On group-valued continuous functions: k -groups and reflexivity G´ abor Luk´ acs lukacs@topgroups.ca Halifax, Nova Scotia, Canada 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.0/23
Notations For X,Y ∈ Haus and G,H ∈ Ab ( Haus ) : C ( X,Y ):= cts functions, with compact-open topology. C ( X,G ) is a top. group with pointwise operations. H ( G,H ):= C ( G,H ) ∩ hom( G,H ) . Put T := R / Z . ˆ G := H ( G, T ) . α G : G → ˆ ˆ G is the evaluation homomorphism, ( α A ( g ))( χ )= χ ( g ) . 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.1/23
Four questions about α G Is α G injective? Is α G surjective? Is α G cts? Is α G open onto its image? 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Four questions about α G Is α G injective? There is a coarser Hausdorff group topology τ on R such that � ( R ,τ )=0 (Nienhuys, 1971). Is α G surjective? Is α G cts? Is α G open onto its image? 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Four questions about α G Is α G injective? There is a coarser Hausdorff group topology τ on R such that � ( R ,τ )=0 (Nienhuys, 1971). Is α G surjective? Z ([0 , 1]) is not onto (Außenhofer, 1999), where α L p L p Z ([0 , 1]):= a.e. integer funcs. in L p ([0 , 1]) , 1 <p< ∞ . Is α G cts? Is α G open onto its image? 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Four questions about α G Is α G injective? There is a coarser Hausdorff group topology τ on R such that � ( R ,τ )=0 (Nienhuys, 1971). Is α G surjective? Z ([0 , 1]) is not onto (Außenhofer, 1999), where α L p L p Z ([0 , 1]):= a.e. integer funcs. in L p ([0 , 1]) , 1 <p< ∞ . Is α G cts? α Z + is not cts, where Z + :=( Z , Bohr topology ) . Is α G open onto its image? 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Four questions about α G Is α G injective? There is a coarser Hausdorff group topology τ on R such that � ( R ,τ )=0 (Nienhuys, 1971). Is α G surjective? Z ([0 , 1]) is not onto (Außenhofer, 1999), where α L p L p Z ([0 , 1]):= a.e. integer funcs. in L p ([0 , 1]) , 1 <p< ∞ . Is α G cts? α Z + is not cts, where Z + :=( Z , Bohr topology ) . Is α G open onto its image? α V is not open onto its image for a non-locally convex topological vector space V . 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Four questions about α G Is α G injective? Is α G surjective? Is α G cts? Is α G open onto its image? Terminology: G is reflexive if α G is a topological isomorphism. G is almost reflexive if α G is an open isomorphism. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.2/23
Pontryagin duality for LCA For L ∈ LCA : ˆ L ∈ LCA . L is reflexive. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.3/23
Pontryagin duality for LCA For L ∈ LCA : ˆ L ∈ LCA . L is reflexive. If H ≤ L is a closed subgroup, then: � L/H ∼ H ∼ = H ⊥ , and ˆ = ˆ L/H ⊥ , where H ⊥ := { χ ∈ ˆ L | χ ( H )=0 } ; and 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.3/23
Pontryagin duality for LCA For L ∈ LCA : ˆ L ∈ LCA . L is reflexive. If H ≤ L is a closed subgroup, then: � L/H ∼ H ∼ = H ⊥ , and ˆ = ˆ L/H ⊥ , where H ⊥ := { χ ∈ ˆ L | χ ( H )=0 } ; and if H is compact, then H ⊥ is open in ˆ L . 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.3/23
Pontryagin duality for LCA For L ∈ LCA : ˆ L ∈ LCA . L is reflexive. If H ≤ L is a closed subgroup, then: � L/H ∼ H ∼ = H ⊥ , and ˆ = ˆ L/H ⊥ , where H ⊥ := { χ ∈ ˆ L | χ ( H )=0 } ; and if H is compact, then H ⊥ is open in ˆ L . c ( L ) ⊥ = B (ˆ L ) and B ( L ) ⊥ = c (ˆ L ) , where: c ( L ):= connected component of 0 in L . B ( L ):= { x ∈ L |� x � is compact } . 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.3/23
Observations and motivation Let X ∈ Haus and G ∈ Ab ( Haus ) . If α G is injective, then so is α C ( X,G ) . If α G is an embedding, then α C ( X,G ) is open onto its image ( G being LQC implies that C ( X,G ) is so). 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.4/23
Observations and motivation Let X ∈ Haus and G ∈ Ab ( Haus ) . If α G is injective, then so is α C ( X,G ) . If α G is an embedding, then α C ( X,G ) is open onto its image ( G being LQC implies that C ( X,G ) is so). Motivation Is C ( X,G ) (almost) reflexive? � What does C ( X,G ) look like? 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.4/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . α G is k -continuous. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . α G is k -continuous. X is a k -space if every k -cts map on X is cts. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . α G is k -continuous. X is a k -space if every k -cts map on X is cts. If X is LC or metrizable, then it is a k -space. If X is a k -space and Y is locally compact, then X × Y is a k -space. If X is a k -space and G is complete, then C ( X,G ) is complete. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . α G is k -continuous. X is a k -space if every k -cts map on X is cts. X is hemicompact if its family of compact subsets contains a countable cofinal family (cobase). 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Hausdorff k -spaces Let X,Y ∈ Haus and G ∈ Ab ( Haus ) . f : X → Y is k -cts if f | K is cts for every compact K ⊆ X . α G is k -continuous. X is a k -space if every k -cts map on X is cts. X is hemicompact if its family of compact subsets contains a countable cofinal family (cobase). If X is hemicompact and G is metrizable, then C ( X,G ) is metrizable. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.5/23
Special cases Let X be a Tychonoff k -space. C ( X, T ) is almost reflexive (Außenhofer, 1999). C ( X, R ) is almost reflexive (because it is a complete locally convex vector space). C ( X,D ) is almost reflexive for every discrete group D (because it is complete and has a linear topology). Thus, C ( X,G ) is almost reflexive for every abelian Lie group ( G = R n × T k × D ). 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.6/23
Special cases Let X be a hemicompact k -space. C ( X, T ) is reflexive (Außenhofer, 1999). C ( X, R ) is reflexive (because it is a complete metrizable locally convex vector space). C ( X,D ) is reflexive for every discrete group D (because it is complete, metrizable, and has a linear topology). Thus, C ( X,G ) is reflexive for every abelian Lie group ( G = R n × T k × D ). 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.6/23
Theorems (GL, 2015) If X is a Tychonoff k -space and G ∈ LCA , then: C ( X,G ) is almost reflexive; � � C ( X,G ) ≈ lim C ( K,G/C ) , where K ⊆ X is compact → and C ≤ G is compact such that G/C is a Lie group. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.7/23
Theorems (GL, 2015) If X is a Tychonoff k -space and G ∈ LCA , then: C ( X,G ) is almost reflexive; � � C ( X,G ) ≈ lim C ( K,G/C ) , where K ⊆ X is compact → and C ≤ G is compact such that G/C is a Lie group. If X is a hemicompact k -space, G ∈ LCA , and G is metrizable, then C ( X,G ) is reflexive. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.7/23
Theorems (GL, 2015) If X is a Tychonoff k -space and G ∈ LCA , then: C ( X,G ) is almost reflexive; � � C ( X,G ) ≈ lim C ( K,G/C ) , where K ⊆ X is compact → and C ≤ G is compact such that G/C is a Lie group. If X is a hemicompact k -space, G ∈ LCA , and G is metrizable, then C ( X,G ) is reflexive. If X is compact metrizable and zero-dimensional, and G ∈ LCA , then C ( X,G ) is reflexive. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.7/23
k -groups of Noble (1970) Let G ∈ Grp ( Haus ) . G is a k -group if every k -cts homomorphism on G is cts. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.8/23
k -groups of Noble (1970) Let G ∈ Grp ( Haus ) . G is a k -group if every k -cts homomorphism on G is cts. If G is a k -space, then it is a k -group. 12th Topological Symposium, Prague, Czech Republic, July 25-29, 2016 – p.8/23
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