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, ∞]. RP := Cu(UHFp) = 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, ∞]. RP := Cu(UHFp) = N[ 1

p] ∪ (0, ∞].

Cu(II1-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) ∼ = Cu

• γ(S), T
• .

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) ∼ = Cu

• γ(S), T
• .

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: ≺ has interpolation: a≺ is upward directed.

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: ≺ has interpolation: a≺ is upward directed. + preserves ≺: a≺ + b≺ ⊆ (a + b)≺.

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: ≺ has interpolation: a≺ is upward directed. + preserves ≺: a≺ + b≺ ⊆ (a + b)≺. + is continuous: a≺ + b≺ ⊆ (a + b)≺ 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: ≺ has interpolation: a≺ is upward directed. + preserves ≺: a≺ + b≺ ⊆ (a + b)≺. + is continuous: a≺ + b≺ ⊆ (a + b)≺ is cofinal. 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: ≺ has interpolation: a≺ is upward directed. + preserves ≺: a≺ + b≺ ⊆ (a + b)≺. + is continuous: a≺ + b≺ ⊆ (a + b)≺ is cofinal. 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: ≺ has interpolation: a≺ is upward directed. + preserves ≺: a≺ + b≺ ⊆ (a + b)≺. + is continuous: a≺ + b≺ ⊆ (a + b)≺ is cofinal. 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 is continuous: f(s≪, t≪) ⊆ f(s, t)≪ is cofinal.

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 is continuous: f(s≪, t≪) ⊆ f(s, t)≪ is cofinal. 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 is continuous: f(s≪, t≪) ⊆ f(s, t)≪ is cofinal. Approach: First define ⊗ 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.

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.

• n 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.

• n tensor product S ⊗alg T of monoids, let ≺ be induced by

Definition

• i s′

i ⊗ t′ i ≺0 i si ⊗ ti

⇔ s′

i ≺ si, t′ i ≺ ti

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.

• n tensor product S ⊗alg T of monoids, let ≺ be induced by

Definition

• i s′

i ⊗ t′ i ≺0 i si ⊗ ti

⇔ s′

i ≺ si, t′ i ≺ ti

Lemma S ⊗W T := (S ⊗alg T, ≺) is W-semigroup. S × T → S ⊗W T is W-bimorphism with universal property: W(S ⊗W T, R)

∼ =

BiW(S × 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

Tensor product in Cu

The tensor product of Cu-semigroups S and T is: S ⊗Cu T := γ(S ⊗W T). Theorem S ⊗Cu T linearizes Cu-bimorphisms: BiCu(S × T, R) ∼ = Cu(S ⊗Cu T, R).

7 / 11

Tensor product in Cu

The tensor product of Cu-semigroups S and T is: S ⊗Cu T := γ(S ⊗W T). Theorem S ⊗Cu T linearizes Cu-bimorphisms: BiCu(S × T, R) ∼ = Cu(S ⊗Cu T, R). N = {0, 1, 2, . . . , ∞} is tensor unit: N ⊗Cu S ∼ = S ∼ = S ⊗Cu N.

7 / 11

Tensor product in Cu

The tensor product of Cu-semigroups S and T is: S ⊗Cu T := γ(S ⊗W T). Theorem S ⊗Cu T linearizes Cu-bimorphisms: BiCu(S × T, R) ∼ = Cu(S ⊗Cu T, R). N = {0, 1, 2, . . . , ∞} is tensor unit: N ⊗Cu S ∼ = S ∼ = S ⊗Cu N. Cu is a symmetric, monoidal category. Proof. BiCu(S × T, R) ∼ = BiW(S × T, R) ∼ = W(S ⊗W T, R) ∼ = Cu(γ(S ⊗W T), R) = Cu(S ⊗Cu T, R).

7 / 11

Examples of tensor products

For Rp := Cu(UHFp) = N[ 1

p] ∪ (0, ∞] have Rp ⊗ Rq ∼

= Rpq.

8 / 11

Examples of tensor products

For Rp := Cu(UHFp) = N[ 1

p] ∪ (0, ∞] have Rp ⊗ Rq ∼

= Rpq. S ∼ = Rp ⊗ S if and only if S is p-divisible and p-unperforated (pa ≤ pb ⇒ a ≤ b)

8 / 11

Examples of tensor products

For Rp := Cu(UHFp) = N[ 1

p] ∪ (0, ∞] have Rp ⊗ Rq ∼

= Rpq. S ∼ = Rp ⊗ S if and only if S is p-divisible and p-unperforated (pa ≤ pb ⇒ a ≤ b) For Z := Cu(Z) = N ∪ (0, ∞] have Z ⊗ Z ∼ = Z.

8 / 11

Examples of tensor products

For Rp := Cu(UHFp) = N[ 1

p] ∪ (0, ∞] have Rp ⊗ Rq ∼

= Rpq. S ∼ = Rp ⊗ S if and only if S is p-divisible and p-unperforated (pa ≤ pb ⇒ a ≤ b) For Z := Cu(Z) = N ∪ (0, ∞] have Z ⊗ Z ∼ = Z. S ∼ = Z ⊗ S if and only if S is almost divisible and almost unperforated.

8 / 11

Examples of tensor products

For Rp := Cu(UHFp) = N[ 1

p] ∪ (0, ∞] have Rp ⊗ Rq ∼

= Rpq. S ∼ = Rp ⊗ S if and only if S is p-divisible and p-unperforated (pa ≤ pb ⇒ a ≤ b) For Z := Cu(Z) = N ∪ (0, ∞] have Z ⊗ Z ∼ = Z. S ∼ = Z ⊗ S if and only if S is almost divisible and almost unperforated. {0, ∞} ⊗ S ∼ = Lat(S) - lattice of ideals in S (Scott-closed submonoids).

8 / 11

Cu is closed

Category Q such that Cu ⊆ Q full, hereditary. Functor τ : Q → Cu that is coreflection: Q(T, P) ∼ = Cu(T, τ(P)). Q admits right adjoint for its bimorphism functor: BiQ(S × T, P) ∼ = Q(S, [T, P]).

9 / 11

Cu is closed

Category Q such that Cu ⊆ Q full, hereditary. Functor τ : Q → Cu that is coreflection: Q(T, P) ∼ = Cu(T, τ(P)). Q admits right adjoint for its bimorphism functor: BiQ(S × T, P) ∼ = Q(S, [T, P]). The internal hom of Cu-semigroups S and T is: S, T := τ([S, T]).

9 / 11

Cu is closed

Category Q such that Cu ⊆ Q full, hereditary. Functor τ : Q → Cu that is coreflection: Q(T, P) ∼ = Cu(T, τ(P)). Q admits right adjoint for its bimorphism functor: BiQ(S × T, P) ∼ = Q(S, [T, P]). The internal hom of Cu-semigroups S and T is: S, T := τ([S, T]). Theorem T, is right adjoint to ⊗Cu T: Cu(S × T, R) ∼ = Cu(S, T, R). Cu is a closed, symmetric, monoidal category.

9 / 11

Examples of internal homs

For Rp = N[ 1

p] ∪ (0, ∞] have Rp, Rq = Rq if p divides q,

and Rp, Rq = R+ otherwise.

10/ 11

Examples of internal homs

For Rp = N[ 1

p] ∪ (0, ∞] have Rp, Rq = Rq if p divides q,

and Rp, Rq = R+ otherwise. For Z = N ∪ (0, ∞] , have Z, Z = Z.

10/ 11

Examples of internal homs

For Rp = N[ 1

p] ∪ (0, ∞] have Rp, Rq = Rq if p divides q,

and Rp, Rq = R+ otherwise. For Z = N ∪ (0, ∞] , have Z, Z = Z. For M := Cu(II1-factor), have R+, R+ = M, M, M = M.

10/ 11

Thank you.

11/ 11