The Gelfand spectrum of a noncommutative C*-algebraecoas/ecoas_slides/Landsman... · 2010. 10. 28. · Klaas Landsman Radboud Universiteit Nijmegen Functional Analysis Seminar Dartmouth

The Gelfand spectrum of a noncommutative C*-algebra

A topos-theoretic approach

Klaas LandsmanRadboud Universiteit Nijmegen

Functional Analysis SeminarDartmouth College, 25 October 2010

LiteratureI. M. Gelfand, M. A. Naimark, On the imbedding of normed rings into the ring of operators on a Hilbert space, Math. Sbornik 12 (2), 197–217 (1943)

S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, New York, 1994)

J. Butterfield and C. Isham, A topos perspective on the Kochen-Specker Theorem. I., International Journal of Theoretical Physics 37, 2669-2733 (1998)

B. Banaschewski & C. J. Mulvey, A globalisation of the Gelfand duality theorem Annals of Pure and Applied Logic 137, 62-103 (2006)

C. Heunen, N. P. Landsman, B. Spitters, A topos for algebraic quantum theory Communications in Mathematical Physics 291, 63-110 (2009), arXiv:0709.4364

C. Heunen, N. P. Landsman, B. Spitters, Bohrification of operator algebras and quantum logic, arXiv:0905.2275, Synthese (2011)

C. Heunen, N. P. Landsman, B. Spitters, S. Wolters, The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach, arXiv:1010.2050 Journal of the Australian Mathematical Society (2011)

Gelfand duality

Compact space X ☞ C(X) ≡ C(X, ℂ) as commutative C*-algebra i.e. complex Banach space + associative involutive algebra, ||ab|| ≤ ||a|| ||b||, ||a*a|| =||a||

Commutative unital C*-algebra A ☞ spectrum Σ(A) as space Σ(A) realized either as characters A → ℂ or as pure states on A, both with weak*-topology

➜ compact Hausdorff spaces ≃ (unital commutative C*-algebras)categorical duality - C and Σ functors, inverse mod isomorphism: Σ(C(X)) ≅ X, C(Σ(A)) ≅ A

‘quantum’ jump: “noncommutative spaces” ≃ (C*-algebras)

Amazing fact: by Gelfand-Naimark Theorem, A ⊆ B(H) ➜ “noncommutative spaces” relate to Hilbert spaces H

Why ℂ ? Alternative ways to capture spaces algebraically?

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Order-theoretic approachSpace X ☞ topology O(X) as lattice, U ≤ W iff U ⊆ W (lattice = partially ordered set with finite infima ∧ (g.l.b) and suprema ∨ (l.u.b))

Fine structure: O(X) is special lattice called frame i.e. complete lattice such that U⋀⋁i {Wi} = ⋁i {U ⋀ Wi} (‘infinite’ version of distributivity: U∧(V∨W)=(U∧V)∨ (U∧W), cf. classical logic)

Point of frame F is frame map p: F → 2 ≡ {0,1} = O(⊛)

Points(F) topologized by open sets {p| p(U)=1}, U ∈ F

Frame F is called spatial if F ≌ O(Points(F))

Space X is called sober if X ≌ Points(O(X))

➡ “Sober” duality: Sober spaces ≃ (spatial frames) (Stone duality: Stone spaces ≃ (Boolean algebras) is nontrivial special case)

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Pointfree spaces and logic

“Sober” duality: sober spaces ≃ (spatial frames)

Leap of faith: “pointfree spaces”(locales) ≃ (frames)

Surprising fact: pointfree spaces relate to logic

Heyting algebra is (distributive) lattice H with top ⊤, bottom ⊥, and map ⇒: H → H such that A ≤ (B ⇒ C) iff (A ∧ B) ≤ C

Heyting algebras describe intuitionistic propositional logic, with negation ¬ A ≔ (A ⇒ ⊥): typically A ∨ ¬ A ≠ ⊤, ¬ ¬A ≠ A (Boolean algebras are Heyting, with (A ⇒ B) = (¬ A ∨ B) defined by complementation ¬)

Frame ↔ complete Heyting algebra: (B ⇒ C) = ⋁{A | (A ⋀ B) ≤ C}

So: sober spaces ☞ spatial frames ☞ pointfree spaces ☞ logic (cpt spaces ☞ comm. C*-algebra’s ☞ noncommutative spaces ☞ Hilbert space)

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Constructive Gelfand duality• Constructive mathematics: no Axiom of Choice or Law of Excluded Middle (‘P or not-P’)

and hence no proof by contradiction/reductio ad absurdum (L. E. J. Brouwer, E. Bishop)

(For me), use of constructive methods is not matter of philosophical taste: non-constructive proofs are valid in set theory, but problematic in topos theory

• Many classical results not valid constructively, e.g. Gelfand Duality A ≌ C(X, ℂ) But: classically equivalent to constructively valid “pointfree” result A ≌ “C”(“X”, “ℂ”)

1. Constructive Gelfand spectrum “X” of commutative C*-algebra A is pointfree space i.e. frame (particular lattice) seen as object in category (frames) = locales

2. “C”(“X”, “Y”) := Frm(“Y”, “X”) i.e. {Frame maps from “X” to “Y”}

Classically (i.e. in Set Theory): “X” = O(X) for usual Gelfand spectrum X = Σ(A)

Classically: for sober spaces X, Y: Frm(O(X), O(Y)) ≌ C(X,Y), (f ↔ f )

Hence classically “C”(“X”, “ℂ”) ≌ C(X, ℂ) which recovers usual Gelfand duality

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Intermezzo: topos theory

Invented by Grothendieck (1960) as language for algebraic geometry

Axiomatized (and hence generalized) by Lawvere & Tierney (1970) as language for all of mathematics: topos is category with terminal object (∀S ∃! S → *),

pullbacks (×), exponentials (X ), subobject classifier (A ⊆ S ↔ χA: S → {0,1})

Topos theory is generalization of set theory in following sense: Topos is category with exactly the structure required to “interpret” constructive mathematics, e.g. notion of group in category Top (Grothendieck, Kan)

Theorem from set theory is valid in arbitrary topos iff proof is constructive

Example: C*-algebras, frames, locales, constructive Gelfand duality: OK

Applications to foundations of: mathematics (Lawvere, Joyal, Moerdijk), classical mechanics (Lawvere, Bell), quantum mechanics (Isham et al., L., et al.)

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Noncommutative Gelfand spectrum

A = unital C*-algebra (in Sets, or even in some other topos)

Poset C(A) of unital commutative C*-subalgebras of A

Topos Sets of functors F: C(A) → Sets [C(A) seen as category] ≌

topos Sh(C(A)) of sheaves [C(A) seen as space in Alexandrov topology]

“Tautological” functor A: C↦ C (on arrows, C ≤ D ↦ i: C ↪ D)

This functor A is a unital commutative C*-algebra in the topos T(A)

• Operator algebra question: to what extent is A determined by A, i.e., by its unital commutative C*-subalgebras ordered by inclusion?

➡ A has (pointfree) Gelfand spectrum Σ(A) in Sh(C(A)): Σ(A) is a sheaf

C(A)

External description

C*-algebra A determines commutative C*-algebra A in topos Sh(C(A))

A has ‘internal’ pointfree Gelfand spectrum Σ(A) in Sh(C(A)): hard to grasp

But: pointfree spaces X in sheaf toposes Sh(“Y”) have “external description” in set theory (Fourman-Scott, Joyal-Tierney, Johnstone, 1980):

X in Sh(Y) ≌ continuous map “X” → “Y” in Sets (for pointfree space “X”)

➜ Gelfand spectrum Σ(A) of C*-algebra A in topos Sh(C(A)) is equivalent to “continuous” map “ΣA” → “C(A)” in Sets, with “C(A)” = O(C(A))

Surprise: frame “ΣA”= Σ(A)(C(A)) is spatial i.e. “ ΣA” = O(ΣA)

➜ “External Gelfand spectrum” of A is just a fibration ΣA → C(A) in Sets

(Cf. Dauns-Hofmann Theorem: A realized as sheaf over its center Z(A), often trivial)

Explicit resultΣA = ∐C∈ C(A) Σ(C) topologized by declaring U ⊂ ΣA open iff:

1. UC ≡ U ∩ Σ(C) ∈ O(Σ(C))

2. For all D ⊇ C: if λ ∈ Σ (D) satisfies λ| C ∈ UC , then λ ∈ UD

Projection ΣA → C(A) is obvious one: Σ (C) ∋ λ ↦ C ∈ C(A)

This external Gelfand spectrum is highly non-Hausdorff!

Simplification if projection lattice P(A) of A generates A (e.g., A is von Neumann or Rickart C*-algebra - take A = n × n matrices)

O(ΣA) ≌ {S: C(A) → P(A) | S(C) ∈ P(C), S(C) ≦ S(D) if C ⊆ D}

lattice w.r.t. pointwise order i.e. S ≤ T iff S(C) ≦ S(T) in P(A)

Interpretation in quantum logic: new notion of “proposition”

Summary

Noncommutative geometry: spaces ☞ commutative C*-algebras

☞ general C*-algebras ≃ noncommutative spaces ☞ Hilbert space

Pointfree topology: spaces ☞ spatial frames ☞ general frames

≃ pointfree spaces (locales) ☞ logic ☞ topos theory

Connection between noncommutative and pointfree spaces:

C*-algebra A defines commutative C*-algebra A in topos Sh(C(A))

A has internal Gelfand spectrum Σ(A) as pointfree space in Sh(C(A))

Sheaf Σ(A) has external description as (fibered) space in Sets

External Gelfand spectrum ΣA → C(A) of A is explicitly computable