Functorial spectra anddiscretization of C*-algebras

Chris Heunen

1 / 13

Introduction

Equivalence cCstar KHausopHom(−,C)

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,

cCstar KHausop

Cstar Cop

Spec

F

I

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

3 / 13

Obstruction

Theorem: If C has strict initial object ∅ and I continuous,

cCstar KHausop

Cstar Cop

Spec

F

I

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

Proof :

1. define K : cCstar→ Cop 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,

cCstar KHausop

Cstar Cop

Spec

F

I

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,

cCstar KHausop

Cstar Cop

Spec

F

I

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

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

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]C(A) ' C(B) =⇒ (A, ◦) ' (B, ◦) for a ◦ b = 12(ab+ ba)

I Quasi-Jordan: if A not C2 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 BC(A) ' C(B) and dim(A)

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14]projection valued measures compatible when commuteany 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 marginalsany 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 onesscattered: 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

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14]projection valued measures compatible when commuteany 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 marginalsany 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 onesscattered: 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

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14]projection valued measures compatible when commuteany 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 marginalsany 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 onesscattered: 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

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14]projection valued measures compatible when commuteany 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 marginalsany 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 onesscattered: 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

Combinatorial structure

I C(A) can encode graphs: [H & Fritz & Reyes 14]projection valued measures compatible when commuteany 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 marginalsany 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 onesscattered: 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

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

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

Active lattices

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

I May replace AWstarC→ Poset with AWstar Proj−→ Poset

Not full and faithful

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

AWstar

Poset Group

Proj U

7 / 13

Active lattices

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

I May replace AWstarC→ Poset with AWstar Proj−→ Poset

Not full and faithful

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

AWstar

Poset Group

Proj U

7 / 13

Active lattices

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

I May replace AWstarC→ Poset with AWstar Proj−→ Poset

Not full and faithful

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

AWstar

Poset Group

Proj U

7 / 13

Active lattices

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

I May replace AWstarC→ Poset with AWstar Proj−→ Poset

Not full and faithful

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

AWstar

Poset Group

Proj U

p

upu∗1− 2pu

7 / 13

Active lattices

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

I May replace AWstarC→ Poset with AWstar Proj−→ Poset

Not full and faithful

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

AWstar

Poset Group

Proj U

p

upu∗1− 2pu

ActLat

7 / 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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

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

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

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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

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

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

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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

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

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

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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

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

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

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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

I Definition: an active lattice is a complete orthomodular latticeP , together with a group G generated by P ' Proj(F (P )), andan 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

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)I if A = Mn(C) type I≥2, then Sym(A) = det

−1(Sym(C))I If A type I∞, II, III, then Sym(A) = U(A)

I A piecewise C*-algebra is reflexive symmetric � ⊆ A×A withpartial structure (addition, multiplication) such that S ⊆ A withS2 ⊆ � extends to commutative C*-algebra T with T 2 ⊆ �.

I There is equivalence pAWstar pCBool COrthoProj

F

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

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

8 / 13

Discretization

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

Definition: a discretization of a C*-algebra A is a morphism

A M

C(X) `∞(X)

φ

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 WstarI Mn(C(X)) ↪→Mn(`∞(X)) is faithful functorial into WstarI Mn(C(X)) ↪→Mn(CX) is faithful functorial into proCstarI A 7→ limI A/I faithful functorial into Wstar or proCstar

for residually finite-dimensional subhomogeneous A

9 / 13

Discretization

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

Definition: a discretization of a C*-algebra A is a morphism

A M

C(X) `∞(X)

φ

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 WstarI Mn(C(X)) ↪→Mn(`∞(X)) is faithful functorial into WstarI Mn(C(X)) ↪→Mn(CX) is faithful functorial into proCstarI A 7→ limI A/I faithful functorial into Wstar or proCstar

for residually finite-dimensional subhomogeneous A

9 / 13

Discretization

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

Definition: a discretization of a C*-algebra A is a morphism

A M

C(X) `∞(X)

φ

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 WstarI Mn(C(X)) ↪→Mn(CX) is faithful functorial into proCstarI A 7→ limI A/I faithful functorial into Wstar or proCstar

for residually finite-dimensional subhomogeneous A

9 / 13

Discretization

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

Definition: a discretization of a C*-algebra A is a morphism

A M

C(X) `∞(X)

φ

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 WstarI Mn(C(X)) ↪→Mn(`∞(X)) is faithful functorial into WstarI Mn(C(X)) ↪→Mn(CX) is faithful functorial into proCstar

I A 7→ limI A/I faithful functorial into Wstar or proCstarfor residually finite-dimensional subhomogeneous A

9 / 13

Discretization

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

Definition: a discretization of a C*-algebra A is a morphism

A M

C(X) `∞(X)

φ

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 WstarI Mn(C(X)) ↪→Mn(`∞(X)) is faithful functorial into WstarI Mn(C(X)) ↪→Mn(CX) is faithful functorial into proCstarI A 7→ limI A/I faithful functorial into Wstar or proCstar

for residually finite-dimensional subhomogeneous A

9 / 13

Discretization: another obstruction

Definition: State∫−dµ : C(X)→ C diffuse when µ has no atoms.

Pair C,D ∈ C(A) is relatively diffuse when every extension of purestate of D to A restricts to diffuse state on C.

Example: C = L∞[0, 1], D = `∞(N), A = B(L2[0, 1]) [Kadison-Singer]

Theorem: If C '

D '

C(X)

A

C(Y )

`∞(X)

M

`∞(Y )

φφC

φD

then φC(δx)φD(δy) = 0.

[H & Reyes, 2016]

Corollary: Functorial discretizations Cstar→ AWstar map A to 0

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

10 / 13

Discretization: another obstruction

Definition: State∫−dµ : C(X)→ C diffuse when µ has no atoms.

Pair C,D ∈ C(A) is relatively diffuse when every extension of purestate of D to A restricts to diffuse state on C.

Example: C = L∞[0, 1], D = `∞(N), A = B(L2[0, 1]) [Kadison-Singer]

Theorem: If C '

D '

C(X)

A

C(Y )

`∞(X)

M

`∞(Y )

φφC

φD

then φC(δx)φD(δy) = 0.

[H & Reyes, 2016]

Corollary: Functorial discretizations Cstar→ AWstar map A to 0

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

10 / 13

Discretization: another obstruction

Definition: State∫−dµ : C(X)→ C diffuse when µ has no atoms.

Pair C,D ∈ C(A) is relatively diffuse when every extension of purestate of D to A restricts to diffuse state on C.

Example: C = L∞[0, 1], D = `∞(N), A = B(L2[0, 1]) [Kadison-Singer]

Theorem: If C '

D '

C(X)

A

C(Y )

`∞(X)

M

`∞(Y )

φφC

φD

then φC(δx)φD(δy) = 0.

[H & Reyes, 2016]

Corollary: Functorial discretizations Cstar→ AWstar map A to 0

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

10 / 13

Discretization: another obstruction

Definition: State∫−dµ : C(X)→ C diffuse when µ has no atoms.

Pair C,D ∈ C(A) is relatively diffuse when every extension of purestate of D to A restricts to diffuse state on C.

Example: C = L∞[0, 1], D = `∞(N), A = B(L2[0, 1]) [Kadison-Singer]

Theorem: If C '

D '

C(X)

A

C(Y )

`∞(X)

M

`∞(Y )

φφC

φD

then φC(δx)φD(δy) = 0.

[H & Reyes, 2016]

Corollary: Functorial discretizations Cstar→ AWstar map A to 0

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

10 / 13

Discretization: another obstruction

Definition: State∫−dµ : C(X)→ C diffuse when µ has no atoms.

Pair C,D ∈ C(A) is relatively diffuse when every extension of purestate of D to A restricts to diffuse state on C.

Example: C = L∞[0, 1], D = `∞(N), A = B(L2[0, 1]) [Kadison-Singer]

Theorem: If C '

D '

C(X)

A

C(Y )

`∞(X)

M

`∞(Y )

φφC

φD

then φC(δx)φD(δy) = 0.

[H & Reyes, 2016]

Corollary: Functorial discretizations Cstar→ AWstar map A to 0

Lesson: Discretization needs other global coherence structure ofprojections 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’

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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

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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):category whose objects behave a lot like setsin 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)

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