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$: Non-commutative Gel'fand-Na mark Duality · Background Basic Constructions Gel’fand-Na mark Duality What Are the \Spectra" of C*-algebras? Outline I Background I Review of Commutative$
BackgroundBasic Constructions

Gel’fand-Naımark DualityWhat Are the “Spectra” of C*-algebras?

Non-commutative Gel’fand-Naımark 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-Naımark Duality



Abstract 1

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




Abstract 2

This work is a joint collaboration with:

I Natee Pitiwan (Chulalongkorn University)

I Roberto Conti (Sapienza Universita di Roma),




Outline

I BackgroundI Review of Commutative Gel’fand-Naımark DualityI Previous Attempts

I Basic ConstructionsI The Transition Amplitude Bundle of a C*-algebraI The Transition Amplitude Space of a C*-algebraI Non-commutative Spaceoids

I Non-commutative Gel’fand-Naımark DualityI The Section Functor ΓI The Spectrum Functor ΣI The Gel’fand Transform GI The Evaluation Transform EI Commutative and Non-commutative Gel’fand Duality

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




Review of Commutative Gel’fand-Naımark DualityPrevious/Other Attempts

• Background

I Commutative Gel’fand-Naımark Duality:I Categories of Commutative (Unital) C*-algebrasI Categories of Compact Hausdorff SpacesI Categorical Duality

I Previous Attempts:I Sectional RepresentationsI Convex Spaces of States as DualsI QuantalesI TopoiI Other Approaches





• Review of CommutativeGel’fand-Naımark Duality





C*-algebras 1

Defined by I.Gel’fand-M.Naımark in 1943, are a “rigid” blend ofalgebra and topology, basic in functional analysis,non-commutative geometry and quantum physics.

A complex unital C*-algebra (C, , ∗,+, ·, ‖ ‖) 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 ‖ ‖ : C→ R such that:I completeness: (C,+, ·, ‖ ‖) is a Banach space,I submultiplicativity: ‖x y‖ ≤ ‖x‖ · ‖y‖, for all x , y ∈ C,I C*-property: ‖x∗ x‖ = ‖x‖2, for all x ∈ C.

A C*-algebra is Abelian (or commutative) if x y = y x .





C*-algebras 2

I A unital ∗-homomorphism is a map φ : A→ B betweenunital 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:

I L(H) set of linear continous maps HT−→ H in a complex

Hilbert space H with composition, adjunction and operatornorm ‖T‖ := infk ∈ R+ | ‖T (h)‖H ≤ k‖h‖H ,∀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

set of complex-valued continuous functions Xf−→ C over a

compact Hausdorff topological space X with pointwisemultiplication and conjugation and norm ‖f ‖ := supp∈X |f (p)|.





CategoriesA category C consists of:

I a quiver: a pair of source/target maps C 0 s←− C 1 t−→ C 0

from a class C 1 of morphisms to a class C 0 of objects,I an identity map C 0 ι−→ C 1 that to every object A ∈ C 0

associates its identity morphisms ιA ∈ C 1 such thats(ιA) = A = t(ιA),

I a partially defined composition map that to every pair ofmorphisms f , g ∈ C 1 such that t(g) = s(f ) associates a newmorphism 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 ,whenever f ∈ HomC (A,B) := x ∈ C1 | s(x) = A, t(x) = B.





Functors

A covariant functor SΓ−→ A between the categories S and A is

a pair of maps Γ1 : S 1 → A 1, Γ0 : S 0 → A 0 makingcommutative the following diagrams:

S 1 Γ1//

sS

A 1

sA

S 0 Γ0// A 0

S 1 Γ1//

tS

A 1

tA

S 0 Γ0// A 0

and that also satisfy the following unital homomorphim axioms:

Γ1(f S g) = Γ1(f ) A Γ1(g), Γ1(ιSA ) = ιAΓ0(A).

A contravariant functor will intertwine sources with targets in thediagrams and will satisfy axioms for unital anti-homomorphism.





Natural Transformations

A natural transformation ΓG−→ Ξ between two covariant (or

contravariant) functors SΓ, Ξ−−→ A consists of a map

F : S 0 → A 1 such that for every morphism Af−→ B in S 1 the

following diagram in A commutes:1

Γ0(A)

Γ1(f )

FA // Ξ0(A)

Ξ1(f )

Γ0(B)FB

// Ξ0(B)

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-Naımark Duality




DualityAn (anti-)isomorphism of categories is given by a pair of

co(ntra)variant functors SΓ ''

Σ

gg A such that Σ Γ = IS and

Γ Σ = IA , where IC denotes the identity functor of C .

An equivalence between categories is a pair of covariant functors

SΓ ''

Σ

gg A with two natural isomorphisms

IAG−→ Γ Σ, IS

E−→ Σ Γ.

When the above functors Γ,Σ are contravariant, we say that wehave a duality between the categories S and A .





Gel’fand-Naımark Duality 1

Theorem (2)

There is a duality between the categories:

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

So of continuous maps betwen compact Hausdorff topologicalspaces.

2Gel’fand I (1941) Normierte Ringe Mat Sbornik N S 51(9):3-24Gel’fand I, Naımark M (1943) On the Embedding of Normed Rings into

the Ring of Operators in Hilbert Space Math Sbornik 12:197-213Paolo Bertozzini Non-commuative Gel’fand-Naımark Duality





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

I The functor Σo : Ao → So associates to every commutativeunital C*-algebra A its Gel’fand spectrumΣo(A) := ω : A→ C | ω is a unital ∗-homomorphismequipped with the (compact Hausdorff) weak∗-topology: theweakest topology making continuous for all x ∈ A theGel’fand transforms x : Σo(A)→ C, x(ω) := ω(x).






I The Gel’fand transform IAo

Go

−→ Γo Σo is the naturalisomorphism that, for every A ∈ A 0

o , associates the unital

∗-isomorphism of C*-algebras AGo

A−−→ Γo Σo(A) given by:Go

A : x 7→ x , for x ∈ A,

I The evaluation transform ISo

Eo

−→ Σo Γo is the naturalisomorphism that, for every X ∈ S 0

o , associates the

homeomorphism XEoX−−→ Σo Γo(X ) given by: Eo

X : p 7→ evp,for p ∈ X , where evp : Γo(X )→ C is the p-evaluation mapevp : σ 7→ σ(p), for all σ ∈ Γo(X ).






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

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

I we can work in the “dual” category of unital C*-algebras, as asubstitute for a missing category of “quantum compactHausdorff topological spaces”.

Without the intention of undermining the basic usefulness of such“dual” point of view, it is the purpose of the present research workto provide a “geometrical/spectral” counterpart tonon-commutative unital C*-algebras.





C*-categoriesHorizontal categorification of C*-algebras defined by J.Roberts.

A C*-category (C, , ∗,+, ·, ‖ ‖) is given by:I an involutive algebroid (C, , ∗,+, ·) over C:

I a category (C, ), with identities C0 ⊂ C,I a contravariant functor ∗ : C→ C acting trivially on C0,I ∀A,B ∈ C0, (CAB ,+, ·), CAB := HomC(B,A), are complex

vector spaces on which is bilinear and ∗ is conjugate-linear,

I equipped with a norm ‖ ‖ : C→ R such that:I completeness: (CAB ,+, ·) are Banach spaces, ∀A,B ∈ C0,I submultiplicativity: ‖x y‖ ≤ ‖x‖ · ‖y‖,I C*-property: ‖x∗ x‖ = ‖x‖2, for all x ∈ C,I positivity: for all x ∈ C, the element x∗ x is positive in the

unital C*-algebra Cs(x)s(x), where s(x)x−→ t(x).

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

are full.





Fell Bundles

A “bundle version” of C*-categories developed by J.Fell.

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,

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

I ‖x y‖ ≤ ‖x‖ · ‖y‖, for all composable x , y ∈ E,

I ‖x∗ x‖ = ‖x‖2, for all x ∈ E and

I x∗ x is positive whenever it belongs to a C*-algebraic fiber.3

A Fell bundle is saturated if Ep Eq is dense in Epq.

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).





Banach Bundles

A Banach bundle is a bundle (E, π,X), i.e. a continous opensurjective 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 subsetE×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” ‖ · ‖ : E→ R, such that:I for all x ∈ X, the fiber Ex := π−1(x) is a complex Banach

space (Ex ,+, ·) with the norm ‖ · ‖,I for all xo ∈ X, the family UO,ε

xo = e ∈ E | ‖e‖ < ε, π(e) ∈ O,where O ⊂ X is an open set containing xo ∈ X and ε > 0, is afundamental system of neighbourhoods of 0 ∈ Exo .

back





• Previous/Other Attempts

Several lines of approach to a non-commutative extension ofGel’fand-Naımark duality have been attempted.

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





Bundle Duals / Sectional Representations

J.M.G.Fell probably initiated the trend of sectional reconstructionof a C*-algebra A using, as dual, a suitable bundle over its spectralspace A (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 calledFell-bundles.6

4Fell JMG (1961) The Structure of Algebras of Operator Fields Acta Math106:233-280

5Tomiyama J (1962) Topological Representations of C*-algebras TohokuMath J 14(2):187-204

6Fell J, Doran R (1998) Representations of C*-algebras, Locally CompactGroups and Banach ∗-algebraic Bundles Vol 1-2 Academic Press





Bundle Duals / Sectional Representations

In the celebrated (but vastly ignored) J.Dauns-K.H.Hofmanntheorem7 the algebra is reconstructed via continous sections of abundle simple C*-algebras (see also J.Migda8).

7Dauns J, Hofmann K-H (1968) Representations of Rings by Sections MemAmer Math Soc 83 AMS

Hofmann K H (1972) Representation of Algebras by Continuous SectionsBull Amer Math Soc 78(3):291-373

Hofmann K H (1972) Some Bibliograpical Remarks on: “Representation ofAlgebras by Continuous Sections” Recent Advances in the RepresentationTheory of Rings and C*-Algebras 177-182 (eds) Hofmann K H, Liukkonen J RMemoirs Amer Math Soc 148 (1974)

Hofmann K H (2011) The Dauns-Hofmann Theorem Revisited Journal ofAlgebra and Its Applications 10(1):29-37

8Midga J (1993) Non-commutative Gelfand-Naimark Theorem CommentMath Univ Carolin 34(2):253-255





Structured Pure State Space Duals / Functional

Functional representations of C*-algebras via continuous functionson generalized spectra consisting of (pure) states equipped withextra structures (transition probability, Poisson, . . . ) started withR.Kadison9 and were subsquently considered by F.Schultz,10

P.Kruszynski-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 CommutativeTopological Algebra Memoires Amer Math Soc 7

10Schultz F (1982) Pure States as a Dual Object for C*-algebras CommunMath Phys 82:497-509

11Kruszynski P, Woronowicz S (1982) A Noncommutative Gelfand NaimarkTheorem J Operator Theory 8:361-389

12Fujimoto I (1998) A Gelfand-Naimark Theorem for C*-algebras Pacific JMath 184(1):95-119

13Landsman N (1997) Poisson Spaces with a Transition Probability Reviewsin Mathematical Physics 9(1):29-57





Bundle of Pure States Duals / Functional

In the mostly ignored work by R.Cirelli-A.Mania-L.Pizzocchero14

spectra are projective Kahler bundles over the spectrum A.

14Cirelli R, Lanzavecchia P, Mania A (1983) Normal Pure States of the vonNeumann Algebra of Bounded Operators as Kahler Manifold J Phys A: MathGen 16:3829-3835

Cirelli R, Lanzavecchia P (1984) Hamiltonian Vector Fields in QuantumMechanics Il Nuovo Cimento 79(2):271-283

Abbati M-C, Cirelli R, Lanzavecchia P, Mania A (1984) Il Nuovo CimentoB 83(1):43-60

Cirelli R, Mania A, Pizzocchero L (1990) Quantum Mechanics as anInfinite-dimensional Hamiltonian System with Uncertainty Structure: Part I JMath Phys 31:2891-2897

Cirelli R, Mania A, Pizzocchero L (1990) Quantum Mechanics as anInfinite-dimensional Hamiltonian System with Uncertainty Structure: Part II JMath Phys 31:2898-2903

Cirelli R, Mania A, Pizzocchero L (1994) A Functional Representation forNon-commutative C*-algebras Rev Math Phys 6(5):675-697





NC-topology on Maximal Ideals / Functional

C.Akemann15 was probably the first to describe the dual of aC*-algebra via maximal left-ideals with a “non-commutative” formof topology.Reformulations of commutative Gel’fand-Naımark duality vialocales by B.Banaschewski-C.Mulvey16 (for constructive versionssee also T.Coquand-B.Spitters17) further inpired the usage ofquantales as duals of non-commutative C*-algebras.18

15Akemann C (1971) A Gelfand Representation Theory for C*-algebrasPacific Journal of Mathematics 39 (1):1-11

16Banaschewski B, Mulvey CJ (2000) The Spectral Theory of CommutativeC*-algebras Quaestiones Mathematicae 23:425-464

17Coquand T, Spitters B (2009) Constructive Gelfand Duality for C*-algebrasMathematical Proceedings of the Cambridge Philosophical Society 147:339-344

18Banaschewski B, Mulvey CJ (2006) A Globalisation of the Gelfand DualityTheorem Annals of Pure and Applied Logic 137:62-103





Quantales as Duals

The study of “point free” non-commutative topologies viaquantales, as duals of C*-algebras, has been further pursued byF.Borceux-J.Rosicky-G.Van den Bossche19

C.J.Mulvey-J.W.Pelletier20 andD.Kruml-J.W.Pelletier-P.Resende-J.Rosicky.21

19Borceux F, Rosicky J, Van den Bossche G (1989) Quantales andC*-algebras J London Math Soc 40:398-404

20Mulvey CJ, Pelletier JW (2002) On the Quantisation of Spaces J PureAppl Algebra 175:289-325

21Kruml D, Pelletier J W, Resende P, Rosicky J (2003) On Quantales andSpectra of C*-algebras Appl Categ Structures 11:543-560

Kruml D, Resende P (2004) On Quantales that Classify C*-algebras CahTopol Geom Differ Categ 45:287-296

Resende P (2007) Etale Groupoids and their Quantales Advances inMathematics 208(1):147-209





Topos Theory Approaches

Several works suggest to reconstruct C*-algebras via theGrothendieck topoi of their commutative subalgebras

I A.Doring22 (based on works by C.Isham-J.Butterfield).

I C.Heunen-N.Landsman-B.Spitters-S.Wolters23

22Doring A (2012) Generalised Gelfand Spectra of Nonabelian UnitalC*-Algebras arXiv:1212.2613Doring A (2012) Flows on Generalised Gelfand Spectra of Nonabelian UnitalC*-Algebras and Time Evolution of Quantum Systems arXiv:1212.4882

23Heunen C, Landsman N, Spitters B (2009) A Topos for AlgebraicQuantum Theory Communications in Mathematical Physics 291:63-110

Heunen C, Landsman NP, Spitters B, Wolters S (2011) The GelfandSpectrum of a Noncommutative C*-algebra: a Topos-theoretic Approach JAustr Math Soc 90:39-52Wolters S (2013) A Comparison of Two Topos-theoretic Approaches toQuantum Theory Communications in Mathematical Physics 317(1):3-53





Other Categorical Sheaves / Topos Theory Approaches

Among the ongoing recent efforts towards Gel’fand-Naımarkduality using topoi or sheaves we mention:

I S.Henry24

I C.Flori-T.Fritz25

24Henry S (2014) Localic Metric Spaces and the Localic Gelfand DualityarXiv:1411.0898 [math.CT]

Henry S (2014) Constructive Gelfand Duality for Non-unital CommutativeC*-algebras arXiv:1412.2009 [math.CT]

Henry S (2015) Toward a Non-commutative Gelfand Duality: BooleanLocally Separated Toposes and Monoidal Monotone Complete C*-categoriesarXiv:1501.07045 [math.CT]

25Flori C, Fritz T (2015) (Almost) C*-algebras as Sheaves with Self ActionarXiv:1512.01669





Other Approaches

Apart from the extremely vast literature on classification ofC*-algebras, some quite recent attempts to produceGel’fand-Naımark dualities, at least for some reasonable classes ofC*-algebras have been put forward by:

I C.Heunen-M.Reyes26

I N.de Silva27

26Heunen C, Reyes ML (2014) Active Lattices Determine AW*-algebrasJournal of Mathematical Analysis and Applications 416:289-313

Heunen C, Reyes ML On Discretization of C*-algebras arXiv:1412.172127de Silva N (2014) From Topology to Noncommutative Geomtery: K-theory

arXiv:1408.1170Paolo Bertozzini Non-commuative Gel’fand-Naımark Duality




The Present Approach

In our proposed approach, C*-algebras are reconstructed bysections of a bundle in the same tradition of J.M.G.Fell,J.Dauns-K.H.Hofmann, R.Cirelli-A.Mania-L.Pizzocchero andN.Landsman.

Contrary to these previous cases,

I our bundles have only one-dimensional fibers,

I all differential geometric features (Kahler / Poisson structures)are eliminated from the spectrum and “substituted” by ahorizontal 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.




The Transition Amplitude BundleThe Transition Amplitude SpaceNon-commutative Spaceoids

• Basic Constructions

I The Transition Amplitude Bundle

I The Transition Amplitude Space

I Non-commutative Spaceoids





• The Transition Amplitude Bundle





The Bundle of Pure StatesI The family SA of states of unital C*-algebra A consists of all

the linear maps ω : A→ C that areI positive: ω(x∗ x) ∈ C+, for all x ∈ A,I normalized: ω(1A) = 1C.

I By Gel’fand-Naımark-Segal theorem, every state ω induces arepresentation πω : A→ L(Hω) and a unit vector ξω ∈ Hωsuch that ω(x) = 〈ξω | πω(x)ξω〉Hω , for all x ∈ A.

I Let PA ⊂ SA denote the family of pure states of A: these arethose states ω ∈ SA such that πω is irreducible.

I PA ⊂ SA ⊂ A∗ is equipped with the weak∗-topology: theweakest topology making continuous all the mapsx : ω 7→ ω(x), for all x ∈ A.

I PA is a bundle over A, the usual “spectrum of A”, that is thequotient space of PA under the equivalence relation of unitaryequivalence of irreducible GNS-representations.





The Base Spectrum

The base space of our spectra is just a fiberwise horizontalcategorification of the previous bundle PA over A:instead of considering only the points ω ∈ PA, we consider allpossible 1-arrows (ordered pairs) between pure states in the samefiber of bundle PA.

The fiberwise product

PA ×APA := (ω, ρ) | [πω] = [πρ] ⊂ PA × PA,

with the topology induced by the product of the weak∗-topologyon PA, is a bundle over A with pair groupoids as fibers.

PA ×APA is a bundle gerbe of pair groupoids over A and is the

base spectrum of A.





The Von Neumann Enveloping BundleI If ω ∈ PA, the GNS-representation πω is irreducible, henceπω(A)′′ = L(Hω). Furthermore, if [ω] = [ρ], ∃U : Hω → Hρunitary such that πρ(x) = Uπω(x)U∗, for all x ∈ A.

I U is unique up to a phase, since U1∗U2 ∈ πω(A)′ = C · 1Hω ,

but the unital ∗-isomorphim πω(A)′′AdU−−→ πρ(A)′′ is unique.

I For every p ∈ A consider the pair groupoid with objectsL(Hω), with [ω] = p, and 1-arrows the unique unital∗-isomorphisms AdU induced by the unitaries U intertwiningthe given representations and construct the W*-algebra A′′p oforbits of such groupoid (A′′p ' L(Hω), ∀ω ∈ p).

I The bundle of (type I factors) W*-algebras A′′p over A is theVon Neumann enveloping bundle of A.⊕p∈AA

′′p coincides with the atomic part of the Arens

W*-envelope A∗∗ of A.





The Total Spectrum

I There is a natural embedding of (PA)p into A′′p that to everyω ∈ p associates | ω 〉〈ω | the (orbit of the) one-dimensionalprojector | ξω 〉〈 ξω | in L(Hω) ' A′′p.

I For every ω, ρ ∈ (P)p, the “corner space”| ω 〉〈ω | A′′p | ρ 〉〈 ρ | is one-dimensional.

I The total spectrum EA of A is the disjoint union⊎(ω,ρ)∈PA×A

PA| ω 〉〈ω | A′′p | ρ 〉〈 ρ | as a bundle over the

base spectrum PA ×APA.

I EA is an involutive category: in each fiber (EA)p thecomposition and the involution are, for x , y ∈ A′′p, ω, ρ, η ∈ p:

(| ω 〉〈ω | x | ρ 〉〈 ρ |) (| ρ 〉〈 ρ | y | η 〉〈 η |):=| ω 〉〈ω | x | ρ 〉〈 ρ | y | η 〉〈 η |,

(| ω 〉〈ω | x | ρ 〉〈 ρ |)∗ :=| ρ 〉〈 ρ | x∗ | ω 〉〈ω | .





The Non-commutative Gel’fand Transform

I Every x ∈ A determines a section of the Von Neumannenveloping bundle x 7→ πp(x) (the orbit of πω(x)).

I Every x ∈ A induces a section x of the total spectrum bundleEA over PA ×A

PA given, for all p ∈ A and for all ω, ρ ∈ p by:

x(ω, ρ) :=| ω 〉〈ω | πp(x) | ρ 〉〈 ρ | .

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





The Transition Amplitude Bundle

I The total spectrum EA of A becomes a Fell bundle with thetopology induced by the bases of “tubular neighborhoods”Ux ,εO of any point eo ∈ (EA)(ωo ,ρo) defined as follows:

for every open neighborhood O ⊂ PA ×APA of (ωo , ρo), for

every x ∈ A such that x(ωo , ρo) = eo , for every ε > 0,

Ux ,εO := e ∈ EA | ∀(ω, ρ) ∈ O, ‖x(ω, ρ)− e‖ < ε.

The Fell bundle EA is the transition amplitude bundle of A.

I For every p ∈ A, the sub-bundle (EA)p is actually aone-dimensional C*-category with objects (PA)p, hence thetotal spectrum of A can be described alternatively as a bundleof one-dimensional C*-categories.





Bundle Uniformity on the Transition Amplitude BundleSince the spaces EA and BA := PA ×A

PA are not usuallycompact, we will need to consider uniformly continuous sections.

I There is a standard uniform structure UBAon the space

BA := PA ×APA ⊂ PA × PA ⊂ A∗ ×A∗ obtained by

restricting the product of the uniform structure induced by theweak*-topology on A∗.

I A bundle uniformity UEAfor the transition amplitude bundle

is given by the filter generated on EA × EA by this filterbaseof subsets of EA ×BA

EA:

Uε := (e1, e2) ∈ EA ×BAEA | ‖e1 − e2‖EA

< ε, ε > 0.

Note that this uniformity does not induce the already definedtopology on the total space E of the transition amplitudebundle, hence the total space is not itself a uniform space!





• The Transition Amplitude Space





The Gauge Section of a Unital C*-algebra

Every unital C*-algebra determines a very special gauge sectionof its transition amplitudes bundle EA: the Gel’fand transform 1Aof the identity element 1A ∈ A:

1A : (ω, ρ) 7→| ω 〉〈ω | 1A | ρ 〉〈 ρ |, ∀p ∈ A, ∀ω, ρ ∈ (PA)p.

The gauge section is Hermitian: 1A(ω, ρ)∗ = 1A(ρ, ω).In general it is not a subcategory of EA.

Informally, the gauge section allows the specification of bundlegerbe acting as symmetry morphisms of a “horizontal categorified”site of “gauge blocks” inside the transition amplitudes bundle; astructure that is crucial in the reconstruction of a C*-algebra(isomorphic to A) form EA.





The Gauge Transition Amplitude Space

The gauge section 1A induces on the set (PA)p a structure oftotal Fell bundle-valued transition amplitude space28:

I for all ω ∈ (PA)p, 1A(ω, ω) =| ω 〉〈ω |= 1(EA)(ω,ω),

I 1A(ω, ρ)∗ = 1A(ρ, ω), for all ω, ρ ∈ (PA)p,

I there exists at least one frame i.e. a subset F ⊂ (PA)p suchthat, for all ω, ρ ∈ (PA)p,1A(ω, ρ) =

∑θ∈F 1A(ω, θ) 1A(θ, ρ).

I ⊂ (PA)p is orthonormal if ω 6= ρ ∈ F, ⇒ 1A(ω, ρ) = 0(EA)(ω,ρ).

Frames are maximal othonormal.

The transition amplitude space is total if every maximalorthonormal set is a frame.

28For complex-valued transition amplitude spaces, see section 4.5 in:S.Gudder (1988) Quantum Probability, Elsevier.





The Gauge Bundle Gerbe

I Our gauge transition amplitude space is total, hence for everyp ∈ A, gauge frame blocks coincide with maximalorthonormal sets of pure states i.e. sets F ⊂ (PA)p such thatthe “matrix” 1A|F×F, for ω, ρ ∈ F, is the “identity”:

1A(ω, ρ) =

| ω 〉〈ω |= 1(EA)(ω,ω)

, ω = ρ,

0(EA)(ω,ρ), ω 6= ρ.

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





The Spectral Bundle Gerbe

Restricting the transition amplitude bundle EA onto each one ofthe gauge blocks F1 × F2 of the pair groupoid FA of gauge blockframes we obtain a bundle gerbe enriched in the Morita 2-groupoidof 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 bimodulesbetween 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-Naımark Duality




The Transition Amplitude Uniformity

Every Fell bundle-valued transition amplitude space (P, γ)determines a uniformity on P: the family of entourages of ∆P isgiven by the family of subsets given, for 0 < ε < 1, by:U ε := (ω, ρ) ∈ P × P | ‖γ(ω, ρ)‖ > 1− ε ⊂ P × P.

It will be a requirement for our spaceoids to assume that fibrewisethe uniform structure of P coincides with the uniform structureinduced by its Fell bundle-valued transition amplitude spacestructure. This requirement is always satisfied for spectralspaceoids and it is the counterpart, in our setting, of thecoincidence between the weak*-uniformity and the Kahler metricuniformity in the phase-space of a quantum system.





The Enveloping C*-algebra of a C*-category

Given a C*-category C, its C*-envelope is a unital C*-algebraΞ(C) with a ∗-functor ι : C→ Ξ(C) that satisfies the followinguniversal factorization property: for any other ∗-functor φ : C→ A

into a unital C*-algebra A, there exists a unique unital∗-homomorphism φ : Ξ(C)→ A such that φ = φ ι.

Proposition

Every C*-category C admits a C*-envelope Cι−→ Ξ(C).

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





• Non-commutative Spaceoids





Non-commutative Spaceoids 1

A non-commutative spaceoid (E , π,P, χ,X , γ) is a saturatedunital Fell line-bundle (E , π,P ×X P) over a topological bundle ofpair groupoids (P ×X P, χ,X ), where χ : P → X is a surjectiveopen continuous projection from a uniform Hausdorff space P ontothe quotient space X ; equipped with a continuous sectionγ : P ×X P → E inducing on P a structure of Fell bundle-valuedsaturated full transition amplitude space that is compatible withthe 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);





Non-commutative Spaceoids 2I P

χ−→ X is a surjective map onto the set X hence, when X isequipped with the quotient topology, the projection map χ iscontinuous and open; as a further consequence, whenequipped with the restriction of the product topologyP ×X P := (ω, ρ) ∈ P × P | χ(ω) = χ(ρ) ⊂ P × P, is atopological groupoid, and so P ×X P → X is a topologicalbundle of pair groupoids over X ;

I (E , π,P ×X P) is a saturated unital Fell line-bundle over thetopological groupoid P ×X P;

I P ×X Pγ−→ E is a continuous section of the previous bundle;

I (P, γ) is a Fell bundle-valued transition amplitude space thatis total and algebraically saturated; Isbell reflective subcategory

I the uniform structure on P is fibrewise “compatible” with itsuniform structure as a transition amplitude space (P, γ).





Morphisms of Non-commutative Spaceoids

A morphism of non-commutative spaceoids

(E 1, π1,P1, χ1,X 1, γ1)(λ,Λ)−−−→ (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:

I λ : P1 → P2 is a uniformly continuous map such thatχ2 λ = χ1, hence it induces a necessarily continuousquotient map [λ] : X 1 → X 2 and(λ, λ) : P1 ×X 1 P1 → P2 ×X 2 P2 is necessarily a continuoushomomorphism of bundles of pair groupoids;

I Λ : λ•(E 2)→ E 1 is a morphism of Fell bundles from the(λ, λ)-pull-back of E 2 to E 1;

I (λ,Λ) : (P1, γ2)→ (P2, γ2) is a morphism of Fellbundle-valued transition amplitude spacesi.e. Λ(γ2(λ(p), λ(q))) = γ1(p, q), for all p, q ∈ P1.





The Category S of Non-commutative Spaceoids

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

(λ′,Λ′) (λ,Λ) := (λ′ λ, Λ λ•(Λ′) ΘE3

λ′,λ)

where E 1 (λ,Λ)−−−→ E 2 (λ′,Λ′)−−−−→ E 3 and whereΘE3

λ′,λ : (λ′ λ)•(E 3)→ λ•((λ′)•(E 3)) is the canonical isomorphismbetween standard pull-backs of bundles.





The Category A of Unital C*-algebras

For us a morphism of unital C*-algebras will be a

∗-homomorphism A1φ−→ A2 whose pull-back φ• : SA2 → SA1 is

pure state preserving: φ•(PA2) ⊂ PA1 .30

Unital ∗-epimorphism are a special case.

We have a category A of unital C*-algebras with the usualcomposition of such pure state preserving unital ∗-homomorphisms.

30This choice let us recover continuous maps in the usual commutativeGel’fand duality and simultaneously allows to limit the study to spaceoidsdefined using only pure states. It is possible to consider more general classes ofmorphism.




The Section and Spectrum FunctorsThe Gel’fand IsomorphismThe Evaluation IsomorphismCommutative and Non-commutative Gel’fand-Naımark Duality

• Gel’fand-Naımark Duality





The Section Functor Γ

The section functor Γ : S → A , associates to everynon-commutative spaceoid (E , π,P, χ,X , γ) the unital C*-algebraΓ(E ) consisting of all the sections σ : P ×X P → E such that:

I σ is continuous and its restriction σ|∆Pto the diagonal

∆P := (p, p) | p ∈ P of P is uniformly continuous;31

I σ is gauge invariant i.e. for every pair of γ-orthonormal framesF1,F2 ⊂ P, σ|F1 and σ|F2 are related byσ|F2 = Adγ|F2×F1

(σ|F1) = γ|F2×F1 σ|F1 γ|?F2×F1,

where and ? denote here the convolution product and theadjoint involution in the C*-category with objects themaximal orthonormal γ-frames of P and 1-arrows therestrictions σ|F1×F2 .

31Note that since E |∆P is trivial line-bundle equipped with a uniformity, thiscondition is perfectly defined.





The Algebraic Saturation Condition (tentative)

Given a point o ∈ X , let Fo be the family of γ-frames in thetransition amplitude space χ−1

o ⊂ P ×X P.

Consider for all possible choice functions (UF )F∈Fo , where UF is aunitary section of E |F , the family (Uγ

F )F∈Fo of their γ-orbitsdenoted by Uγ

F ∈ Ξ∗∗γ (E |χ−1(o)).

The spaceoid is algebraically saturated if, for all o ∈ X , thereexists at least a choice map (UF )F∈Fo such that (Uγ

F )F∈Fo is thefamily of all unitaries in Ξ∗∗γ (E |χ−1(o)).

This condition identifies the full reflective subcategory of thecategory of algebraic spaceoids that is in Isbell duality with thecategory of atomic W*-bundles.

back to spaceoids





The Section Functor Γ on Morphisms

The section functor Γ associates to a morphism of spaceoids

E 1 (η,Ω)−−−→ E 2 the usual pull-back of sections:σ 7→ Γ(η,Ω)(σ) ∈ Γ(E 1), for all σ ∈ Γ(E 2), whereΓ(η,Ω)(σ) : (p, q) 7→ Ω(σ(η(p), η(q))) ∈ E 1, for all p, q ∈ P1.

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.





The Spectrum Functor ΣThe spectrum functor Σ : A → S associates to every unitalC*-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 theweak*-uniformity,

I χA : PA → XA := A is the quotient map onto the usualspectrum of A (the family of equivalence classes of untarilyequivalent representations),

I (EA, πA,PA ×XAPA) is the transition amplitude Fell

line-bundle of A, as already constructed,

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





The Spectrum Functor Σ on Morphisms

The spectrum functor associates to every unital ∗-epimorphism ofunital C*-algebras φ : A1 → A2 a morphism of non-commutative

spaceoids Σ(A2)(λφ,Λφ)−−−−−→ Σ(A1) where:

I λφ : PA2 → PA1 is the usual φ-pull-back of pure states:λφ : ω 7→ ω φ, for all ω ∈ PA2 ;32

I Λφ : λ•φ(EA1)→ EA2 is the disjoint union, for ω, ρ ∈ PA2 , ofthe fiberwise linear relations(Λφ)(ω,ρ)λ

•φ(EA1)(ω,ρ) → (EA2)(ω,ρ), given for x ∈ A1 by:

| λφ(ω) 〉〈λφ(ω) | x | λφ(ρ) 〉〈λφ)(ρ) | 7→ | ω 〉〈ω | φ(x) | ρ 〉〈 ρ | .

The spectrum functor Σ : A → S is contravariant.

32The pure state preserving property of φ is necessary condition in order todefine this map.





The Gel’fand Transform

The non-commutative Gel’fand transformGA : A→ Γ Σ(A) = Γ(EA) is simply the map that to everyx ∈ A associates the γA-invariant section x : PA ×XA

PA → EA

given by x(ω, ρ) :=| ω 〉〈ω | x | ρ 〉〈 ρ |, for all ω, ρ ∈ PA.

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





The Gel’fand Isomorphism

TheoremThe Gel’fand transform G : IA → Γ Σ is a natural isomorphism.





The Evaluation Transform

The evaluation transform EE : E → Σ Γ(E ) from the spaceoidE to the spectral spaceoid of the unital C*-algebra Γ(E ) consistsof a morphism of spaceoids EE := (ηE ,ΩE ) as follows:

I ηE : P → PΓ(E) associates to every p ∈ P the map

ηEp := ζEp evEp : Γ(E )→ C, σ 7→ ζEp (σ(p, p)) obtained

composing the function evEp : Γ(E )→ Γ(E )(p,p) that evaluatesevery section σ ∈ Γ(E ) in the point (p, p) ∈ P ×X P with thecanonical Gel’fand-Mazur isomorphism ζEp : Γ(E )(p,p) → Cbetween one-dimensional unital C*-algebras;

I ΩE : (ηE )•(EΓ(E))→ E is the relation fiberwise defined as

ΩE(p,q) : e 7→ σ(p, q), for every σ ∈ Γ(E ) such that

σ(ηE (p), ηE (q)) = e, where p, q ∈ P ande ∈ Σ Γ(E )(ηE (p,q)) = (ηE )•(EΓ(E))(p,q).





The Evaluation Isomorphism Theorem

TheoremThe evaluation transform is a natural isomorphism IS

E−→ Σ Γ.

Crucial ingredients here are the saturation conditions on thespaceoid. The proof makes use of the non-commutativeStone-Weierstrass theorem by J.Glimm:

A unital ∗-subalgebra B of a unital C*-algebra A that separatesthe states in the weak*-closure PA ⊂ SA of the set of pure statesof A is norm dense in A.





Commutative / Non-commutative Gel’fand Duality 1

We have the following pair of commutative diagrams of functors:

SΓ // A

TΓc

//

F

OO

C

I

OO S AΣoo

T

F

OO

C

I

OO

Σcoo

where TΓc

&&

Σc

gg C is the commutative Gel’fand-Naımark duality.

I : C → A is the usual inclusion functor.






F : T → S is the covariant functor that associates to everycompact Hausdorff space T the topological spaceoid(ET , πT ,PT , χT ,XT , γT ) with XT := T , PT := T , χT := ιT theidentity map of T , (ET , πT ,PT ×XT PT ) the trivial complexline-bundle on the diagonal ∆T = T ×T T of T and γT theidentity constant section of ET .33

F associates to every continuous map f : T1 → T2 of compactHausdorff topological spaces the morphism of spaceoids(λf ,Λf ) := (f ,F ), where F is the canonical isomorphism betweentrivial complex line-bundles f •(ET2)→ ET1 over T1 (thef -pull-back of a trivial bundle is trivial).

33Notice that every compact Hausdorff space is equipped with a uniqueuniformity inducing its topology (enturages are the just neighborhoods of thediagonal of X ) and continuous maps are uniform for such uniformity.






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

SES

// S

TET

//

F

OO

C

F

OO AGA

// A

C

I

OO

GC// C

I

OO

finally prove that our non-commutative Gel’fand-Naımark duality isa natural extension of the usual commutative case (and reduces toit by restricting to the embedded full sub-categories F(T ) ⊂ Sand I(C ) ⊂ A ).




• What Are the “Spectra” ofC*-algebras?




The Two “Souls” of Geometry

space

vv ((∩

descent // ? quantum space ? covarianceoo

Descent theory (gluing) Covariance (transport)

sheaves/stacks Klein-Cartan geometriesGrothendieck 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.




Non-commutative Klein-Cartan Geometries?

Klein’s Erlangen program characterizes geometry from its group ofsymmetries: Klein’s geometries are homogeneous spaces.34

Cartan dealt with local symmetries: Cartan’s geometries arebundles of homogeneus spaces with a connection.35

We need an understanding of non-commutative covariance andtransport (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-Naımark Duality



Non-commutative Topoi / Descent Theory?We probably need a version non-commutative topos theorywhere categories of tensor products and involutions of bimodulestake the place of the usual Cartesian closed categories environment.

I P.Cartier36 (see also37)

I F. van Oystaeyen38 (see also39)

I M.Kontsevich, A.Rosenberg (NC descent theory)40

I C.Flori, T.Fritz (Gleaves)41

36A Mad Day’s Work: from Grothendieck to Connes and Kontsevich: TheEvolution of Concepts of Space and Symmetry (2001) Bull Amer Math Soc38:389-408.

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.




Thank You for Your Kind Attention!

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

We acknowledge the partial support from

I the Department of Mathematics and Statistics in ThammasatUniversity and

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


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