∗ uniform algebra subalgebra A that generates the C -algebra B, 1 2 A. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary A non-commutative version of the Shilov boundary was proposed by Arveson [’70, Acta Math.]: ∗ C(X) unital C -algebras B (noncom. topological spaces) K ⊂ X, closed closed ideals in B ∗ uniform algebra subalgebra A that generates the C -algebra B, 1 2 A. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary A non-commutative version of the Shilov boundary was proposed by Arveson [’70, Acta Math.]: ∗ C(X) unital C -algebras B (noncom. topological spaces) K ⊂ X, closed closed ideals in B ∗ uniform algebra subalgebra A that generates the C -algebra B, 1 2 A. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary A non-commutative version of the Shilov boundary was proposed by Arveson [’70, Acta Math.]: ∗ C(X) unital C -algebras B (noncom. topological spaces) K ⊂ X, closed closed ideals in B ∗ uniform algebra subalgebra A that generates the C -algebra B, 1 2 A. Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary Let D be a C∗-algebra and A ⊆ D a unital subalgebra that generates D as a C∗-algebra. Definition An ideal J in D is called a boundary ideal of D relative A if the canonical map j : D ! D=J is a complete isometry when restricted to A. A boundary ideal is called a Shilov boundary ideal if it contains any other boundary ideal. Shilov boundary ideal exists and unique (Arveson ’70, Hamana ’79). Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary Let D be a C∗-algebra and A ⊆ D a unital subalgebra that generates D as a C∗-algebra. Definition An ideal J in D is called a boundary ideal of D relative A if the canonical map j : D ! D=J is a complete isometry when restricted to A. A boundary ideal is called a Shilov boundary ideal if it contains any other boundary ideal. Shilov boundary ideal exists and unique (Arveson ’70, Hamana ’79). Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary Definition of Shilov Boundary Let D be a C∗-algebra and A ⊆ D a unital subalgebra that generates D as a C∗-algebra.