Tensor products of Cuntz semigroups Hannes Thiel (joint work with - PowerPoint PPT Presentation

Tensor products of Cuntz semigroups Hannes Thiel (joint work with Ramon Antoine, Francesc Perera) University of M¨ unster, Germany 26. June 2017 TACL, Prague 1 / 11

The category Cu of abstract Cuntz semigroup Recall: Cu -semigroup is domain with monoid structure such that addition is jointly Scott continuous and ≪ -preserving: a ′ ≪ a , b ′ ≪ b a ′ + b ′ ≪ a + b . ⇒ Cu -morphism f : S → T is additive, ≪ -preserving Scott continuous map: a ′ ≪ a f ( a ′ ) ≪ f ( a ) . ⇒ 2 / 11

The category Cu of abstract Cuntz semigroup Recall: Cu -semigroup is domain with monoid structure such that addition is jointly Scott continuous and ≪ -preserving: a ′ ≪ a , b ′ ≪ b a ′ + b ′ ≪ a + b . ⇒ Cu -morphism f : S → T is additive, ≪ -preserving Scott continuous map: a ′ ≪ a f ( a ′ ) ≪ f ( a ) . ⇒ Examples: N := { 0 , 1 , 2 , . . . , ∞} . 2 / 11

The category Cu of abstract Cuntz semigroup Recall: Cu -semigroup is domain with monoid structure such that addition is jointly Scott continuous and ≪ -preserving: a ′ ≪ a , b ′ ≪ b a ′ + b ′ ≪ a + b . ⇒ Cu -morphism f : S → T is additive, ≪ -preserving Scott continuous map: a ′ ≪ a f ( a ′ ) ≪ f ( a ) . ⇒ Examples: N := { 0 , 1 , 2 , . . . , ∞} . Z := Cu ( Z ) = N ∪ ( 0 , ∞ ] . 2 / 11

The category Cu of abstract Cuntz semigroup Recall: Cu -semigroup is domain with monoid structure such that addition is jointly Scott continuous and ≪ -preserving: a ′ ≪ a , b ′ ≪ b a ′ + b ′ ≪ a + b . ⇒ Cu -morphism f : S → T is additive, ≪ -preserving Scott continuous map: a ′ ≪ a f ( a ′ ) ≪ f ( a ) . ⇒ Examples: N := { 0 , 1 , 2 , . . . , ∞} . Z := Cu ( Z ) = N ∪ ( 0 , ∞ ] . R P := Cu ( UHF p ) = N [ 1 p ] ∪ ( 0 , ∞ ] . 2 / 11

The category Cu of abstract Cuntz semigroup Recall: Cu -semigroup is domain with monoid structure such that addition is jointly Scott continuous and ≪ -preserving: a ′ ≪ a , b ′ ≪ b a ′ + b ′ ≪ a + b . ⇒ Cu -morphism f : S → T is additive, ≪ -preserving Scott continuous map: a ′ ≪ a f ( a ′ ) ≪ f ( a ) . ⇒ Examples: N := { 0 , 1 , 2 , . . . , ∞} . Z := Cu ( Z ) = N ∪ ( 0 , ∞ ] . R P := Cu ( UHF p ) = N [ 1 p ] ∪ ( 0 , ∞ ] . Cu ( II 1 -factor ) = [ 0 , ∞ ) ∪ ( 0 , ∞ ] . 2 / 11

Goals and strategy Problem Define S ⊗ Cu T and show that Cu is closed, monoidal category. 3 / 11

Goals and strategy Problem Define S ⊗ Cu T and show that Cu is closed, monoidal category. Strategy: Define category W of ‘pre-completed Cu -semigroups’. 3 / 11

Goals and strategy Problem Define S ⊗ Cu T and show that Cu is closed, monoidal category. Strategy: Define category W of ‘pre-completed Cu -semigroups’. Define ⊗ W . 3 / 11

Goals and strategy Problem Define S ⊗ Cu T and show that Cu is closed, monoidal category. Strategy: Define category W of ‘pre-completed Cu -semigroups’. Define ⊗ W . Completion functor γ : W → Cu that is reflection: W ( S , T ) ∼ � � γ ( S ) , T = Cu . 3 / 11

Goals and strategy Problem Define S ⊗ Cu T and show that Cu is closed, monoidal category. Strategy: Define category W of ‘pre-completed Cu -semigroups’. Define ⊗ W . Completion functor γ : W → Cu that is reflection: W ( S , T ) ∼ � � γ ( S ) , T = Cu . Reflection functors transfer monoidal structure. 3 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: 4 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: a ≺ + b ≺ ⊆ ( a + b ) ≺ . + preserves ≺ : 4 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: a ≺ + b ≺ ⊆ ( a + b ) ≺ . + preserves ≺ : a ≺ + b ≺ ⊆ ( a + b ) ≺ is cofinal. + is continuous: 4 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: a ≺ + b ≺ ⊆ ( a + b ) ≺ . + preserves ≺ : a ≺ + b ≺ ⊆ ( a + b ) ≺ is cofinal. + is continuous: W -morphism preserves 0 , + , ≺ and is continuous: f ( a ≺ ) ⊆ f ( a ) ≺ is cofinal. 4 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: a ≺ + b ≺ ⊆ ( a + b ) ≺ . + preserves ≺ : a ≺ + b ≺ ⊆ ( a + b ) ≺ is cofinal. + is continuous: W -morphism preserves 0 , + , ≺ and is continuous: f ( a ≺ ) ⊆ f ( a ) ≺ is cofinal. Cu -semigroup S � W -semigroup ( S , ≪ ) . W -semigroup ( S , ≺ ) � round-ideal completion γ ( S , ≺ ) . 4 / 11

Category W of pre-completed Cuntz semigroups The predecessor set: a ≺ := { x | x ≺ a } . Definition W -semigroup is monoid with transitive relation ≺ such that: a ≺ is upward directed. ≺ has interpolation: a ≺ + b ≺ ⊆ ( a + b ) ≺ . + preserves ≺ : a ≺ + b ≺ ⊆ ( a + b ) ≺ is cofinal. + is continuous: W -morphism preserves 0 , + , ≺ and is continuous: f ( a ≺ ) ⊆ f ( a ) ≺ is cofinal. Cu -semigroup S � W -semigroup ( S , ≪ ) . W -semigroup ( S , ≺ ) � round-ideal completion γ ( S , ≺ ) . Theorem Cu is a full, reflective subcategory of W . Have completion functor γ : W → Cu . 4 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . 5 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . Definition Cu -bimorphism is f : S × T → R such that: f is additive in each variable. 5 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . Definition Cu -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≪ -preserving: f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ . 5 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . Definition Cu -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≪ -preserving: f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ . f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ is cofinal. f is continuous: 5 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . Definition Cu -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≪ -preserving: f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ . f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ is cofinal. f is continuous: Approach: First define ⊗ in W. 5 / 11

Bimorphisms ⊗ Cu should linearize bilinear maps: BiCu ( S × T , R ) ∼ = Cu ( S ⊗ Cu T , R ) . Definition Cu -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≪ -preserving: f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ . f ( s ≪ , t ≪ ) ⊆ f ( s , t ) ≪ is cofinal. f is continuous: Approach: First define ⊗ in W. Definition W -bimorphism is f : S × T → R such that: f is additive in each variable. f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ . f is jointly ≺ -preserving: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ cofinal. f is continuous: 5 / 11

Tensor product in W Definition W -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≺ -preserving: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ . f is continuous: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ cofinal. 6 / 11

Tensor product in W Definition W -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≺ -preserving: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ . f is continuous: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ cofinal. on tensor product S ⊗ alg T of monoids, let ≺ be induced by 6 / 11

Tensor product in W Definition W -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≺ -preserving: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ . f is continuous: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ cofinal. on tensor product S ⊗ alg T of monoids, let ≺ be induced by Definition i ≺ 0 � � i s ′ i ⊗ t ′ s ′ i ≺ s i , t ′ i s i ⊗ t i ⇔ i ≺ t i 6 / 11

Tensor product in W Definition W -bimorphism is f : S × T → R such that: f is additive in each variable. f is jointly ≺ -preserving: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ . f is continuous: f ( s ≺ , t ≺ ) ⊆ f ( s , t ) ≺ cofinal. on tensor product S ⊗ alg T of monoids, let ≺ be induced by Definition i ≺ 0 � � i s ′ i ⊗ t ′ s ′ i ≺ s i , t ′ i s i ⊗ t i ⇔ i ≺ t i Lemma S ⊗ W T := ( S ⊗ alg T , ≺ ) is W -semigroup. S × T → S ⊗ W T is W -bimorphism with universal property: ∼ = � BiW ( S × T , R ) . W ( S ⊗ W T , R ) 6 / 11

Tensor product in Cu The tensor product of Cu -semigroups S and T is: S ⊗ Cu T := γ ( S ⊗ W T ) . 7 / 11