Sufficient conditions for isomorphisms between function algebras Thomas Tonev The University of Montana, Missoula Göteborg, July-August, 2013 T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 1 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Theorem 1 Norm-multiplicative maps Let A ⊂ C ( X ) be a function algebra on a loc. compact Hausdorff space X , i.e. A is uniformly closed and strongly separates the points of X . We assume that X contains the Shilov boundary ∂ A of A , and together - its Choquet boundary δ A . By � f � = max x ∈ X | f ( x ) | we denote the uniform norm of f ∈ A . Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If � Tf Tg � = � fg � for all f , g ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y )) | for all y ∈ δ B and f ∈ A. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17
Norm-multiplicative maps Corollary 1 The theorem holds also if A , B are (uniformly) dense subalgebras of function algebras such that δ A = p ( A ) , δ B = p ( B ) where p ( A ) , p ( B ) are the sets of p -points, i.e. the strong boundary points of A , B . In particular, it holds for semisimple algebras A with δ A = p ( A ) by the way of the Gelfand algebras � A . Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. � Tf � = � f � for all f ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y ) | for all y ∈ δ B and f ∈ A. Indeed, in this case � Tf Tg � = � T ( fg ) � = � fg � and the result follows directly from Theorem 1. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17
Norm-multiplicative maps Corollary 1 The theorem holds also if A , B are (uniformly) dense subalgebras of function algebras such that δ A = p ( A ) , δ B = p ( B ) where p ( A ) , p ( B ) are the sets of p -points, i.e. the strong boundary points of A , B . In particular, it holds for semisimple algebras A with δ A = p ( A ) by the way of the Gelfand algebras � A . Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. � Tf � = � f � for all f ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y ) | for all y ∈ δ B and f ∈ A. Indeed, in this case � Tf Tg � = � T ( fg ) � = � fg � and the result follows directly from Theorem 1. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17
Norm-multiplicative maps Corollary 1 The theorem holds also if A , B are (uniformly) dense subalgebras of function algebras such that δ A = p ( A ) , δ B = p ( B ) where p ( A ) , p ( B ) are the sets of p -points, i.e. the strong boundary points of A , B . In particular, it holds for semisimple algebras A with δ A = p ( A ) by the way of the Gelfand algebras � A . Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. � Tf � = � f � for all f ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y ) | for all y ∈ δ B and f ∈ A. Indeed, in this case � Tf Tg � = � T ( fg ) � = � fg � and the result follows directly from Theorem 1. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17
Norm-multiplicative maps Corollary 1 The theorem holds also if A , B are (uniformly) dense subalgebras of function algebras such that δ A = p ( A ) , δ B = p ( B ) where p ( A ) , p ( B ) are the sets of p -points, i.e. the strong boundary points of A , B . In particular, it holds for semisimple algebras A with δ A = p ( A ) by the way of the Gelfand algebras � A . Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. � Tf � = � f � for all f ∈ A then there is a homeomorphism ψ : δ B → δ A so that | ( Tf )( y ) | = | f ( ψ ( y ) | for all y ∈ δ B and f ∈ A. Indeed, in this case � Tf Tg � = � T ( fg ) � = � fg � and the result follows directly from Theorem 1. T. Tonev (University of Montana) Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17
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