Non-Commutative Gel'fand-Naımark Duality

Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras?

Non-commutative Gel’fand-Na˘ımark Duality

Paolo Bertozzini

Department of Mathematics and Statistics - Thammasat University - Bangkok.

Mahidol International College Seminar 28 March 2018 dedicated to the memory of John E.Roberts and Renzo Cirelli

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? Abstract 1

We outline a new attempt to obtain a non-commutative generalization of the well-known Gel’fand-Na˘ımark duality (between compact Hausdorﬀ topological spaces and commutative unital C*-algebras), where “geometric spectra” for unital non-commutative C*-algebras can be described via “non-commutative spaceoids”: suitable bundles of one-dimensional full C*-categories, equipped with a transition amplitude structure, satisfying certain saturation and uniformity conditions.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? Abstract 2

This work is a joint collaboration with:

I Natee Pitiwan (Chulalongkorn University)

I Roberto Conti (Sapienza Universit`adi Roma),

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? Outline

I Background

I Review of Commutative Gel’fand-Na˘ımark Duality I Previous Attempts I Basic Constructions

I The Transition Amplitude Bundle of a C*-algebra I The Transition Amplitude Space of a C*-algebra I Non-commutative Spaceoids I Non-commutative Gel’fand-Na˘ımark Duality

I The Section Functor Γ I The Spectrum Functor Σ I The Gel’fand Transform G I The Evaluation Transform E I Commutative and Non-commutative Gel’fand Duality

I What are the “Spectra” of C*-algebras?

• Background

I Commutative Gel’fand-Na˘ımark Duality:

I Categories of Commutative (Unital) C*-algebras I Categories of Compact Hausdorﬀ Spaces I Categorical Duality I Previous Attempts:

I Sectional Representations I Convex Spaces of States as Duals I Quantales I Topoi I Other Approaches

• Review of Commutative Gel’fand-Na˘ımark Duality

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? C*-algebras 1 Deﬁned by I.Gel’fand-M.Na˘ımark in 1943, are a “rigid” blend of algebra and topology, basic in functional analysis, non-commutative geometry and quantum physics.

A complex unital C*-algebra (C, ◦, ∗, +, ·, k k) is given by: I a complex associative unital involutive algebra: I a vector space (C, +, ·) over C, I an associative unital bilinear multiplication ◦ : C × C → C, I a conjugate-linear antimutiplicative involution ∗ : C → C, I a norm k k : C → R such that: I completeness: (C, +, ·, k k) is a Banach space, I submultiplicativity: kx ◦ yk ≤ kxk · kyk, for all x, y ∈ C, ∗ 2 I C*-property: kx ◦ xk = kxk , for all x ∈ C. A C*-algebra is Abelian (or commutative) if x ◦ y = y ◦ x.

I A unital ∗-homomorphism is a map φ : A → B between unital C*-algebras A, B such that for all x, y ∈ A:

φ(x ◦A y) = φ(x) ◦B φ(y), and φ(1A) = 1B.

I Examples of unital C*-algebras: T I L(H) set of linear continous maps H −→ H in a complex Hilbert space H with composition, adjunction and operator norm kT k := inf{k ∈ R+ | kT (h)kH ≤ kkhkH , ∀h ∈ H}. Every unital C*-algebra is an operator-norm-closed unital ∗-subalgebra of L(H) for a certain H. I Every Abelian unital C*-algebra is of the form C(X ; C): the f set of complex-valued continuous functions X −→ C over a compact Hausdorﬀ topological space X with pointwise multiplication and conjugation and norm kf k := supp∈X |f (p)|.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Categories A category C consists of: 0 s 1 t 0 I a quiver: a pair of source/target maps C ←− C −→ C from a class C 1 of morphisms to a class C 0 of objects, 0 ι 1 0 I an identity map C −→ C that to every object A ∈ C 1 associates its identity morphisms ιA ∈ C such that s(ιA) = A = t(ιA), I a partially defined composition map that to every pair of morphisms f , g ∈ C 1 such that t(g) = s(f ) associates a new morphism f ◦ g ∈ C 1 with s(f ◦ g) = s(g), t(f ◦ g) = t(f ), that further satisfies the following algebraic axioms: I associativity:(f ◦ g) ◦ h = f ◦ (g ◦ h), whenever (one of) the two terms are defined,

I unitality: f ◦ ιA = f = ιB ◦ f , 1 whenever f ∈ HomC (A, B) := {x ∈ C | s(x) = A, t(x) = B}.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Functors Γ A covariant functor S −→ A between the categories S and A is a pair of maps Γ1 : S 1 → A 1,Γ0 : S 0 → A 0 making commutative the following diagrams:

Γ1 Γ1 S 1 / A 1 S 1 / A 1

sS sA tS tA Γ0 Γ0 S 0 / A 0 S 0 / A 0 and that also satisfy the following unital homomorphim axioms:

1 1 1 1 S A Γ (f ◦S g) = Γ (f ) ◦A Γ (g), Γ (ιA ) = ιΓ0(A). A contravariant functor will intertwine sources with targets in the diagrams and will satisfy axioms for unital anti-homomorphism.

A natural transformation Γ −→G Ξ between two covariant (or Γ, Ξ contravariant) functors S −−→ A consists of a map f F : S 0 → A 1 such that for every morphism A −→ B in S 1 the following diagram in A commutes:1

FA Γ0(A) / Ξ0(A)

Γ1(f ) Ξ1(f ) Γ0(B) / Ξ0(B) FB

A natural isomorphism is a natural transformation such that FA is invertible in A 1, for all A ∈ S 0.

1In the contravariant case the direction of vertical arrows is reversed. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Duality An (anti-)isomorphism of categories is given by a pair of Γ ' co(ntra)variant functors S A such that Σ ◦ Γ = I and g S Σ Γ ◦ Σ = IA , where IC denotes the identity functor of C .

An equivalence between categories is a pair of covariant functors

Γ ' S A with two natural isomorphisms g Σ

G E IA −→ Γ ◦ Σ, IS −→ Σ ◦ Γ. When the above functors Γ, Σ are contravariant, we say that we have a duality between the categories S and A . Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Gel’fand-Na˘ımark Duality 1

Theorem (2) There is a duality between the categories:

Ao of unital ∗-homomorphisms between commutative unital C*-algebras,

So of continuous maps betwen compact Hausdorﬀ topological spaces.

2Gel’fand I (1941) Normierte Ringe Mat Sbornik N S 51(9):3-24 Gel’fand I, Na˘ımark M (1943) On the Embedding of Normed Rings into the Ring of Operators in Hilbert Space Math Sbornik 12:197-213 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Gel’fand-Na˘ımark Duality 2

I The functor Γo : So → Ao associates to every compact Hausdorﬀ space X the commutative unital C*-algebra Γo(X ) of continuous complex-valued functions on X (with pointwise multipication and conjugation and maximum modulus norm).

I The functor Σo : Ao → So associates to every commutative unital C*-algebra A its Gel’fand spectrum Σo(A) := {ω : A → C | ω is a unital ∗-homomorphism} equipped with the (compact Hausdorﬀ) weak∗-topology: the weakest topology making continuous for all x ∈ A the Gel’fand transforms xˆ :Σo(A) → C,x ˆ(ω) := ω(x).

Go I The Gel’fand transform IAo −→ Γo ◦ Σo is the natural 0 isomorphism that, for every A ∈ Ao , associates the unital o GA ∗-isomorphism of C*-algebras A −−→ Γo ◦ Σo(A) given by: o GA : x 7→ xˆ, for x ∈ A, Eo I The evaluation transform ISo −→ Σo ◦ Γo is the natural 0 isomorphism that, for every X ∈ So , associates the Eo X o homeomorphism X −−→ Σo ◦ Γo(X ) given by: EX : p 7→ evp, for p ∈ X , where evp :Γo(X ) → C is the p-evaluation map evp : σ 7→ σ(p), for all σ ∈ Γo(X ).

This topological version of Descartes’s algebraization of geometry is the usual starting point of non-commutative geometry:

I since a commutative unital C*-algebra “is” a compact Hausdorﬀ topological space, we will think of non-commutative C*-algebras as (duals of) “quantum topological spaces”,

I we can work in the “dual” category of unital C*-algebras, as a substitute for a missing category of “quantum compact Hausdorﬀ topological spaces”. Without the intention of undermining the basic usefulness of such “dual” point of view, it is the purpose of the present research work to provide a “geometrical/spectral” counterpart to non-commutative unital C*-algebras.

A C*-category (C, ◦, ∗, +, ·, k k) is given by: I an involutive algebroid (C, ◦, ∗, +, ·) over C: 0 I a category (C, ◦), with identities C ⊂ C, 0 I a contravariant functor ∗ : C → C acting trivially on C , 0 I ∀A, B ∈ C ,(CAB , +, ·), CAB := HomC(B, A), are complex vector spaces on which ◦ is bilinear and ∗ is conjugate-linear, I equipped with a norm k k : C → R such that: 0 I completeness: (CAB , +, ·) are Banach spaces, ∀A, B ∈ C , I submultiplicativity: kx ◦ yk ≤ kxk · kyk, ∗ 2 I C*-property: kx ◦ xk = kxk , for all x ∈ C, ∗ I positivity: for all x ∈ C, the element x ◦ x is positive in the x unital C*-algebra Cs(x)s(x), where s(x) −→ t(x).

A C*-category is full if all the bimodules CAA (CAB )CBB are full.

A Fell bundle is a Banach bundle ref (E, π, X) such that:

I (E, ◦, ∗) and (X, ◦, ∗) are topological involutive categories,

I π : E → X is a ∗-functor, −1 I restricted to the ﬁbers Ep := π (p), for p ∈ X, ◦ is bilinear and ∗ is conjugate-linear,

I kx ◦ yk ≤ kxk · kyk, for all composable x, y ∈ E, ∗ 2 I kx ◦ xk = kxk , for all x ∈ E and ∗ 3 I x ◦ x is positive whenever it belongs to a C*-algebraic ﬁber.

A Fell bundle is saturated if Ep ◦ Eq is dense in Ep◦q.

3If (X, ◦, ∗) is inverse involutive category (p ◦ p∗ ◦ p = p ∈ X) or a groupoid, ∗ simply require x ◦ x positive in the C*-algebra Eπ(x∗◦x). Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Banach Bundles A Banach bundle is a bundle (E, π, X), i.e. a continous open surjective map π : E → X, whose total space is equipped with:

I a partially deﬁned continuous binary operation of addition + : E ×X E → E, with domain the subset E ×X E := {(x, y) ∈ E × E | π(x) = π(y)}, I a continuous operation of multiplication by scalars · : K × E → E, I a continuous “norm” k · k : E → R, such that: −1 I for all x ∈ X, the ﬁber Ex := π (x) is a complex Banach space (Ex , +, ·) with the norm k · k, O, I for all x ∈ X, the family U = {e ∈ E | kek < , π(e) ∈ O}, o xo where O ⊂ X is an open set containing xo ∈ X and > 0, is a

fundamental system of neighbourhoods of 0 ∈ Exo .

back

• Previous/Other Attempts

Several lines of approach to a non-commutative extension of Gel’fand-Na˘ımark duality have been attempted.

We give here a brief guide to the complex literature on this topic.

J.M.G.Fell probably initiated the trend of sectional reconstruction of a C*-algebra A using, as dual, a suitable bundle over its spectral space Aˆ (the set of equivalence classes of unitarily equivalent ∗-representations).4 Results in similar directions were produced by J.Tomiyama.5 J.M.G.Fell also pioneered the deﬁnition of the now called Fell-bundles.6

4Fell JMG (1961) The Structure of Algebras of Operator Fields Acta Math 106:233-280 5Tomiyama J (1962) Topological Representations of C*-algebras Tohoku Math J 14(2):187-204 6Fell J, Doran R (1998) Representations of C*-algebras, Locally Compact Groups and Banach ∗-algebraic Bundles Vol 1-2 Academic Press Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Bundle Duals / Sectional Representations

In the celebrated (but vastly ignored) J.Dauns-K.H.Hofmann theorem7 the algebra is reconstructed via continous sections of a bundle simple C*-algebras (see also J.Migda8).

7Dauns J, Hofmann K-H (1968) Representations of Rings by Sections Mem Amer Math Soc 83 AMS Hofmann K H (1972) Representation of Algebras by Continuous Sections Bull Amer Math Soc 78(3):291-373 Hofmann K H (1972) Some Bibliograpical Remarks on: “Representation of Algebras by Continuous Sections” Recent Advances in the Representation Theory of Rings and C*-Algebras 177-182 (eds) Hofmann K H, Liukkonen J R Memoirs Amer Math Soc 148 (1974) Hofmann K H (2011) The Dauns-Hofmann Theorem Revisited Journal of Algebra and Its Applications 10(1):29-37 8Midga J (1993) Non-commutative Gelfand-Naimark Theorem Comment Math Univ Carolin 34(2):253-255 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Structured Pure State Space Duals / Functional Functional representations of C*-algebras via continuous functions on generalized spectra consisting of (pure) states equipped with extra structures (transition probability, Poisson, . . . ) started with R.Kadison9 and were subsquently considered by F.Schultz,10 P.Kruszy´nski-S.Woronowicz,11 (see also I.Fujimoto).12 A duality was essentially obtained by N.Landsman.13

9Kadison R-V (1951) A Representation Theory for Commutative Topological Algebra Memoires Amer Math Soc 7 10Schultz F (1982) Pure States as a Dual Object for C*-algebras Commun Math Phys 82:497-509 11KruszyńskiP, Woronowicz S (1982) A Noncommutative Gelfand Naimark Theorem J Operator Theory 8:361-389 12Fujimoto I (1998) A Gelfand-Naimark Theorem for C*-algebras Pacific J Math 184(1):95-119 13Landsman N (1997) Poisson Spaces with a Transition Probability Reviews in Mathematical Physics 9(1):29-57 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Bundle of Pure States Duals / Functional In the mostly ignored work by R.Cirelli-A.Maniá-L.Pizzocchero14 spectra are projective Kählerbundles over the spectrum Aˆ.

14Cirelli R, Lanzavecchia P, ManiàA (1983) Normal Pure States of the von Neumann Algebra of Bounded Operators as KählerManifold J Phys A: Math Gen 16:3829-3835 Cirelli R, Lanzavecchia P (1984) Hamiltonian Vector Fields in Quantum Mechanics Il Nuovo Cimento 79(2):271-283 Abbati M-C, Cirelli R, Lanzavecchia P, ManiáA (1984) Il Nuovo Cimento B 83(1):43-60 Cirelli R, ManiàA, Pizzocchero L (1990) Quantum Mechanics as an Infinite-dimensional Hamiltonian System with Uncertainty Structure: Part I J Math Phys 31:2891-2897 Cirelli R, ManiàA, Pizzocchero L (1990) Quantum Mechanics as an Infinite-dimensional Hamiltonian System with Uncertainty Structure: Part II J Math Phys 31:2898-2903 Cirelli R, ManiàA, Pizzocchero L (1994) A Functional Representation for Non-commutative C*-algebras Rev Math Phys 6(5):675-697 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? NC-topology on Maximal Ideals / Functional

C.Akemann15 was probably the ﬁrst to describe the dual of a C*-algebra via maximal left-ideals with a “non-commutative” form of topology. Reformulations of commutative Gel’fand-Na˘ımark duality via locales by B.Banaschewski-C.Mulvey16 (for constructive versions see also T.Coquand-B.Spitters17) further inpired the usage of quantales as duals of non-commutative C*-algebras.18

15Akemann C (1971) A Gelfand Representation Theory for C*-algebras Paciﬁc Journal of Mathematics 39 (1):1-11 16Banaschewski B, Mulvey CJ (2000) The Spectral Theory of Commutative C*-algebras Quaestiones Mathematicae 23:425-464 17Coquand T, Spitters B (2009) Constructive Gelfand Duality for C*-algebras Mathematical Proceedings of the Cambridge Philosophical Society 147:339-344 18Banaschewski B, Mulvey CJ (2006) A Globalisation of the Gelfand Duality Theorem Annals of Pure and Applied Logic 137:62-103 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Quantales as Duals

The study of “point free” non-commutative topologies via quantales, as duals of C*-algebras, has been further pursued by F.Borceux-J.Rosick´y-G.Van den Bossche19 C.J.Mulvey-J.W.Pelletier20 and D.Kruml-J.W.Pelletier-P.Resende-J.Rosick´y.21

19Borceux F, RosickýJ, Van den Bossche G (1989) Quantales and C*-algebras J London Math Soc 40:398-404 20Mulvey CJ, Pelletier JW (2002) On the Quantisation of Spaces J Pure Appl Algebra 175:289-325 21Kruml D, Pelletier J W, Resende P, RosickýJ (2003) On Quantales and Spectra of C*-algebras Appl Categ Structures 11:543-560 Kruml D, Resende P (2004) On Quantales that Classify C*-algebras Cah Topol Geom Differ Categ 45:287-296 Resende P (2007) Etale´ Groupoids and their Quantales Advances in Mathematics 208(1):147-209 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Topos Theory Approaches

Several works suggest to reconstruct C*-algebras via the Grothendieck topoi of their commutative subalgebras 22 I A.D¨oring (based on works by C.Isham-J.Butterﬁeld). 23 I C.Heunen-N.Landsman-B.Spitters-S.Wolters

22D¨oringA (2012) Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras arXiv:1212.2613 D¨oringA (2012) Flows on Generalised Gelfand Spectra of Nonabelian Unital C*-Algebras and Time Evolution of Quantum Systems arXiv:1212.4882 23Heunen C, Landsman N, Spitters B (2009) A Topos for Algebraic Quantum Theory Communications in Mathematical Physics 291:63-110 Heunen C, Landsman NP, Spitters B, Wolters S (2011) The Gelfand Spectrum of a Noncommutative C*-algebra: a Topos-theoretic Approach J Austr Math Soc 90:39-52 Wolters S (2013) A Comparison of Two Topos-theoretic Approaches to Quantum Theory Communications in Mathematical Physics 317(1):3-53 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Other Categorical Sheaves / Topos Theory Approaches

Among the ongoing recent eﬀorts towards Gel’fand-Na˘ımark duality using topoi or sheaves we mention: 24 I S.Henry 25 I C.Flori-T.Fritz

24Henry S (2014) Localic Metric Spaces and the Localic Gelfand Duality arXiv:1411.0898 [math.CT] Henry S (2014) Constructive Gelfand Duality for Non-unital Commutative C*-algebras arXiv:1412.2009 [math.CT] Henry S (2015) Toward a Non-commutative Gelfand Duality: Boolean Locally Separated Toposes and Monoidal Monotone Complete C*-categories arXiv:1501.07045 [math.CT] 25Flori C, Fritz T (2015) (Almost) C*-algebras as Sheaves with Self Action arXiv:1512.01669 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? Other Approaches

Apart from the extremely vast literature on classiﬁcation of C*-algebras, some quite recent attempts to produce Gel’fand-Na˘ımark dualities, at least for some reasonable classes of C*-algebras have been put forward by: 26 I C.Heunen-M.Reyes 27 I N.de Silva

26Heunen C, Reyes ML (2014) Active Lattices Determine AW*-algebras Journal of Mathematical Analysis and Applications 416:289-313 Heunen C, Reyes ML On Discretization of C*-algebras arXiv:1412.1721 27de Silva N (2014) From Topology to Noncommutative Geomtery: K-theory arXiv:1408.1170 Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Review of Commutative Gel’fand-Na˘ımark Duality Gel’fand-Na˘ımark Duality Previous/Other Attempts What Are the “Spectra” of C*-algebras? The Present Approach In our proposed approach, C*-algebras are reconstructed by sections of a bundle in the same tradition of J.M.G.Fell, J.Dauns-K.H.Hofmann, R.Cirelli-A.Mani´a-L.Pizzocchero and N.Landsman. Contrary to these previous cases,

I our bundles have only one-dimensional ﬁbers,

I all differential geometric features (Kähler/ Poisson structures) are eliminated from the spectrum and “substituted” by a horizontal categorification of the base of the bundle. Direct inspirational input of this project comes from:

I W.Heisenberg / A.Connes,

I R.Feynmann / L.Crane.

• Basic Constructions

I The Transition Amplitude Bundle

I The Transition Amplitude Space

I Non-commutative Spaceoids

• The Transition Amplitude Bundle

I The family SA of states of unital C*-algebra A consists of all the linear maps ω : A → C that are ∗ I positive: ω(x ◦ x) ∈ C+, for all x ∈ A, I normalized: ω(1A) = 1C. I By Gel’fand-Na˘ımark-Segal theorem, every state ω induces a representation πω : A → L(Hω) and a unit vector ξω ∈ Hω

such that ω(x) = hξω | πω(x)ξωiHω , for all x ∈ A. I Let PA ⊂ SA denote the family of pure states of A: these are those states ω ∈ SA such that πω is irreducible. ∗ ∗ I PA ⊂ SA ⊂ A is equipped with the weak -topology: the weakest topology making continuous all the maps xˆ : ω 7→ ω(x), for all x ∈ A. ˆ I PA is a bundle over A, the usual “spectrum of A”, that is the quotient space of PA under the equivalence relation of unitary equivalence of irreducible GNS-representations.

The base space of our spectra is just a fiberwise horizontal ˆ categorification of the previous bundle PA over A: instead of considering only the points ω ∈ PA, we consider all possible 1-arrows (ordered pairs) between pure states in the same fiber of bundle PA. The fiberwise product

PA ×Aˆ PA := {(ω, ρ) | [πω] = [πρ]} ⊂ PA × PA, with the topology induced by the product of the weak∗-topology ˆ on PA, is a bundle over A with pair groupoids as ﬁbers. ˆ PA ×Aˆ PA is a bundle gerbe of pair groupoids over A and is the base spectrum of A.

I If ω ∈ PA, the GNS-representation πω is irreducible, hence 00 πω(A) = L(Hω). Furthermore, if [ω] = [ρ], ∃U : Hω → Hρ ∗ unitary such that πρ(x) = Uπω(x)U , for all x ∈ A. ∗ 0 I U is unique up to a phase, since U1 U2 ∈ πω(A) = C · 1Hω , 00 AdU 00 but the unital ∗-isomorphim πω(A) −−→ πρ(A) is unique. I For every p ∈ Aˆ consider the pair groupoid with objects L(Hω), with [ω] = p, and 1-arrows the unique unital ∗-isomorphisms AdU induced by the unitaries U intertwining 00 the given representations and construct the W*-algebra Ap of 00 orbits of such groupoid (Ap ' L(Hω), ∀ω ∈ p). 00 ˆ I The bundle of (type I factors) W*-algebras Ap over A is the Von Neumann enveloping bundle of A. 00 ⊕p∈AˆAp coincides with the atomic part of the Arens W*-envelope A∗∗ of A.

I Every x ∈ A determines a section of the Von Neumann enveloping bundle x 7→ πp(x) (the orbit of πω(x)). I Every x ∈ A induces a sectionx ˆ of the total spectrum bundle ˆ EA over PA ×Aˆ PA given, for all p ∈ A and for all ω, ρ ∈ p by:

xˆ(ω, ρ) :=| ω ih ω | πp(x) | ρ ih ρ | .

The sectionx ˆ is the Gel’fand transform of x ∈ A.

I The total spectrum EA of A becomes a Fell bundle with the topology induced by the bases of “tubular neighborhoods” x, UO of any point eo ∈ (EA)(ωo ,ρo ) deﬁned as follows: for every open neighborhood O ⊂ PA ×Aˆ PA of (ωo, ρo), for every x ∈ A such thatx ˆ(ωo, ρo) = eo, for every > 0,

x, UO := {e ∈ EA | ∀(ω, ρ) ∈ O, kxˆ(ω, ρ) − ek < }.

The Fell bundle EA is the transition amplitude bundle of A. ˆ I For every p ∈ A, the sub-bundle (EA)p is actually a one-dimensional C*-category with objects (PA)p, hence the total spectrum of A can be described alternatively as a bundle of one-dimensional C*-categories.

Since the spaces EA and BA := PA ×Aˆ PA are not usually compact, we will need to consider uniformly continuous sections.

I There is a standard uniform structure UBA on the space ∗ ∗ BA := PA ×Aˆ PA ⊂ PA × PA ⊂ A × A obtained by restricting the product of the uniform structure induced by the weak*-topology on A∗.

I A bundle uniformity UEA for the transition amplitude bundle is given by the ﬁlter generated on EA × EA by this ﬁlterbase

of subsets of EA ×BA EA: U := {(e1, e2) ∈ EA ×BA EA | ke1 − e2kEA < }, > 0. Note that this uniformity does not induce the already deﬁned topology on the total space E of the transition amplitude bundle, hence the total space is not itself a uniform space!

• The Transition Amplitude Space

Every unital C*-algebra determines a very special gauge section of its transition amplitudes bundle EA: the Gel’fand transform 1ˆA of the identity element 1A ∈ A: ˆ 1ˆA :(ω, ρ) 7→| ω ih ω | 1A | ρ ih ρ |, ∀p ∈ A, ∀ω, ρ ∈ (PA)p.

∗ The gauge section is Hermitian: 1Â(ω, ρ) = 1Â(ρ, ω). In general it is not a subcategory of EA. Informally, the gauge section allows the specification of bundle gerbe acting as symmetry morphisms of a “horizontal categorified” site of “gauge blocks” inside the transition amplitudes bundle; a structure that is crucial in the reconstruction of a C*-algebra (isomorphic to A) form EA.

The gauge section 1Â induces on the set (PA)p a structure of total Fell bundle-valued transition amplitude space28: for all ω ∈ (P ) , 1ˆ (ω, ω) =| ω ih ω |= 1 , I A p A (EA)(ω,ω) ∗ I 1Â(ω, ρ) = 1Â(ρ, ω), for all ω, ρ ∈ (PA)p,

I there exists at least one frame i.e. a subset F ⊂ (PA)p such that, for all ω, ρ ∈ (PA)p, ˆ P ˆ ˆ 1A(ω, ρ) = θ∈F 1A(ω, θ) ◦ 1A(θ, ρ). I ⊂ (P ) is orthonormal if ω 6= ρ ∈ F, ⇒ 1ˆ (ω, ρ) = 0 . A p A (EA)(ω,ρ) Frames are maximal othonormal. The transition amplitude space is total if every maximal orthonormal set is a frame. 28For complex-valued transition amplitude spaces, see section 4.5 in: S.Gudder (1988) Quantum Probability, Elsevier. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Transition Amplitude Bundle Basic Constructions The Transition Amplitude Space Gel’fand-Na˘ımark Duality Non-commutative Spaceoids What Are the “Spectra” of C*-algebras? The Gauge Bundle Gerbe

I Our gauge transition amplitude space is total, hence for every p ∈ Aˆ, gauge frame blocks coincide with maximal orthonormal sets of pure states i.e. sets F ⊂ (PA)p such that the “matrix” 1ˆA|F×F, for ω, ρ ∈ F, is the “identity”: ( | ω ih ω |= 1 , ω = ρ, (EA)(ω,ω) 1ˆA(ω, ρ) = 0 , ω 6= ρ. (EA)(ω,ρ)

I For two gauge frame blocks F1, F2, the “oﬀ-diagonal matrix” ˆ ˆ 1A|F1×F2 is a “unitary” with inverse 1A|F2×F1 ; hence we obtain ˆ a gauge bundle gerbe FA of gauge frame blocks over A.

Restricting the transition amplitude bundle EA onto each one of the gauge blocks F1 × F2 of the pair groupoid FA of gauge block frames we obtain a bundle gerbe enriched in the Morita 2-groupoid of full 1-C*-categories:

I Each bundle EA|F×F is a full 1-C*-category,

I The bundles EA|F1×F2 are Morita isomorphism bimodules

between the two C*-categories EA|Fj ×Fj , for j = 1, 2. Possible links with Flori-Fritz’s gleaves must be explored.29

29Compositories and Gleaves, arXiv:1308.6548. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Transition Amplitude Bundle Basic Constructions The Transition Amplitude Space Gel’fand-Na˘ımark Duality Non-commutative Spaceoids What Are the “Spectra” of C*-algebras? The Transition Amplitude Uniformity

Every Fell bundle-valued transition amplitude space (P, γ) determines a uniformity on P: the family of entourages of ∆P is given by the family of subsets given, for 0 < < 1, by: U := {(ω, ρ) ∈ P × P | kγ(ω, ρ)k > 1 − } ⊂ P × P. It will be a requirement for our spaceoids to assume that fibrewise the uniform structure of P coincides with the uniform structure induced by its Fell bundle-valued transition amplitude space structure. This requirement is always satisfied for spectral spaceoids and it is the counterpart, in our setting, of the coincidence between the weak*-uniformity and the Kählermetric uniformity in the phase-space of a quantum system.

Given a C*-category C, its C*-envelope is a unital C*-algebra Ξ(C) with a ∗-functor ι : C → Ξ(C) that satisﬁes the following universal factorization property: for any other ∗-functor φ : C → A into a unital C*-algebra A, there exists a unique unital ∗-homomorphism φˆ : Ξ(C) → A such that φ = φˆ ◦ ι. Proposition Every C*-category C admits a C*-envelope C −→ι Ξ(C).

We will denote by Ξ∗∗(C) the W*-envelope.

• Non-commutative Spaceoids

A non-commutative spaceoid (E, π, P, χ, X , γ) is a saturated unital Fell line-bundle (E, π, P ×X P) over a topological bundle of pair groupoids (P ×X P, χ, X ), where χ : P → X is a surjective open continuous projection from a uniform Hausdorﬀ space P onto the quotient space X ; equipped with a continuous section γ : P ×X P → E inducing on P a structure of Fell bundle-valued saturated full transition amplitude space that is compatible with the uniform topology of P. Let us spell in detail the deﬁnition.

I P is a uniform space;

I the uniform completion of P is compact Hausdorﬀ (topological saturation condition);

I (E, π, P ×X P) is a saturated unital Fell line-bundle over the topological groupoid P ×X P; γ I P ×X P −→ E is a continuous section of the previous bundle; I (P, γ) is a Fell bundle-valued transition amplitude space that is total and algebraically saturated; Isbell reﬂective subcategory I the uniform structure on P is ﬁbrewise “compatible” with its uniform structure as a transition amplitude space (P, γ).

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Transition Amplitude Bundle Basic Constructions The Transition Amplitude Space Gel’fand-Na˘ımark Duality Non-commutative Spaceoids What Are the “Spectra” of C*-algebras? Morphisms of Non-commutative Spaceoids A morphism of non-commutative spaceoids (λ,Λ) (E 1, π1, P1, χ1, X 1, γ1) −−−→ (E 2, π2, P2, χ2, X 2, γ2) is given by a pair of maps λ : P1 → P2 and Λ : λ•(E 2) → E 1 such that: 1 2 I λ : P → P is a uniformly continuous map such that χ2 ◦ λ = χ1, hence it induces a necessarily continuous quotient map [λ]: X 1 → X 2 and 1 1 2 2 (λ, λ): P ×X 1 P → P ×X 2 P is necessarily a continuous homomorphism of bundles of pair groupoids; • 2 1 I Λ: λ (E ) → E is a morphism of Fell bundles from the (λ, λ)-pull-back of E 2 to E 1; 1 2 2 2 I (λ, Λ) : (P , γ ) → (P , γ ) is a morphism of Fell bundle-valued transition amplitude spaces i.e. Λ(γ2(λ(p), λ(q))) = γ1(p, q), for all p, q ∈ P1.

We have a category of non-commutative spaceoids where composition of morphisms is deﬁned as:

0 0 0 • 0 E 3 (λ , Λ ) ◦ (λ, Λ) := (λ ◦ λ, Λ ◦ λ (Λ ) ◦ Θλ0,λ)

(λ,Λ) (λ0,Λ0) where E 1 −−−→ E 2 −−−−→ E 3 and where E 3 0 • 3 • 0 • 3 Θλ0,λ :(λ ◦ λ) (E ) → λ ((λ ) (E )) is the canonical isomorphism between standard pull-backs of bundles.

For us a morphism of unital C*-algebras will be a φ • ∗-homomorphism A1 −→ A2 whose pull-back φ : SA2 → SA1 is • 30 pure state preserving: φ (PA2 ) ⊂ PA1 . Unital ∗-epimorphism are a special case. We have a category A of unital C*-algebras with the usual composition of such pure state preserving unital ∗-homomorphisms.

30This choice let us recover continuous maps in the usual commutative Gel’fand duality and simultaneously allows to limit the study to spaceoids deﬁned using only pure states. It is possible to consider more general classes of morphism. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality

• Gel’fand-Na˘ımark Duality

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality The Section Functor Γ The section functor Γ: S → A , associates to every non-commutative spaceoid (E, π, P, χ, X , γ) the unital C*-algebra Γ(E) consisting of all the sections σ : P ×X P → E such that:

I σ is continuous and its restriction σ|∆P to the diagonal 31 ∆P := {(p, p) | p ∈ P} of P is uniformly continuous; I σ is gauge invariant i.e. for every pair of γ-orthonormal frames

F1, F2 ⊂ P, σ|F1 and σ|F2 are related by ? σ|F = Ad (σ|F ) = γ|F ×F σ|F γ| , 2 γ|F2×F1 1 2 1 1 F2×F1 where and ? denote here the convolution product and the adjoint involution in the C*-category with objects the maximal orthonormal γ-frames of P and 1-arrows the

restrictions σ|F1×F2 .

31 Note that since E|∆P is trivial line-bundle equipped with a uniformity, this condition is perfectly deﬁned. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality The Algebraic Saturation Condition (tentative)

Given a point o ∈ X , let Fo be the family of γ-frames in the −1 transition amplitude space χo ⊂ P ×X P.

Consider for all possible choice functions (UF )F ∈Fo , where UF is a γ unitary section of E|F , the family (UF )F ∈Fo of their γ-orbits γ ∗∗ denoted by UF ∈ Ξγ (E|χ−1(o)). The spaceoid is algebraically saturated if, for all o ∈ X , there γ exists at least a choice map (UF )F ∈Fo such that (UF )F ∈Fo is the ∗∗ family of all unitaries in Ξγ (E|χ−1(o)). This condition identiﬁes the full reﬂective subcategory of the category of algebraic spaceoids that is in Isbell duality with the category of atomic W*-bundles.

back to spaceoids

The section functor Γ associates to a morphism of spaceoids (η,Ω) E 1 −−−→ E 2 the usual pull-back of sections: 1 2 σ 7→ Γ(η,Ω)(σ) ∈ Γ(E ), for all σ ∈ Γ(E ), where 1 1 Γ(η,Ω)(σ):(p, q) 7→ Ω(σ(η(p), η(q))) ∈ E , for all p, q ∈ P .

We see that Γ(η,Ω) is indeed well-deﬁned and a unital ∗-homomorphism of unital C*-algebras that preserves pure states. The functor Γ : S → A is contravariant.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality The Spectrum Functor Σ The spectrum functor Σ: A → S associates to every unital C*-algebra A its spectral non-commutative spaceoid (EA, πA, PA, χA, XA, γA), where

I PA is the family of pure states of A equipped with the weak*-uniformity, ˆ I χA : PA → XA := A is the quotient map onto the usual spectrum of A (the family of equivalence classes of untarily equivalent representations),

I (EA, πA, PA ×XA PA) is the transition amplitude Fell line-bundle of A, as already constructed,

I γA := 1ˆA is the Gel’fand transform of the identity element of A making (PA, γA) into a full Fell line bundle-valued transition amplitude space that is topologically and algebraically saturated.

The spectrum functor associates to every unital ∗-epimorphism of unital C*-algebras φ : A1 → A2 a morphism of non-commutative (λφ,Λφ) spaceoids Σ(A2) −−−−−→ Σ(A1) where:

32The pure state preserving property of φ is necessary condition in order to deﬁne this map. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality The Gel’fand Transform

The non-commutative Gel’fand transform GA : A → Γ ◦ Σ(A) = Γ(EA) is simply the map that to every

x ∈ A associates the γA-invariant sectionx ˆ : PA ×XA PA → EA given byx ˆ(ω, ρ) :=| ω ih ω | x | ρ ih ρ |, for all ω, ρ ∈ PA.

GA is actually a unital ∗-homomorphism between C*-algebras.

Theorem The Gel’fand transform G : IA → Γ ◦ Σ is a natural isomorphism.

The evaluation transform EE : E → Σ ◦ Γ(E) from the spaceoid E to the spectral spaceoid of the unital C*-algebra Γ(E) consists E E of a morphism of spaceoids EE := (η , Ω ) as follows: E I η : P → PΓ(E) associates to every p ∈ P the map E E E E ηp := ζp ◦ evp : Γ(E) → C, σ 7→ ζp (σ(p, p)) obtained E composing the function evp : Γ(E) → Γ(E)(p,p) that evaluates every section σ ∈ Γ(E) in the point (p, p) ∈ P ×X P with the E canonical Gel’fand-Mazur isomorphism ζp : Γ(E)(p,p) → C between one-dimensional unital C*-algebras; E E • I Ω :(η ) (EΓ(E)) → E is the relation ﬁberwise deﬁned as E Ω(p,q) : e 7→ σ(p, q), for every σ ∈ Γ(E) such that σ(ηE (p), ηE (q)) = e, where p, q ∈ P and E • e ∈ Σ ◦ Γ(E)(ηE (p,q)) = (η ) (EΓ(E))(p,q).

Theorem E The evaluation transform is a natural isomorphism IS −→ Σ ◦ Γ. Crucial ingredients here are the saturation conditions on the spaceoid. The proof makes use of the non-commutative Stone-Weierstrass theorem by J.Glimm: A unital ∗-subalgebra B of a unital C*-algebra A that separates the states in the weak*-closure PA ⊂ SA of the set of pure states of A is norm dense in A.

We have the following pair of commutative diagrams of functors:

Γ Σ S / A SAo O O O O F I F I T / C T o C Γc Σc

Γc where T & C is the commutative Gel’fand-Na˘ımark duality. g Σc I : C → A is the usual inclusion functor.

F : T → S is the covariant functor that associates to every compact Hausdorﬀ space T the topological spaceoid T T T T T T T T T (E , π , P , χ , X , γ ) with X := T , P := T , χ := ιT the T T T T identity map of T ,(E , π , P ×X T P ) the trivial complex T line-bundle on the diagonal ∆T = T ×T T of T and γ the identity constant section of E T .33

F associates to every continuous map f : T1 → T2 of compact Hausdorﬀ topological spaces the morphism of spaceoids (λf , Λf ) := (f , F ), where F is the canonical isomorphism between • T T trivial complex line-bundles f (E 2 ) → E 1 over T1 (the f -pull-back of a trivial bundle is trivial).

33Notice that every compact Hausdorﬀ space is equipped with a unique uniformity inducing its topology (enturages are the just neighborhoods of the diagonal of X ) and continuous maps are uniform for such uniformity. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background The Section and Spectrum Functors Basic Constructions The Gel’fand Isomorphism Gel’fand-Na˘ımark Duality The Evaluation Isomorphism What Are the “Spectra” of C*-algebras? Commutative and Non-commutative Gel’fand-Na˘ımark Duality Commutative / Non-commutative Gel’fand Duality 3

The following commutative diagrams between the commutative and non-commutative Gel’fand and evaluation transforms:

ES GA S / S A / A O O O O F F I I T / C C / C ET GC

ﬁnally prove that our non-commutative Gel’fand-Na˘ımark duality is a natural extension of the usual commutative case (and reduces to it by restricting to the embedded full sub-categories F(T ) ⊂ S and I(C ) ⊂ A ).

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras?

• What Are the “Spectra” of C*-algebras?

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? The Two “Souls” of Geometry

space

∩ v ( descent / ? quantum space ? o covariance

Descent theory (gluing) Covariance (transport) sheaves/stacks Klein-Cartan geometries Grothendieck topoi Ehresmann connections Grothendieck categories category theory (higher) homotopy/holonomy In a quantum space: (a) relations between points are primary concepts, (b) transport depends on a transition amplitude structure.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? Non-commutative Klein-Cartan Geometries?

Klein’s Erlangen program characterizes geometry from its group of symmetries: Klein’s geometries are homogeneous spaces.34 Cartan dealt with local symmetries: Cartan’s geometries are bundles of homogeneus spaces with a connection.35 We need an understanding of non-commutative covariance and transport (and how they merge with the descent data of the space).

34F.Klein (1872) arXiv:0807.3161. 35See the book: R.W.Sharpe (1997) Diﬀerential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Springer. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras? Non-commutative Topoi / Descent Theory? We probably need a version non-commutative topos theory where categories of tensor products and involutions of bimodules take the place of the usual Cartesian closed categories environment.

36 37 I P.Cartier (see also ) 38 39 I F. van Oystaeyen (see also ) 40 I M.Kontsevich, A.Rosenberg (NC descent theory) 41 I C.Flori, T.Fritz (Gleaves) 36A Mad Day’s Work: from Grothendieck to Connes and Kontsevich: The Evolution of Concepts of Space and Symmetry (2001) Bull Amer Math Soc 38:389-408. 37T.Maszczyk, arXiv:math/0611806. 38Virtual Topology and Functor Geometry (2007) CRC. 39K.Cvetko-Vah, J.Hemelaer, L.Le Bruyn arXiv:1705.02831. 40arXiv:math/9812158; see also S.Mahanta arXiv:math/0501166. 41arXiv:1308.6548. Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality Background Basic Constructions Gel’fand-Na˘ımark Duality What Are the “Spectra” of C*-algebras?

Thank You for Your Kind Attention!

This ﬁle has been realized using the “beamer” LATEX package of the TEX-live distribution and TEXstudio editor on Ubuntu Linux.

We acknowledge the partial support from

I the Department of Mathematics and Statistics in Thammasat University and

I the Thammasat University Research Grant n. 2/15/2556: “Categorical Non-commutative Geometry”.

Paolo Bertozzini Non-commuative Gel’fand-Na˘ımark Duality