Functorial spectra and discretization of C*-algebras

Chris Heunen

1 / 13 Introduction

Hom(−, C) Equivalence cCstar KHausop Hom(−, C)

1. Many attempts at noncommutative version, none functorial

2. Idea: noncommutative space = set of commutative subspaces

3. Active lattices: ‘functions’ on noncommutative space

4. Discretization: ‘continuous’ functions on noncommutative space

2 / 13 Obstruction Theorem: If C has strict initial object ∅ and I continuous, Spec cCstar KHausop

I Cstar Cop F

then F (Mn(C)) = ∅ for all n > 2. [Berg & H, 2014]

3 / 13 Obstruction Theorem: If C has strict initial object ∅ and I continuous, Spec cCstar KHausop

I Cstar Cop F

then F (Mn(C)) = ∅ for all n > 2. [Berg & H, 2014] Proof: op 1. define K : cCstar → C by A 7→ limC⊆A I(Spec(C)) 2. then K(C) = I(Spec(C)) for commutative C 3. K is final with this property

4. I ◦ Spec preserves limits, so K(A) = I(Spec(colimC⊆A C))

5. Kochen-Specker: colimC⊆Mn(C) Proj(C) is Boolean algebra 1 6. so F (Mn(C)) → K(Mn(C)) = ∅

3 / 13 Obstruction Theorem: If C has strict initial object ∅ and I continuous, Spec cCstar KHausop

I Cstar Cop F

then F (Mn(C)) = ∅ for all n > 2. [Berg & H, 2014] Remarks:

I Rules out sets, schemes, locales, quantales, ringed toposes, ... I Not just Mn(C): W*-algebras without summands C or M2(C) I Not just Gelfand duality: also Stone, Zariski, Pierce

I Remarkable that physics theorem affects all rings

I Ways out: different limit behaviour, square not commutative

3 / 13 Obstruction Theorem: If C has strict initial object ∅ and I continuous, Spec cCstar KHausop

I Cstar Cop F

then F (Mn(C)) = ∅ for all n > 2. [Berg & H, 2014] Remarks:

I Rules out sets, schemes, locales, quantales, ringed toposes, ... I Not just Mn(C): W*-algebras without summands C or M2(C) I Not just Gelfand duality: also Stone, Zariski, Pierce

I Remarkable that physics theorem affects all rings

I Ways out: different limit behaviour, square not commutative Lesson: Set of commutative subalgebras important

3 / 13 I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba) 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B

Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

4 / 13 I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba) 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B

Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

4 / 13 I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba) 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B

Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

4 / 13 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B

Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba)

4 / 13 I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B

Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba) 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

4 / 13 Commutative subalgebras Definition: for C*-algebra A, let C(A) = {C ⊆ A commutative} partially ordered by inclusion. [H & Landsman & Spitters 09]

How much does C(A) know about A?

I Not everything: [Connes 75] there is A 6' Aop, but C(A) 'C(Aop)

I Everything commutative: if A, B commutative, [Mendivil 99] C(A) 'C(B) =⇒ A ' B

I Jordan: if A, B are W* have no I2 summand, [Harding & Doering 10] 1 C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 2 (ab + ba) 2 I Quasi-Jordan: if A not C or M2(C), [Hamhalter 11] C(A) 'C(B) =⇒ (A, ◦) ' (B, ◦) quasi-linear

I Type and dimension: [Lindenhovius 15] C(A) 'C(B) and A is W*/AW* =⇒ so is B C(A) 'C(B) and dim(A) < ∞ =⇒ A ' B 4 / 13 I C(A) can encode simplicial complexes: [Kunjwal & H & Fritz 14] positive operator valued measures compatible when marginals any simplical complex can be realised as in some C(A)

I C(A) domain ⇐⇒ A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C ∈ C(A) scattered; isolated points dense

Then C(A) is compact Hausdorff in Lawson ; C(C(A))?

Lesson: C(A) has lots of structure, interesting to study

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14] projection valued measures compatible when commute any graph can be realised as PVMs in some C(A)

5 / 13 I C(A) domain ⇐⇒ A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C ∈ C(A) scattered; isolated points dense

Then C(A) is compact Hausdorff in Lawson topology; C(C(A))?

Lesson: C(A) has lots of structure, interesting to study

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14] projection valued measures compatible when commute any graph can be realised as PVMs in some C(A)

I C(A) can encode simplicial complexes: [Kunjwal & H & Fritz 14] positive operator valued measures compatible when marginals any simplical complex can be realised as POVMs in some C(A)

5 / 13 Then C(A) is compact Hausdorff in Lawson topology; C(C(A))?

Lesson: C(A) has lots of structure, interesting to study

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14] projection valued measures compatible when commute any graph can be realised as PVMs in some C(A)

I C(A) can encode simplicial complexes: [Kunjwal & H & Fritz 14] positive operator valued measures compatible when marginals any simplical complex can be realised as POVMs in some C(A)

I C(A) domain ⇐⇒ A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C ∈ C(A) scattered; isolated points dense

5 / 13 Lesson: C(A) has lots of structure, interesting to study

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14] projection valued measures compatible when commute any graph can be realised as PVMs in some C(A)

I C(A) can encode simplicial complexes: [Kunjwal & H & Fritz 14] positive operator valued measures compatible when marginals any simplical complex can be realised as POVMs in some C(A)

I C(A) domain ⇐⇒ A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C ∈ C(A) scattered; isolated points dense

Then C(A) is compact Hausdorff in Lawson topology; C(C(A))?

5 / 13 Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14] projection valued measures compatible when commute any graph can be realised as PVMs in some C(A)

I C(A) can encode simplicial complexes: [Kunjwal & H & Fritz 14] positive operator valued measures compatible when marginals any simplical complex can be realised as POVMs in some C(A)

I C(A) domain ⇐⇒ A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C ∈ C(A) scattered; isolated points dense

Then C(A) is compact Hausdorff in Lawson topology; C(C(A))?

Lesson: C(A) has lots of structure, interesting to study

5 / 13 Lesson: Not just partial order C(A) important, also action

Characterization

When is a partially ordered set of the form C(A)?

If A has weakly terminal abelian subalgebra C(X): [H 14] 1. C(A) 'C(C(X))

2. C(C(X)) ' P (X) o S(X)

3. Axiomatization known for partition lattice P (X) [Firby 73]

4. Axiomatize monoid S(X) of epimorphisms X  X 5. Axiomatize semidirect product of posets and monoids

6 / 13 Characterization

When is a partially ordered set of the form C(A)?

If A has weakly terminal abelian subalgebra C(X): [H 14] 1. C(A) 'C(C(X))

2. C(C(X)) ' P (X) o S(X)

3. Axiomatization known for partition lattice P (X) [Firby 73]

4. Axiomatize monoid S(X) of epimorphisms X  X 5. Axiomatize semidirect product of posets and monoids

Lesson: Not just partial order C(A) important, also action

6 / 13 C Proj I May replace AWstar → Poset with AWstar −→ Poset Not full and faithful

I Use action to make it full and faithful [H & Reyes 14]

AWstar Proj U Poset Group

Active lattices

I Restrict to ‘noncommutative sets and functions’ AW*-algebras: abundance of projections [Kaplansky 51]

7 / 13 I Use action to make it full and faithful [H & Reyes 14]

AWstar Proj U Poset Group

Active lattices

I Restrict to ‘noncommutative sets and functions’ AW*-algebras: abundance of projections [Kaplansky 51]

C Proj I May replace AWstar → Poset with AWstar −→ Poset Not full and faithful

7 / 13 Active lattices

I Restrict to ‘noncommutative sets and functions’ AW*-algebras: abundance of projections [Kaplansky 51]

C Proj I May replace AWstar → Poset with AWstar −→ Poset Not full and faithful

I Use action to make it full and faithful [H & Reyes 14]

AWstar Proj U Poset Group

7 / 13 Active lattices

I Restrict to ‘noncommutative sets and functions’ AW*-algebras: abundance of projections [Kaplansky 51]

C Proj I May replace AWstar → Poset with AWstar −→ Poset Not full and faithful

I Use action to make it full and faithful [H & Reyes 14]

AWstar Proj U Poset Group p 1 − 2p upu∗ u

7 / 13 Active lattices

I Restrict to ‘noncommutative sets and functions’ AW*-algebras: abundance of projections [Kaplansky 51]

C Proj I May replace AWstar → Poset with AWstar −→ Poset Not full and faithful

I Use action to make it full and faithful [H & Reyes 14]

AWstar Proj U Poset ActLat Group p 1 − 2p upu∗ u

7 / 13 I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ . Proj I There is equivalence pAWstar pCBool COrtho F

I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful Lesson: ‘Noncommutative sets’ have hidden actionn

Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

8 / 13 Proj I There is equivalence pAWstar pCBool COrtho F

I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful Lesson: ‘Noncommutative sets’ have hidden actionn

Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean ring Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ .

8 / 13 I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful Lesson: ‘Noncommutative sets’ have hidden actionn

Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean ring Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ . Proj I There is equivalence pAWstar pCBool COrtho F

8 / 13 Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful Lesson: ‘Noncommutative sets’ have hidden actionn

Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean ring Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ . Proj I There is equivalence pAWstar pCBool COrtho F

I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

8 / 13 Lesson: ‘Noncommutative sets’ have hidden actionn

Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean ring Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ . Proj I There is equivalence pAWstar pCBool COrtho F

I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful

8 / 13 Active lattices: details

I Symmetry group Sym(A) ⊆ U(A) generated by {1 − 2p}

I if A commutative, then Sym(A) is Boolean ring Proj(A) −1 I if A = Mn(C) type I≥2, then Sym(A) = det (Sym(C)) I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric ⊆ A × A with partial structure (addition, multiplication) such that S ⊆ A with S2 ⊆ extends to commutative C*-algebra T with T 2 ⊆ . Proj I There is equivalence pAWstar pCBool COrtho F

I Definition: an active lattice is a complete orthomodular lattice P , together with a group G generated by P ' Proj(F (P )), and an action of G on P with induced action on F (P ) conjugation.

Theorem: A 7→ (Proj(A), Sym(A), conjugation) is full and faithful Lesson: ‘Noncommutative sets’ have hidden actionn

8 / 13 I Free products gives faithful φ into Cstar, but not functorial

I Colimits give functorial φ into Cstar, but not faithful I C ⊕ K(H) ,→ B(H) is faithful functorial into Wstar ∞ I Mn(C(X)) ,→ Mn(` (X)) is faithful functorial into Wstar X I Mn(C(X)) ,→ Mn(C ) is faithful functorial into proCstar I A 7→ limI A/I faithful functorial into Wstar or proCstar for residually finite-dimensional subhomogeneous A

Discretization How go from ‘noncommutative sets’ to ‘noncommutative ’?

Definition: a discretization of a C*-algebra A is a morphism A M φ C(X) `∞(X) Where can M live?

9 / 13 I C ⊕ K(H) ,→ B(H) is faithful functorial into Wstar ∞ I Mn(C(X)) ,→ Mn(` (X)) is faithful functorial into Wstar X I Mn(C(X)) ,→ Mn(C ) is faithful functorial into proCstar I A 7→ limI A/I faithful functorial into Wstar or proCstar for residually finite-dimensional subhomogeneous A

Discretization How go from ‘noncommutative sets’ to ‘noncommutative topologies’?

Definition: a discretization of a C*-algebra A is a morphism A M φ C(X) `∞(X) Where can M live?

I Free products gives faithful φ into Cstar, but not functorial

I Colimits give functorial φ into Cstar, but not faithful

9 / 13 ∞ I Mn(C(X)) ,→ Mn(` (X)) is faithful functorial into Wstar X I Mn(C(X)) ,→ Mn(C ) is faithful functorial into proCstar I A 7→ limI A/I faithful functorial into Wstar or proCstar for residually finite-dimensional subhomogeneous A

Discretization How go from ‘noncommutative sets’ to ‘noncommutative topologies’?

Definition: a discretization of a C*-algebra A is a morphism A M φ C(X) `∞(X) Where can M live?

I Free products gives faithful φ into Cstar, but not functorial

I Colimits give functorial φ into Cstar, but not faithful I C ⊕ K(H) ,→ B(H) is faithful functorial into Wstar

9 / 13 I A 7→ limI A/I faithful functorial into Wstar or proCstar for residually finite-dimensional subhomogeneous A

Discretization How go from ‘noncommutative sets’ to ‘noncommutative topologies’?

Definition: a discretization of a C*-algebra A is a morphism A M φ C(X) `∞(X) Where can M live?

I Free products gives faithful φ into Cstar, but not functorial

I Colimits give functorial φ into Cstar, but not faithful I C ⊕ K(H) ,→ B(H) is faithful functorial into Wstar ∞ I Mn(C(X)) ,→ Mn(` (X)) is faithful functorial into Wstar X I Mn(C(X)) ,→ Mn(C ) is faithful functorial into proCstar

9 / 13 Discretization How go from ‘noncommutative sets’ to ‘noncommutative topologies’?

Definition: a discretization of a C*-algebra A is a morphism A M φ C(X) `∞(X) Where can M live?

I Free products gives faithful φ into Cstar, but not functorial

I Colimits give functorial φ into Cstar, but not faithful I C ⊕ K(H) ,→ B(H) is faithful functorial into Wstar ∞ I Mn(C(X)) ,→ Mn(` (X)) is faithful functorial into Wstar X I Mn(C(X)) ,→ Mn(C ) is faithful functorial into proCstar I A 7→ limI A/I faithful functorial into Wstar or proCstar for residually finite-dimensional subhomogeneous A

9 / 13 ∞ ∞ 2 Example: C = L [0, 1], D = ` (N), A = B(L [0, 1]) [Kadison-Singer]

∞ Theorem: If C 'C(X) ` (X) then φC (δx)φD(δy) = 0. φ φ C A M φD D ' C(Y ) `∞(Y ) [H & Reyes, 2016]

Corollary: Functorial discretizations Cstar → AWstar map A to 0

Lesson: Discretization needs other global coherence structure of projections than that of AW*-algebras.

Discretization: another obstruction R Definition: State − dµ: C(X) → C diffuse when µ has no atoms. Pair C,D ∈ C(A) is relatively diffuse when every extension of pure state of D to A restricts to diffuse state on C.

10 / 13 ∞ Theorem: If C 'C(X) ` (X) then φC (δx)φD(δy) = 0. φ φ C A M φD D ' C(Y ) `∞(Y ) [H & Reyes, 2016]

Corollary: Functorial discretizations Cstar → AWstar map A to 0

Lesson: Discretization needs other global coherence structure of projections than that of AW*-algebras.

Discretization: another obstruction R Definition: State − dµ: C(X) → C diffuse when µ has no atoms. Pair C,D ∈ C(A) is relatively diffuse when every extension of pure state of D to A restricts to diffuse state on C.

∞ ∞ 2 Example: C = L [0, 1], D = ` (N), A = B(L [0, 1]) [Kadison-Singer]

10 / 13 Corollary: Functorial discretizations Cstar → AWstar map A to 0

Lesson: Discretization needs other global coherence structure of projections than that of AW*-algebras.

Discretization: another obstruction R Definition: State − dµ: C(X) → C diffuse when µ has no atoms. Pair C,D ∈ C(A) is relatively diffuse when every extension of pure state of D to A restricts to diffuse state on C.

∞ ∞ 2 Example: C = L [0, 1], D = ` (N), A = B(L [0, 1]) [Kadison-Singer]

∞ Theorem: If C 'C(X) ` (X) then φC (δx)φD(δy) = 0. φ φ C A M φD D ' C(Y ) `∞(Y ) [H & Reyes, 2016]

10 / 13 Lesson: Discretization needs other global coherence structure of projections than that of AW*-algebras.

Discretization: another obstruction R Definition: State − dµ: C(X) → C diffuse when µ has no atoms. Pair C,D ∈ C(A) is relatively diffuse when every extension of pure state of D to A restricts to diffuse state on C.

∞ ∞ 2 Example: C = L [0, 1], D = ` (N), A = B(L [0, 1]) [Kadison-Singer]

∞ Theorem: If C 'C(X) ` (X) then φC (δx)φD(δy) = 0. φ φ C A M φD D ' C(Y ) `∞(Y ) [H & Reyes, 2016]

Corollary: Functorial discretizations Cstar → AWstar map A to 0

10 / 13 Discretization: another obstruction R Definition: State − dµ: C(X) → C diffuse when µ has no atoms. Pair C,D ∈ C(A) is relatively diffuse when every extension of pure state of D to A restricts to diffuse state on C.

∞ ∞ 2 Example: C = L [0, 1], D = ` (N), A = B(L [0, 1]) [Kadison-Singer]

∞ Theorem: If C 'C(X) ` (X) then φC (δx)φD(δy) = 0. φ φ C A M φD D ' C(Y ) `∞(Y ) [H & Reyes, 2016]

Corollary: Functorial discretizations Cstar → AWstar map A to 0

Lesson: Discretization needs other global coherence structure of projections than that of AW*-algebras.

10 / 13 Conclusion

I It pays to take commutative subalgebras seriously

I Functoriality crucial to ensure they fit together

I Leads to active lattices as ‘noncommutative sets’

I But not good enough for ‘noncommutative topology’

11 / 13 References

I B. van den Berg, C. Heunen ‘Extending obstructions to noncommutative functorial spectra’ Theory and Applications of Categories 29(17):457–474, 2014

I C. Heunen, N. P. Landsman, B. Spitters ‘A topos for algebraic quantum theory’ Communications in Mathematical Physics 291:63–110, 2009

I C. Heunen ‘Characterizations of categories of commutative C*-subalgebras’ Communications in Mathematical Physics 331(1):215-238, 2014

I C. Heunen, M. L. Reyes ‘Active lattices determine AW*-algebras’ Journal of Mathematical Analysis and Applications 416:289-313, 2014

I C. Heunen, A. Lindenhovius ‘Domains of commutative C*-subalgebras’ Logic in Computer Science 450–461, 2015

I C. Heunen, M. L. Reyes ‘Discretization of C*-algebras’ Journal of Operator Theory to appear 2016

12 / 13 Topos trick

I Consider ‘contextual sets’ over C*-algebra A: assignment of set S(C) to each C ∈ C(A) such that C ⊆ D implies S(C) ,→ S(D)

I They form a topos T (A): whose objects behave a lot like sets in particular, it has a logic of its own!

I There is a canonical contextual set A given by C 7→ C

I T (A) believes that A is a commutative C*-algebra

I A has spectrum within T (A) corresponds externally to map into C(A)

13 / 13