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) < 1 =) A ' B Commutative subalgebras Definition: for C*-algebra A, let C(A) = fC ⊆ A commutativeg 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) < 1 =) A ' B Commutative subalgebras Definition: for C*-algebra A, let C(A) = fC ⊆ A commutativeg 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) < 1 =) A ' B Commutative subalgebras Definition: for C*-algebra A, let C(A) = fC ⊆ A commutativeg 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) < 1 =) A ' B Commutative subalgebras Definition: for C*-algebra A, let C(A) = fC ⊆ A commutativeg 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) < 1 =) A ' B Commutative subalgebras Definition: for C*-algebra A, let C(A) = fC ⊆ A commutativeg 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) = fC ⊆ A commutativeg 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) < 1 =) 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 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 2 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) 5 / 13 I C(A) domain () A scattered: [H & Lindenhovius 15] domain: directed suprema, all elements supremum of finite ones scattered: spectrum C 2 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 2 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 2 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 2 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.