the quantum gromov-hausdorff propinquityquantum tori and their finite dimensional approximations...

QuantumPropinquity

FrédéricLatrémolière,

PhD

NoncommutativeMetric Spaces

The QuantumGromov-HausdorffPropinquity

Annex

Thanks!

The Quantum Gromov-Hausdorff Propinquity

Frédéric Latrémolière

Great Plain Operator Theory Symposium 2013University of California, Berkeley

QuantumPropinquity


PhD



Annex

Object of the talk

Problem addressed in this talkWe present a brief survey of our most recent research innoncommutative metric geometry focused on theconstruction of topologies on classes of quantum metric spaces.

The Quantum Gromov-Hausdorff Propinquity, F.Latrémolière, Submitted (2013), 75 pages, ArXiv:1302.4058.Locally Compact Quantum Metric Spaces, F. Latrémolière,Journal of Functional Analysis 264 (2013) 1, 362–402,ArXiv: 1208.2398

QuantumPropinquity


PhD



Annex

Structure of this Presentation

1 Noncommutative Metric SpacesThe Monge Kantorovich distanceCompact Quantum Metric SpacesLocally Compact Quantum Metric Spaces

2 The Quantum Gromov-Hausdorff PropinquityThe quantum Gromov-Hausdorff distanceThe Quantum PropinquityQuantum Tori and their Finite Dimensionalapproximations

QuantumPropinquity


PhD

NoncommutativeMetric SpacesThe MongeKantorovich distance

Compact QuantumMetric Spaces

Locally CompactQuantum MetricSpaces


Annex

GPS



QuantumPropinquity


PhD





Annex

Gel’fand duality and Noncommutative Geometry

Theorem (Gel’fand-Naimark duality)

The category of C*-algebras, with *-morphisms as arrows, is aconcrete realization of the dual category of locally compact spaces,with proper continuous maps as arrows.

Founding Allegory of Noncommutative Geometry

Noncommutative geometry is the study of noncommutativegeneralizations of algebras of functions on spaces.

QuantumPropinquity


PhD





Annex




Founding Allegory of Noncommutative Metric Geometry

Noncommutative metric geometry is the study ofnoncommutative generalizations of algebras of Lipschitzfunctions on metric spaces.

QuantumPropinquity


PhD





Annex




MotivationNoncommutative metric geometry aims at providing afoundation for constructions of approximations in quantumphysics based upon quantum spaces, and provides a newapproach to developing a geometry for quantum spacesfrom the metric geometry of their state spaces. The key toolsare metrics on classes of quantum metric spaces.

QuantumPropinquity


PhD





Annex

Lipschitz Seminorms

A natural dual object to a metric is the Lipschitz seminorm:

Definition

Let (X, m) be a metric space. For any function f : X→ R,define:

L(f ) = sup{|f (x)− f (y)|

m(x, y): x, y ∈ X, x 6= y

}.

Questions

1 Can we recover the metric from its Lipschitzseminorm?

2 What makes a Lipschitz seminorm special among allseminorms?

QuantumPropinquity


PhD





Annex

Lipschitz Seminorms


Definition


L(f ) = sup{|f (x)− f (y)|

m(x, y): x, y ∈ X, x 6= y

}.

Questions1 Can we recover the metric from its Lipschitz

seminorm?

2 What makes a Lipschitz seminorm special among allseminorms?

QuantumPropinquity


PhD





Annex

Lipschitz Seminorms


Definition


L(f ) = sup{|f (x)− f (y)|

m(x, y): x, y ∈ X, x 6= y

}.

Questions1 Can we recover the metric from its Lipschitz

seminorm?2 What makes a Lipschitz seminorm special among all

seminorms?

QuantumPropinquity


PhD





Annex

A distance on the state space

The self-adjoint part of a C*-algebra A is denoted by sa(A)while its state space is denoted by S (A).

Definition

A unital Lipschitz pair (A, L) is a unital C*-algebra A and adensely defined seminorm L on sa(uA) such that{a ∈ sa(A) : L(a) = 0} = R1A.

Definition (Kantorovich (1940), Kantorovich-Rubinstein (1958),Wasserstein (1969), Dobrushin (1970), Connes (1981))The extended Monge-Kantorovich metric mkL on S (A)associated with a Lipschitz pair (A, L) is defined for allϕ, ψ ∈ S (A) by:

mkL(ϕ, ψ) = sup {|ϕ(a)− ψ(a)| : a ∈ sa(A), L(a) 6 1} .

QuantumPropinquity


PhD





Annex

The classical extended Monge-Kantorovich metric

TheoremLet (X, m) be a locally compact metric space and identify X withthe space of Dirac probability measures over X (i.e. the Gel’fandspectrum of C0(X)). Then:

∀x, y ∈ X m(x, y) = mkL(x, y).

The extended Monge-Kantorovich metric is well-behavedwhen working over compact metric spaces:

Theorem (Wasserstein, Dobrushin (1970))Let (X, m) be a compact metric space. The extendedMonge-Kantorovich metric mkL associated with m is a metricwhich metrizes the weak* topology on the state space S (C(X)) ofC(X).

QuantumPropinquity


PhD





Annex

Compact Quantum Metric Spaces

Based on this observation, Rieffel introduced:

Definition (Rieffel, 1998)

A compact quantum metric space (A, L) consists of anorder-unit space A and a seminorm L densely defined on A,satisfying:

{a ∈ A : L(a) = 0} = R1A,

and such that the distance:

mkL : ϕ, ψ ∈ S (A) 7→ sup{|ϕ(a)− ψ(a)| : a ∈ A, L(a) 6 1}

metrizes the weak* topology on the state space S (A). Theseminorm L is then called a Lip-norm.

Compact quantum metric spaces are characterized by normprecompactness of their pointed unit Lip-ball.

QuantumPropinquity


PhD





Annex

Examples: Ergodic Actions of Compact Groupswith continuous Lengths

For any C*-algebra A, let sa(A) be its self-adjoint part and‖ · ‖A be its norm.

Theorem (Rieffel, 1998)

Let α be a strongly continuous action of a compact group G on aunital C*-algebra A and ` be a continuous length function on G.Let e ∈ G be the unit of G. For all a ∈ A, define:

L(a) = sup{‖αg(a)− a‖A

`(g): g ∈ G \ {e}

}.

If {a ∈ A : ∀g ∈ G αg(a) = a} = C1A, then (sa(A), L) is acompact quantum metric space.

This result uses the fact that spectral subspaces for suchactions are finite dimensional (Hoegh-Krohn, Landstad,Stormer, 1981).

QuantumPropinquity


PhD





Annex

Locally Compact Quantum Metric Spaces

The extended Monge-Kantorovich metric is notwell-behaved for non-compact metric spaces, due toescape-at-infinity problems.

Dobrushin introduced in 1970 a metric form oftightness for sets of probabilities. The extendedMonge-Kantorovich metric metrizes the weak*topology on D-tight sets.I introduced in (L., 2013) a noncommutativegeneralization of D-tightness for sets using my notionof topography.I then define and characterize quantum locally compactmetric spaces using pre-compactness of the pointedLip-ball for a new topology on C*-algebras withtopographies.My work uses the multiplication explicitly.

QuantumPropinquity


PhD





Annex


The extended Monge-Kantorovich metric is notwell-behaved for non-compact metric spaces, due toescape-at-infinity problems.Dobrushin introduced in 1970 a metric form oftightness for sets of probabilities. The extendedMonge-Kantorovich metric metrizes the weak*topology on D-tight sets.

I introduced in (L., 2013) a noncommutativegeneralization of D-tightness for sets using my notionof topography.I then define and characterize quantum locally compactmetric spaces using pre-compactness of the pointedLip-ball for a new topology on C*-algebras withtopographies.My work uses the multiplication explicitly.

QuantumPropinquity


PhD





Annex


The extended Monge-Kantorovich metric is notwell-behaved for non-compact metric spaces, due toescape-at-infinity problems.Dobrushin introduced in 1970 a metric form oftightness for sets of probabilities. The extendedMonge-Kantorovich metric metrizes the weak*topology on D-tight sets.I introduced in (L., 2013) a noncommutativegeneralization of D-tightness for sets using my notionof topography.

I then define and characterize quantum locally compactmetric spaces using pre-compactness of the pointedLip-ball for a new topology on C*-algebras withtopographies.My work uses the multiplication explicitly.

QuantumPropinquity


PhD





Annex


The extended Monge-Kantorovich metric is notwell-behaved for non-compact metric spaces, due toescape-at-infinity problems.Dobrushin introduced in 1970 a metric form oftightness for sets of probabilities. The extendedMonge-Kantorovich metric metrizes the weak*topology on D-tight sets.I introduced in (L., 2013) a noncommutativegeneralization of D-tightness for sets using my notionof topography.I then define and characterize quantum locally compactmetric spaces using pre-compactness of the pointedLip-ball for a new topology on C*-algebras withtopographies.

My work uses the multiplication explicitly.

QuantumPropinquity


PhD





Annex


The extended Monge-Kantorovich metric is notwell-behaved for non-compact metric spaces, due toescape-at-infinity problems.Dobrushin introduced in 1970 a metric form oftightness for sets of probabilities. The extendedMonge-Kantorovich metric metrizes the weak*topology on D-tight sets.I introduced in (L., 2013) a noncommutativegeneralization of D-tightness for sets using my notionof topography.I then define and characterize quantum locally compactmetric spaces using pre-compactness of the pointedLip-ball for a new topology on C*-algebras withtopographies.My work uses the multiplication explicitly.

QuantumPropinquity


PhD


The QuantumGromov-HausdorffPropinquityThe quantumGromov-Hausdorffdistance

The QuantumPropinquity

Quantum Tori andtheir FiniteDimensionalapproximations

Annex

GPS



QuantumPropinquity


PhD





Annex

Convergence of Compact Metric Spaces

Definition

Let (X, mX) and (Y, mY) be two compact metric spaces. Adistance m on X ä Y is admissible for (mX, mY) when thecanonical injections (X, mX) ↪→ (X ä Y, m) and(Y, mY) ↪→ (X ä Y, m) are isometries.

NotationThe Hausdorff distance on the compact subsets of a metric space(X, m) is denoted by hm.

Definition (Gromov, 1981)

The Gromov-Hausdorff distance between two compact metricspaces (X, mX) and (Y, mY) is the infimum of the set:

{hm(X, Y) : m is admissible for (mX, mY)} .

QuantumPropinquity


PhD





Annex

Admissible Lip-norms

Let (A, LA) be a compact quantum metric space, B be aunital C*-algebra, and π : sa(A)→ sa(B) be a positivelinear surjection.

1 The quotient of LA for π is the seminorm on sa(B)defined for b ∈ sa(B) by:

LB(b) = inf {LA(a) : a ∈ sa(A), π(a) = b} .

It is a Lip-norm on B.2 There is a dual affine continuous injection

π∗ : S (B)→ S (A).


A Lip-norm L on A⊕B is admissible for (LA, LB) when thequotient seminorm of L on A (resp. B) for the canonicalsurjection A⊕B� A is LA (resp. for the canonicalsurjection A⊕B� B is LB).

QuantumPropinquity


PhD





Annex

The quantum Gromov-Hausdorff distance

Proposition (Rieffel, 1999)

If L is an admissible Lip-norm for (LA, LB) then the canonicalinjections (S (A), mkLA) ↪→ (S (A⊕B), mkL) is an isometry(and similarly with (B, LB)).

We henceforth identify S (A) and S (B) with their imageby the dual of the canonical surjection A⊕B� A andA⊕B� B.


The quantum Gromov-Hausdorff distancedistq((A, LA), (B, LB)) between two compact quantummetric spaces (A, LA) and (B, LB) is the infimum of the set:

{hmkL(S (A), S (B)) : L is admissible for (LA, LB)} .

QuantumPropinquity


PhD





Annex

Basic Properties of distq

Theorem (Rieffel, 2000)

For any three quantum compact metric spaces (A, LA), (B, LB)and (D, LD), we have:

1 diam ((S (A), mkLA) + diam (S (B), mkLB) >distq((A, LA), (B, LB)) = distq((B, LB), (A, LA)) > 0,

2 distq((A, LA), (D, LD)) 6distq((A, LA), (B, LB)) + distq((B, LB), (D, LD)),

3 distq((A, LA), (B, LB)) = 0 iff there exists aorder-unit-space isomorphism from sa(A) to sa(B) whosedual map is an isometry from (S (B), mkLB) to(S (A), mkLA).

QuantumPropinquity


PhD





Annex

Beyond distq

Distance Zero ProblemHow to get *-isomorphism as necessary for distance zero?

1 Replace the state space by 2× 2-matrix-valuedcompletely positive maps: Kerr’s matricialGromov-Hausdorff distance

2 Replace the state space by the graph of themultiplication restricted to the unit Lip-ball: Li’sC*-algebraic distance

3 Work entirely within the C*-algebra category.

First attempt: Li’s nuclear distance.Second attempt: FL’s quantum propinqity for C*-metricspaces.

QuantumPropinquity


PhD





Annex

Beyond distq




3 Work entirely within the C*-algebra category.First attempt: Li’s nuclear distance.

Second attempt: FL’s quantum propinqity for C*-metricspaces.

QuantumPropinquity


PhD





Annex

Beyond distq




3 Work entirely within the C*-algebra category.First attempt: Li’s nuclear distance.Second attempt: FL’s quantum propinqity for C*-metricspaces.

QuantumPropinquity


PhD





Annex

The Leibniz inequality

The main problem of distq is that it does not involve themultiplication at all, and in fact, neither does the definitionof compact quantum metric spaces.

Yet, all importantexamples of quantum locally compact metric space have avery important additional property:

Definition

A seminorm L on a C*-algebra A has the Leibniz propertywhen:

∀a, b ∈ A L(ab) 6 ‖a‖AL(b) + L(a)‖b‖A.

In most cases, the Lip-norms of quantum locally compactmetric space comes from derivations, spectral triples orsimilar constructions which gives the Leibniz property. Thisis a natural connection between metric and multiplicativestructures of quantum locally compact metric space.

QuantumPropinquity


PhD





Annex

The Leibniz inequality

The main problem of distq is that it does not involve themultiplication at all, and in fact, neither does the definitionof compact quantum metric spaces. Yet, all importantexamples of quantum locally compact metric space have avery important additional property:

Definition

A seminorm L on a C*-algebra A has the Leibniz propertywhen:

∀a, b ∈ A L(ab) 6 ‖a‖AL(b) + L(a)‖b‖A.

In most cases, the Lip-norms of quantum locally compactmetric space comes from derivations, spectral triples orsimilar constructions which gives the Leibniz property. Thisis a natural connection between metric and multiplicativestructures of quantum locally compact metric space.

QuantumPropinquity


PhD





Annex

Wish list

Our metric should have the following desirable properties:

1 It only involves the category of C*-algebras,

2 It only involves Leibniz Lip-norms,3 *-isomorphism is a necessary condition for distance

zero,4 It dominates the quantum Gromov-Hausdorff distance,5 It is between GH and qGH on classical compact metric

spaces,6 Its topology is nontrivial on some relevant class of

noncommutative spaces (quantum tori). In particular, itis practical to work with.

QuantumPropinquity


PhD





Annex

Wish list


1 It only involves the category of C*-algebras,2 It only involves Leibniz Lip-norms,

3 *-isomorphism is a necessary condition for distancezero,

4 It dominates the quantum Gromov-Hausdorff distance,5 It is between GH and qGH on classical compact metric



QuantumPropinquity


PhD





Annex

Wish list


1 It only involves the category of C*-algebras,2 It only involves Leibniz Lip-norms,3 *-isomorphism is a necessary condition for distance

zero,

4 It dominates the quantum Gromov-Hausdorff distance,5 It is between GH and qGH on classical compact metric



QuantumPropinquity


PhD





Annex

Wish list



zero,4 It dominates the quantum Gromov-Hausdorff distance,

5 It is between GH and qGH on classical compact metricspaces,

6 Its topology is nontrivial on some relevant class ofnoncommutative spaces (quantum tori). In particular, itis practical to work with.

QuantumPropinquity


PhD





Annex

Wish list




spaces,

6 Its topology is nontrivial on some relevant class ofnoncommutative spaces (quantum tori). In particular, itis practical to work with.

QuantumPropinquity


PhD





Annex

Wish list






QuantumPropinquity


PhD





Annex

Bridges

The internal machinery developed to work with thequantum Gromov-Hausdorff distance involves bridges. Itinspired us to start from the following notion:

Definition (L., 2013)

A bridge (D, πA, πB, ω) from a C*-algebra A to a C*-algebraB is a C*-algebra D, two unital *-monomorphismsπA : A ↪→ D and πB : B ↪→ D and an element ω ∈ D suchthat the set:

L (ω) = {ϕ ∈ S (D) : ϕ(ω·) = ϕ(·ω) = ϕ}

is not empty.

QuantumPropinquity


PhD





Annex

The Bridge Seminorm

Let Γ = (D, πA, πB, ω) be a bridge from (A, LA) to (B, LB).The bridge seminorm of Γ is:

bnγ (LA, LB) : (a, b) ∈ A⊕B 7→ ‖πA(a)ω−ωπB(b)‖D.

We can use this seminorm to build Lip-norms on A⊕B ofthe form:

L(a, b) = max{LA(a), LB(b), λ−1bnγ (LA, LB)(a, b)}

for (a, b) ∈ A⊕B, λ > 0. If LA and LB are (strong) Leibniz,then so is L. This connects our approach to Rieffel’sproximity.

QuantumPropinquity


PhD





Annex

The Bridge Seminorm





for (a, b) ∈ A⊕B, λ > 0.

If LA and LB are (strong) Leibniz,then so is L. This connects our approach to Rieffel’sproximity.

QuantumPropinquity


PhD





Annex

The Bridge Seminorm





for (a, b) ∈ A⊕B, λ > 0. If LA and LB are (strong) Leibniz,then so is L. This connects our approach to Rieffel’sproximity.

QuantumPropinquity


PhD





Annex

The length of a bridge

A bridge provides us with a numerical quantity measuringhow close we may embed two compact quantum metricspaces in another C*-algebra, albeit with some subtlety.


Let (Aj, Lj) (j = 1, 2) be two compact quantum metric spacesand let Γ = (D, π1, π2, ω) be a bridge from A1 to A2.

The height of the bridge Γ is the maximum of theHausdorff distance for mkLj between π∗j (L (ω)) andS (Aj) for j = 1, 2.The reach of Γ is the Hausdorff distance betweenπ1[a : L1(a) 6 1] and π2[a : L2(a) 6 1] for the metrict, s ∈ sa(D) 7→ ‖tω−ωs‖D.The length of Γ is the maximum of its reach and height.

QuantumPropinquity


PhD





Annex





The height of the bridge Γ is the maximum of theHausdorff distance for mkLj between π∗j (L (ω)) andS (Aj) for j = 1, 2.

The reach of Γ is the Hausdorff distance betweenπ1[a : L1(a) 6 1] and π2[a : L2(a) 6 1] for the metrict, s ∈ sa(D) 7→ ‖tω−ωs‖D.The length of Γ is the maximum of its reach and height.

QuantumPropinquity


PhD





Annex





The height of the bridge Γ is the maximum of theHausdorff distance for mkLj between π∗j (L (ω)) andS (Aj) for j = 1, 2.The reach of Γ is the Hausdorff distance betweenπ1[a : L1(a) 6 1] and π2[a : L2(a) 6 1] for the metrict, s ∈ sa(D) 7→ ‖tω−ωs‖D.

The length of Γ is the maximum of its reach and height.

QuantumPropinquity


PhD





Annex





The height of the bridge Γ is the maximum of theHausdorff distance for mkLj between π∗j (L (ω)) andS (Aj) for j = 1, 2.The reach of Γ is the Hausdorff distance betweenπ1[a : L1(a) 6 1] and π2[a : L2(a) 6 1] for the metrict, s ∈ sa(D) 7→ ‖tω−ωs‖D.The length of Γ is the maximum of its reach and height.

QuantumPropinquity


PhD





Annex

The quantum Propinquity

The infimum of the lengths of bridges between twocompact quantum metric spaces fails to satisfy the triangleinequality. We introduce the notion of a trek betweencompact quantum metric spaces in (L., 2013), which is atype of finite family of bridges, and whose length is the sumof the lengths of the component bridges.


The Quantum Gromov-Hausdorff propinquityΛ((A, LA), (B, LB)) between two Leibniz compact quantummetric spaces (A, LA) and (B, LB) is the infimum of all treklengths from (A, LA) to (B, LB).

QuantumPropinquity


PhD





Annex

The Fundamental Theorem about QuantumPropinquity

Theorem (L., 2013)The quantum propinquity is a distance on the classes of*-isomorphic, quantum isometric Leibniz compact quantummetric spaces. It dominates the quantum Gromov-Hausdorffdistance and, when restricted to the Abelian C*-algebras, isdominated by the Gromov-Hausdorff distance.

This result is quite involved. It is proven by, in part, byintroducing the notion of itineraries associated to Lipschitzelements, which are finite sequences of elements along atrek with well-behaved associated bridge seminorms. Acentral proposition is given as follows.

QuantumPropinquity


PhD





Annex

Fundamental Proposition

Proposition (L., 2013)

Let (A, LA) and (B, LB) be two Leibniz compact metric spaces.Let γ be a bridge from A to B and let a ∈ sa(A) withLA(a) < ∞. Define, for any r > LA(a):

t(a, r) = {b ∈ sa(B) : L(b) 6 r and bnγ (a, b) 6 r$ (γ|LA, LB)} .

Then the diameter of t(a, r) for ‖ · ‖B is bounded above by:

2rλ (γ|LA, LB) + ‖a‖A.

(a, b) ∈ A⊕B 7→ ‖πA(a)ω−ωπB(b)‖λ ((D, ω, πA, πB)|LA, LB)

is a bridge in the sense of Rieffel.

QuantumPropinquity


PhD





Annex

Quantum Tori and the Quantum Propinquity

The group of skew-bicharacters of Zd is naturally identifiedwith the d-torus Td as a topological group. Moreover,skew-bicharacters of quotient groups Zd

k of Zd have aunique lift to Zd. With this in mind:

Theorem (L., 2013)

Let d ∈ N \ {0, 1} and σ a skew-bicharacter of Zd. Write∞d = (∞, . . . , ∞) ∈ (N∪ {∞})d. Let l be a continuous lengthfunction on Ud. Then:

lim(c,θ)→(∞d,σ)

Λ((

C∗(Zd

c , θ)

, Ll,c,θ

),(

C∗(Zd, σ

), Ll,∞d,σ

))= 0.

QuantumPropinquity


PhD





Annex

The bridges for quantum tori

The proof of convergence of quantum tori for the quantumpropinquity is very involved. As a general strategy:

1 We represent all quantum and fuzzy tori on `2(Zd) in anatural, non-degenerate, faithful manner.

2 We show that our various representations enjoy anatural form of strong-operator topology continuity“pointwise” wrt the skew bicharacters.

3 We then choose trace class elements ω to take advantageof the SOT continuity of our representations wrt to theskew bicharacters.

4 We must then compute estimates based uponHarmonic analysis as well as our specificrepresentations to compute the length of our bridges.

The proof is significantly more involved than the one fordistq since we must work only with C*-algebras.

QuantumPropinquity


PhD



Annex

Escape at Infinity

For a non-compact locally compact metric space (X, m), theextended Monge-Kantorovich metric is less well-behaved:

1 it is not a metric as it may be infinite,2 it does not metrize the weak* topology, even on closed

balls,3 its topology is not locally compact.

QuantumPropinquity


PhD



Annex

Escape at Infinity


1 it is not a metric as it may be infinite,

2 it does not metrize the weak* topology, even on closedballs,

3 its topology is not locally compact.

Proof.

Let δx denote the Dirac measure at x ∈ R. Let L be theLipschitz seminorm associated with the usual metric on R.

mkL

(δ0, ∑

n∈N2−n−1δ22n

)= ∞.

QuantumPropinquity


PhD



Annex

Escape at Infinity



balls,

3 its topology is not locally compact.

Proof.

Working in R again, we have:

∀n ∈ N mkL

(δ0,

nn + 1

δ0 +1

n + 1δn+1

)= 1

yet(δ0, n

n+1 δ0 +1

n+1 δn+1)

n∈N weak* converges to δ0.

QuantumPropinquity


PhD



Annex

Escape at Infinity



balls,3 its topology is not locally compact.

These problems are attributable to one main feature of thenon-compact case: probability measures can escape at infin-ity.

QuantumPropinquity


PhD



Annex

A first approach


The bounded-Lipschitz distance blL associated with a Lipschitzpair (A, LA) is defined for any ϕ, ψ ∈ S (A) as:

sup {|ϕ(a)− ψ(a)| : a ∈ sa(A), LA(a) 6 1, ‖a‖A 6 1} .

QuantumPropinquity


PhD



Annex

A first approach




Theorem (L., 2007)Let (A, L) be a Lipschitz pair and let:

B = {a ∈ sa(A) : L(a) 6 1 and ‖a‖A 6 1}.

Then the following are equivalent:1 blL metrizes the weak* topology of S (A),2 For some h ∈ A, h > 0 the set hBh is norm precompact,3 For all h ∈ A, h > 0, the set hBh is norm precompact.

QuantumPropinquity


PhD



Annex

A first approach




This notion was used, for instance, by Bellissard,Marcolli, Reihani (2010) for the study of metricproperties of spectral triples over C*-crossed-productsby Z.This notion was also used in mathematical physics (J.Wallet, Cagnache-d’Andrea-Martinetti)

QuantumPropinquity


PhD



Annex

A first approach




However...The bounded-Lipschitz distance only sees the space“locally”, i.e. balls of a radius above 1 are the whole space.This is less than ideal for Gromov-Hausdorff convergence!We are back to: How do we control behavior at infinity? Thiswas unsolved for more than a decade!

QuantumPropinquity


PhD



Annex

Dobrushin’s tightness

Dorbushin discovered a sufficient condition for metrizingthe weak* topology on well-behaved sets of probabilitymeasures:

Theorem (Dobrushin, 1970)Let (X, d) be a (locally compact) metric space. If a subset T ofS (C(X)) satisfies for some x0 ∈ X:

limr→∞

sup{∫

x:d(x,x0)>rd(x0, x) dP(x) : P ∈ T

}= 0

then the weak* topology restricted to T is metrized by theextended Monge-Kantorovich metric associated to the Lipschitzseminorm for d.

It is very challenging to extend this notion to thenoncommutative setting.

QuantumPropinquity


PhD



Annex

Quantum Topographic Spaces


A Lispchitz triple (A, L,M) is a Lipschitz pair (A, L) and anAbelian C*-subalgebra M of A containing an approximateunit for A.

Let K(M) be the collection of all compact subsets of theGelf’and spectrum of M and χK be the indicator function ofK in M.


A subset T of the state space S (A) of a Lipschitz triple(A,M, L) is tame when there exists µ ∈ S (A) andC ∈ K(M) such that µ(χC) = 1 and:

limK∈K(M)

sup{|ϕ(a− χKaχK)| :

ϕ ∈ T , a ∈ sa(A)L(a) 6 1, µ(a) = 0

}= 0.

QuantumPropinquity


PhD



Annex

Quantum Locally Compact Metric Spaces


A quantum locally compact metric space is a Lipschitz triplesuch that:

1 For all K ∈ K(M), the set {ϕ ∈ S (A) : ϕ(χK) = 1} hasfinite radius for mkL,

2 The topology induced on every tame subset of S (A)by mkL is the weak* topology.

Example (L., 2012)

If (C(R2σ), L2(R2)⊗C2, D) is the Gayal, Gracia-Bondia,

Iochum, Schücker, Varilly spectral triple over the Moyalplane C(R2

σ), then (C(R2σ), L,Mσ) is a quantum locally

compact metric space for Mσ generated by the Harmonicoscillator basis projections and L(a) = ‖[D, a]‖ (a ∈ C(R2

σ)).

QuantumPropinquity


PhD



Annex

Characterization of quantum locally compact metricspaces

Theorem (L., 2012)Let (A, L,M) be a Lipschitz triple. The following are equivalent:

1 (A, L,M) is a quantum locally compact metric space,2 There exists a state µ ∈ S (A), C ∈ K(M) with µ(K) = 1

such that for all compactly supported a, b ∈M, the set:

a {c : c ∈ sa(A), L(c) 6 1, µ(c) = 0} b

is norm precompact,3 For all states µ ∈ S (A) for which there exists C ∈ K(M)

with µ(K) = 1, and for all compactly supported a, b ∈M,the set {acb : c ∈ sa(A), L(c) 6 1, µ(c) = 0} is normprecompact.

QuantumPropinquity


PhD



Annex

The role of the Leibniz inequality

The Leibniz inequality plays a central role in Rieffel’srecent work on convergence of vector bundles. Itappears that one should work within the framework ofC*-metric spaces, where Lip-norms are defined onC*-algebras and satisfy a strong form of the Leibnizproperty (cf Rieffel’s work on convergence of matrixalgebras to spheres, for instance).

Yet, the quotient of a Leibniz seminorm is not Leibniz ingeneral. This means that if one asks for admissibleLip-norms to be Leibniz in the definition of distq, oneonly gets a pseudo-semi-metric (Rieffel’s proximity).

ProblemHow does one define a non-trivial metric on *-isomorphic,quantum isometric classes of C*-metric spaces?

QuantumPropinquity


PhD



Annex

The role of the Leibniz inequality

The Leibniz inequality plays a central role in Rieffel’srecent work on convergence of vector bundles. Itappears that one should work within the framework ofC*-metric spaces, where Lip-norms are defined onC*-algebras and satisfy a strong form of the Leibnizproperty (cf Rieffel’s work on convergence of matrixalgebras to spheres, for instance).Yet, the quotient of a Leibniz seminorm is not Leibniz ingeneral. This means that if one asks for admissibleLip-norms to be Leibniz in the definition of distq, oneonly gets a pseudo-semi-metric (Rieffel’s proximity).

ProblemHow does one define a non-trivial metric on *-isomorphic,quantum isometric classes of C*-metric spaces?

the quantum gromov-hausdorff propinquityquantum tori and their finite dimensional approximations...

Documents