Asymptotic dimension and coarse embeddings in the quantum setting
aa r X i v : . [ m a t h . OA ] J un ASYMPTOTIC DIMENSION AND COARSE EMBEDDINGSIN THE QUANTUM SETTING
JAVIER ALEJANDRO CHÁVEZ-DOMÍNGUEZ AND ANDREW T. SWIFT
Abstract.
We generalize the notions of asymptotic dimension andcoarse embeddings from metric spaces to quantum metric spaces in thesense of Kuperberg and Weaver [KW12]. We show that quantum asymp-totic dimension behaves well with respect to metric quotients and directsums, and is preserved under quantum coarse embeddings. Moreover,we prove that a quantum metric space that equi-coarsely contains a se-quence of expanders must have infinite asymptotic dimension. This isdone by proving a quantum version of a vertex-isoperimetric inequalityfor expanders, based upon a previously known edge-isoperimetric onefrom [TKR + Introduction
In [KW12] the authors explore a generalization of metric spaces, calledquantum metric spaces, which is related to the quantum graphs of quan-tum information theory; and they construct many generalizations of familiarmetric space concepts, including a generalization of Lipschtiz map they calla co-Lipschitz morphism. The purpose of this paper is to contribute to thestructure theory of quantum metric spaces by applying ideas coming fromthe large-scale or coarse geometry of classical metric spaces. Specifically,we propose generalizations of coarse embeddings and asymptotic dimension,and then prove some fundamental results about them. Coarse geometry isan important area of mathematics with applications in group theory, Banachspace theory, and computer science. It began as a field of study with thepolynomial growth theorem of Gromov [Gro81], and the definition of coarseembedding and asymptotic dimension are also both due to Gromov [Gro93],although large-scale geometric ideas appear as early as the late 1960’s in theoriginal proof of Mostow’s rigidity theorem [Mos68]. We refer to [NY12] foran excellent introduction to the subject.Some notions from coarse geometry have already been explored in thenoncommutative setting, see e.g. [Ban15, BM16]. There are two importantdifferences between these works and the present paper: One is that they
Mathematics Subject Classification.
Primary: 46L52; Secondary: 54F45, 46L51,46L65, 81R60 .
Key words and phrases.
Quantum metric spaces; Asymptotic dimension; Quantumexpanders.The first-named author was partially supported by NSF grant DMS-1900985. define and study noncommutative versions of coarse spaces whereas we arestudying noncommutative metric spaces in a coarse fashion, but more im-portantly their approach is C ∗ -algebraic while ours (as clearly stated by thetitle of [KW12]) is a von Neumann algebra one. A number of noncommuta-tive notions of topological dimension for C ∗ -algebras have been also studiedin the literature, see e.g. [Rie83, BP91, KW04, Win07, WZ10]. Let us em-phasize in particular the nuclear dimension from [WZ10], which is linked tocoarse geometry: For a discrete metric space of bounded geometry, the nu-clear dimension of the associated uniform Roe algebra is dominated by theasymptotic dimension of the underlying space. Once again, the approach todimension in the present work is significantly different from the aforemen-tioned ones because we are following the von Neumann algebra path.In Section 2 we recall the definitions we need from [KW12], including quan-tum metric, distance, and diameter. In Section 3 we generalize the definitionof coarse embedding to quantum metric spaces using alternative versions ofthe usual moduli of expansion (or uniform continuity) and compression de-fined for classical functions. It is shown that the moduli for classical functionsand their canonically induced quantum functions coincide. In Section 4 wegeneralize the definition of asymptotic dimension to quantum metric spacesand show that asymptotic dimension is preserved under coarse embeddings.We show as a consequence that the asymptotic dimension of a quotient ofa quantum metric space is no greater than the asymptotic dimension of theoriginal space and that the asymptotic dimension of a direct sum of quantummetric spaces is equal to the maximum of the asymptotic dimensions of itssummands. We finish Section 4 by establishing the corresponding inequalityfor arbitrary sums of quantum metric spaces in the case when the sum isa reflexive quantum metric space. In Section 5, we show that quantum ex-panders satisfy a quantum version of a vertex isoperimetric inequality. Thiscan be used to show that a quantum metric space has infinite asymptoticdimension if it equi-coarsely contains a sequence of reflexive quantum ex-panders. In particular, this includes the case when the sequence of quantumexpanders is induced by a sequence of classical expanders.2. Definitions
We use the definitions of quantum metric space and related notions foundin [KW12]. Just as metrics are defined on sets, quantum metrics are definedon von Neumann algebras, and classical metrics on a set X are in a naturalone-to-one correspondence with quantum metrics on ℓ ∞ ( X ) . We view vonNeumann algebras as subsets of some space B ( H ) of bounded linear operatorson a Hilbert space H . Given the von Neumann algebras M ⊆ B ( H ) , N ⊆B ( H ) , we denote by M⊗N their normal spatial tensor product, that is, theweak*-closure of
M ⊗ N in B ( H ⊗ H ) . Given a von Neumann algebra M , we denote the commutant of M by M ′ . An orthogonal projection in avon Neumann algebra will simply be called a projection. UANTUM ASYMPTOTIC DIMENSION 3
Definition 2.1 ([KW12, Definition 2.3]) : A quantum metric on a von Neu-mann algebra M ⊆ B ( H ) is a family V = {V t } t ∈ [0 , ∞ ) of weak*-closed sub-spaces of B ( H ) such that V = M ′ and for all t ∈ [0 , ∞ ) , • V t is self-adjoint. • V s V t ⊆ V s + t for all s ∈ [0 , ∞ ) . • V t = T s>t V s .A quantum metric space is a pair ( M , V ) of a von Neumann algebra M witha quantum metric V defined on it. We will often simply call a quantummetric space by its von Neumann algebra if there is no ambiguity regardingthe quantum metric being considered.Given a metric space ( X, d ) , the canonical quantum metric space associ-ated to it [KW12, Proposition 2.5] is ( ℓ ∞ ( X ) , {V t } t ∈ [0 , ∞ ) ) , where V t = (cid:8) A ∈ B ( ℓ ( X )) | h Ae x , e y i = 0 for all ( x, y ) / ∈ d − [0 , t ] (cid:9) for all t ∈ [0 , ∞ ) . Here ( e x ) x ∈ X is the canonical basis of ℓ ( X ) and ℓ ∞ ( X ) isviewed as a subset of B ( ℓ ( X )) as diagonal operators in the standard way,that is, via the map E : ℓ ∞ → B ( ℓ ( X )) defined by E ( φ )[ f ]( x ) = φ ( x ) f ( x ) for all φ ∈ ℓ ∞ ( X ) , all f ∈ ℓ ( X ) , and all x ∈ X . It is in this way that quan-tum metric spaces are generalizations of classical metric spaces. A naturaldistance function can be defined for projections in a von Neumann algebrathat generalizes the notion of distance between subsets of a classical metricspace. Definition 2.2 ([KW12, Definition 2.6]) : Given a quantum metric V = {V t } t ∈ [0 , ∞ ) on a von Neumann algebra M , the distance between two projec-tions P and Q in M⊗B ( ℓ ) is dist V ( P, Q ) = inf { t | P ( A ⊗ Id) Q = 0 for some A ∈ V t } . Identifying M with M ⊗ Id ⊆ M⊗B ( ℓ ) yields the equivalent formula dist V ( P, Q ) = inf { t | P AQ = 0 for some A ∈ V t } for projections P, Q in M . The subscript will usually be omitted if there isno ambiguity regarding the quantum metric being used.We refer to [KW12, Definition 2.7] and [KW12, Proposition 2.8] for basicproperties of quantum distance. In particular, the distance function associ-ated to a quantum metric satisfies an analog of the triangle inequality. If M is a quantum metric space and P, Q, R are projections in
M⊗B ( ℓ ) , then dist( P, Q ) ≤ dist( P, R ) + sup { dist( ˜ R, Q ) | R ˜ R = 0 } , where ˜ R ranges overprojections in M⊗B ( ℓ ) . Note that the same proof found in [KW12, Propo-sition 2.8] shows that if P, Q, R are projections in M , then ˜ R may be takento range only over projections in M .It is not hard to see that if ( ℓ ∞ ( X ) , V ) is the canonical quantum metricspace associated to metric space ( X, d ) , then dist V ( χ S , χ T ) = d ( S, T ) for any JAVIER ALEJANDRO CHÁVEZ-DOMÍNGUEZ AND ANDREW T. SWIFT subsets
S, T of X . Likewise, the following definitions for diameter and open ε -neighborhood of a projection generalize the corresponding notions for asubset. Definition 2.3 ([KW12, Proposition 2.16]) : Given a quantum metric V = {V t } t ∈ [0 , ∞ ) on a von Neumann algebra M , the diameter of a nonzero pro-jection P in M is diam V ( P ) = sup (cid:8) dist V ( Q, R ) | Q ( P AP ⊗ Id) R = 0 for some A ∈ B ( H ) (cid:9) . The diameter of the zero projection is defined to be zero. The subscript willusually be omitted if there is no ambiguity regarding the quantum metricbeing used.We remark that [KW12] only considers the case of P being the identity of M in the definition above, so our notion is really a generalization of theirs. Definition 2.4 ([KW12, Proposition 2.17]) : Given a quantum metric V = {V t } t ∈ [0 , ∞ ) on a von Neumann algebra M , the open ε -neighborhood of aprojection P in M is the projection ( P ) ε = Id − _ (cid:8) Q ∈ M | dist V ( P, Q ) ≥ ε (cid:9) . It is easy to see from [KW12, Definition 2.14 (b)] that (( P ) ε ) δ ≤ ( P ) ε + δ for all ε, δ > and projections P in a quantum metric space M .Our next definition follows [Kor11]. Definition 2.5:
A function φ : M → N between two von Neumann algebras M and N is called a quantum function if φ is a unital weak*-continuous ∗ -homomorphism.Quantum functions generalize classical functions in the following way: If f : X → Y is a classical function between sets, then φ f : ℓ ∞ ( Y ) → ℓ ∞ ( X ) ,defined by φ f ( g ) = g ◦ f for all g ∈ ℓ ∞ ( Y ) , is the canonical quantum functionassociated to f between the canonical von Neumann algebras associated to Y and X , and is such that φ f ( χ { y } ) = χ { f − ( y ) } for all y ∈ Y .In what follows, we will often identify a von Neumann algebra M with M ⊗ Id ⊆ M⊗B ( ℓ ) and a quantum function φ : M → N with ( φ ⊗ Id) :
M⊗B ( ℓ ) → N ⊗B ( ℓ ) .Recall [KW12, p. 17] that two projections P, Q ∈ M⊗B ( ℓ ) are said tobe unlinkable if there exist ˜ P , ˜ Q ∈ Id ⊗B ( ℓ ) satisfying P ≤ ˜ P , Q ≤ ˜ Q and ˜ P ˜ Q = 0 ; otherwise, they are said to be linkable . By [KW12, Prop. 2.13], fora von Neumann algebra M ⊆ B ( H ) , two projections P, Q ∈ M⊗B ( ℓ ) arelinkable if and only if there exists A ∈ B ( H ) such that P ( A ⊗ Id) Q = 0 .Note that we may generalize these notions to non-projections: two op-erators S, T ∈ M⊗B ( ℓ ) are said to be linkable if and only if there ex-ists A ∈ B ( H ) such that S ( A ⊗ Id) T = 0 ; obviously, this is the same as UANTUM ASYMPTOTIC DIMENSION 5 saying that the source projection J S K of S and the range projection [ T ] of T are linkable. By the aforementioned characterization, two operators S, T ∈ M⊗B ( ℓ ) are not linkable (i.e. unlinkable ) if and only if there existprojections ˜ S, ˜ T ∈ Id ⊗B ( ℓ ) satisfying J S K ≤ ˜ S , [ T ] ≤ ˜ T and ˜ S ˜ T = 0 . Notethat this is closely related to our definition of diameter of a projection: For aprojection P in M , its diameter is the supremum of the distances dist( Q, R ) where Q, R ∈ M⊗B ( ℓ ) are projections such that Q ( P ⊗ Id) and ( P ⊗ Id) R are linkable.The following lemma shows that when φ is a quantum function, φ ⊗ Id maps unlinkable pairs of operators to unlinkable pairs of operators. Lemma 2.6.
Let φ : M → N be a quantum function between von Neumannalgebras. If
S, T ∈ M⊗B ( ℓ ) are unlinkable operators, then ( φ ⊗ Id) S, ( φ ⊗ Id) T ∈ N ⊗B ( ℓ ) are unlinkable as well.Proof. As above, let ˜ S, ˜ T ∈ Id ⊗B ( ℓ ) be projections satisfying J S K ≤ ˜ S , [ T ] ≤ ˜ T , and ˜ S ˜ T = 0 . Note that since φ ⊗ Id is a ∗ -homomorphism and thuspreserves order, ( φ ⊗ Id) ˜ S, ( φ ⊗ Id) ˜ T ∈ Id ⊗B ( ℓ ) are projections and theysatisfy ( φ ⊗ Id) J S K ≤ ( φ ⊗ Id) ˜ S , ( φ ⊗ Id)[ T ] ≤ ( φ ⊗ Id) ˜ T , and ( φ ⊗ Id) ˜ S ( φ ⊗ Id) ˜ T = 0 . The desired result will follow once we prove that ( φ ⊗ Id) J S K = J ( φ ⊗ Id) S K and ( φ ⊗ Id)[ T ] = [( φ ⊗ Id) T ] .By our definition of quantum function, φ is a weak* to weak* continuous ∗ -homomorphism. By [Bla06, Prop. III.2.2.2], φ is normal. Since the tensorproduct of normal completely positive contractions is again a normal positivecontraction [Bla06, III.2.2.5], φ ⊗ Id is normal. By [Bla06, Prop. III.2.2.2]again, φ ⊗ Id is σ -strong to σ -strong continuous from M⊗B ( ℓ ) to N ⊗B ( ℓ ) .It is well-known that the σ -strong and the strong topologies coincide onbounded sets [Bla06, I.3.1.4], so in particular it follows that φ ⊗ Id mapsbounded strongly convergent nets to bounded strongly convergent nets.Now, it is known that ( S ∗ S ) α → J S K strongly as α → [Bla06, I.5.2.1].Since { ( S ∗ S ) α } α ∈ (0 , is norm bounded (which can be easily shown usingfunctional calculus), we conclude that ( φ ⊗ Id) (cid:0) ( S ∗ S ) α (cid:1) −−−→ α → ( φ ⊗ Id) J S K strongly. Since ( φ ⊗ Id) is a unital ∗ -homomorphism we have ( φ ⊗ Id) (cid:0) ( S ∗ S ) α (cid:1) = (cid:0)(cid:0) ( φ ⊗ Id) S (cid:1) ∗ (cid:0) ( φ ⊗ Id) S (cid:1)(cid:1) α (again this can be easily shown using functionalcalculus). Now, (cid:0)(cid:0) ( φ ⊗ Id) S (cid:1) ∗ (cid:0) ( φ ⊗ Id) S (cid:1)(cid:1) α −−−→ α → J ( φ ⊗ Id) S K strongly, and therefore J ( φ ⊗ Id) S K = ( φ ⊗ Id) J S K . The analogous conclusionfor the range projection of T follows from the fact that [ T ] = J T T ∗ K [Bla06,I.5.2.1]. (cid:3) Some of our results will only apply to quantum metric spaces that are(operator) reflexive; we recall the definition below.
JAVIER ALEJANDRO CHÁVEZ-DOMÍNGUEZ AND ANDREW T. SWIFT
Definition 2.7 ([KW12, Defns. 1.5 and 2.23]) : A subspace
V ⊆ B ( H ) is (operator) reflexive if V = (cid:8) B ∈ B ( H ) | P V Q = { } ⇒ P BQ = 0 (cid:9) with P and Q ranging over projections in B ( H ) . A quantum metric V = {V t } t ∈ [0 , ∞ ) is called reflexive if V t is reflexive for each t ∈ [0 , ∞ ) .3. Quantum moduli of expansion and compression
In this section, we define coarse embeddings between quantum metricspaces using moduli and then show how this relates to the definitions ofco-Lipschitz and co-isometric morphisms found in [KW12].Recall that if f : X → Y is a map between metric spaces, we define itsmodulus of expansion ω f by ω f ( t ) = sup (cid:8) d Y ( f ( x ) , f ( y )) | d X ( x, y ) ≤ t (cid:9) and its modulus of compression ρ f by ρ f ( t ) = inf (cid:8) d Y ( f ( x ) , f ( y )) | d X ( x, y ) ≥ t (cid:9) for all t ≥ . We say that f is expanding if lim t →∞ ρ f ( t ) = ∞ , and coarse if ω f ( t ) < ∞ for all t ≥ . We say that f is a coarse embedding if f is bothcoarse and expanding.For our purposes, we will use alternative versions of these moduli. Let ˜ ω f ( t ) = inf (cid:8) d X ( x, y ) | d Y ( f ( x ) , f ( y )) ≥ t (cid:9) and let ˜ ρ f ( t ) = sup (cid:8) d X ( x, y ) | d Y ( f ( x ) , f ( y )) ≤ t (cid:9) . The following observation is surely well-known.
Lemma 3.1.
Let f : X → Y be a map between metric spaces. Then:(a) f is coarse if and only if lim t →∞ ˜ ω f ( t ) = ∞ .(b) f is expanding if and only if ˜ ρ f ( t ) < ∞ for all t ≥ .Proof. Note that ˜ ω is an increasing function, and therefore lim t →∞ ˜ ω f ( t ) = ∞ if and only if ˜ ω f is unbounded.Suppose first that f is not coarse so that by definition there exists t ≥ such that ω f ( t ) = ∞ . Then for each n ∈ N there exist x n , y n ∈ X such that d Y ( f ( x n ) , f ( y n )) ≥ n and d X ( x n , y n ) ≤ t . This implies ˜ ω f ( n ) ≤ t , so ˜ ω f isbounded above by t .Suppose now that ˜ ω f is bounded above by t . Then for each n ∈ N , thereexist x n , y n ∈ X such that d X ( x n , y n ) ≤ t + 1 while d Y ( f ( x n ) , f ( y n )) ≥ n .This implies ω f ( t + 1) = ∞ . That is, f is not coarse. This finishes the proofof (a), and the proof for (b) is analogous. (cid:3) Let us now define corresponding moduli for quantum functions.
Definition 3.2:
Given a quantum function φ : M → N between quantummetric spaces M and N , we define ˜ ω φ and ˜ ρ φ by ˜ ω φ ( t ) = inf (cid:8) dist( φ ( P ) , φ ( Q )) | dist( P, Q ) ≥ t (cid:9) UANTUM ASYMPTOTIC DIMENSION 7 and ˜ ρ φ ( t ) = sup (cid:8) diam( φ ( P )) | diam( P ) ≤ t (cid:9) for all t ≥ , where P, Q range over projections in M .The next proposition shows that the moduli defined above generalize theclassical moduli. Proposition 3.3.
Given metric spaces
X, Y and a function f : X → Y , ˜ ω φ f = ˜ ω f and ˜ ρ φ f = ˜ ρ f .Proof. Let P be a projection in ℓ ∞ ( Y ) . Then P = χ S for some S ⊆ Y , and φ f ( P ) = χ f − [ S ] . Therefore, for t ≥ ω φ f ( t ) = inf { dist( χ f − [ S ] , χ f − [ T ] ) | dist( χ S , χ T ) ≥ t } = inf { d X ( x, y ) | d Y ( f ( x ) , f ( y )) ≥ t } and ˜ ρ φ f ( t ) = sup { diam( χ f − [ S ] ) | diam( χ S ) ≤ t } = sup { d X ( x, y ) | d Y ( f ( x ) , f ( y )) ≤ t } . (cid:3) Proposition 3.3 and Lemma 3.1 justify the following definition. Althoughit would perhaps be in better keeping with [KW12] to use the terminology“co-coarse”, there are two reasons we do not do this. The first reason isthat the inequalities involved concern only a quantum function φ and notits amplification φ ⊗ Id . The second reason is that we are only exploring anotion of coarseness for functions between quantum metric spaces and notfor operators inside of quantum metric spaces. Definition 3.4:
A quantum function φ : M → N between two quantummetric spaces is called a (quantum) coarse embedding if lim t →∞ ˜ ω φ ( t ) = ∞ and ˜ ρ φ ( t ) < ∞ for all t ≥ . Remark 3.5:
In [KW12, Definition 2.27], a quantum function φ : M → N is called a co-Lipschitz morphism if there is some C ≥ such that dist( P, Q ) ≤ C dist(( φ ⊗ Id)( P ) , ( φ ⊗ Id)( Q )) for all projections P, Q ∈ M⊗B ( ℓ ) . It is easily observed that if φ is aco-Lipschitz morphism, then ˜ ω φ ( t ) ≥ t/C for all t ≥ . Remark 3.6:
Also in [KW12, Definition 2.27], a quantum function φ : M →N is called a co-isometric morphism if it is surjective and dist( ˜
P , ˜ Q ) = sup (cid:8) dist( P, Q ) | ( φ ⊗ Id)( P ) = ˜ P , ( φ ⊗ Id)( Q ) = ˜ Q (cid:9) for all projections ˜ P , ˜ Q ∈ N ⊗B ( ℓ ) . If φ is a co-isometric morphism, then inparticular, φ is a co-Lipschitz morphism with constant 1, and so ˜ ω φ ( t ) ≥ t forall t ≥ ; it may be shown that additionally ˜ ρ φ ( t ) ≤ t for all t ≥ . Indeed,if P is a projection in M , and ˜ Q, ˜ R are projections in N ⊗B ( ℓ ) such that JAVIER ALEJANDRO CHÁVEZ-DOMÍNGUEZ AND ANDREW T. SWIFT ˜ Q ( φ ( P ) ⊗ Id) and ( φ ( P ) ⊗ Id) ˜ R are linkable, then since φ is a co-isometricmorphism, by Lemma 2.6, dist( ˜ Q, ˜ R ) = sup (cid:8) dist( Q, R ) | ( φ ⊗ Id)( Q ) = ˜ Q, ( φ ⊗ Id)( R ) = ˜ R (cid:9) ≤ sup (cid:8) dist( Q, R ) | ( φ ⊗ Id) (cid:0) Q ( P ⊗ Id) (cid:1) and ( φ ⊗ Id) (cid:0) ( P ⊗ Id) R (cid:1) are linkable (cid:9) ≤ sup (cid:8) dist( Q, R ) | Q ( P ⊗ Id) and ( P ⊗ Id) R are linkable (cid:9) = diam( P ) . Thus, diam( φ ( P )) ≤ diam( P ) , and therefore ˜ ρ φ ( t ) ≤ t .4. Asymptotic dimension
We will provide a definition of asymptotic dimension that can be appliedgenerally to all quantum metric spaces. Given the definitions that alreadyexist for diameter and ε -neighborhood of a projection, we have chosen to baseour generalization on Part 2 of [BD11, Theorem 2.1.2]. We do not exploregeneralizations of the equivalent formulations of asymptotic dimension foundin [BD11, Theorem 2.1.2]. Definition 4.1:
Let M be a quantum metric space and P a family ofprojections in M . We say that P is a cover for M if Id = W P ∈ P P . We saythat P is r -disjoint if ( P ) r ( Q ) r = 0 for each P, Q ∈ P with P = Q . Wesay that P is uniformly bounded by R if sup P ∈ P diam( P ) ≤ R , and that P is uniformly bounded if it is uniformly bounded by some R > . Definition 4.2:
Let M be a quantum metric space, and let n ∈ N ∪ { } .We say that M has asymptotic dimension less than or equal to n , writtenas asdim( M ) ≤ n , if for every r > there exist uniformly bounded, r -disjoint families of projections P , P , . . . , P n such that S ni =0 P i is a coverfor M . We say that M has asymptotic dimension equal to n , written as asdim( M ) = n , if n = min { m ∈ N ∪ { } | asdim( M ) ≤ m } . Remark 4.3:
Let ( X, d ) be a metric space, and consider the von Neumannalgebra ℓ ∞ ( X ) endowed with the canonical quantum metric induced by d .It is clear that asdim( X ) = asdim( ℓ ∞ ( X )) , since the projections in ℓ ∞ ( X ) are precisely the indicator functions of subsets of X .For classical metric spaces, coarse embeddings are the natural morphismsthat preserve asymptotic dimension because for any r > they map ev-ery R -disjoint, uniformly bounded family of sets to an r -disjoint, uniformlybounded family of sets whenever R > is large enough. This follows eas-ily from the definition of coarse embedding using the moduli of expansionand compression. We show that the same holds true for coarse embeddingsbetween quantum metric spaces. UANTUM ASYMPTOTIC DIMENSION 9
Lemma 4.4.
Let φ : M → N be a quantum function between quantum met-ric spaces. Then for any projection P ∈ M , diam( φ ( P )) ≤ ˜ ρ φ (diam( P )) . In particular, φ maps a family of projections uniformly bounded by R to afamily of projections uniformly bounded by ˜ ρ φ ( R ) .Proof. diam( φ ( P )) ≤ sup { diam( φ ( Q )) | diam( Q ) ≤ diam( P ) } = ˜ ρ φ (diam( P )) . (cid:3) Lemma 4.5.
Let φ : M → N be a quantum function between quantum met-ric spaces, and let r > . Then for any projection P ∈ M , (cid:0) φ ( P ) (cid:1) ˜ ω φ ( r ) ≤ φ (cid:0) ( P ) r (cid:1) . In particular, φ maps an r -disjoint family of projections to an ˜ ω φ ( r ) -disjointfamily of projections.Proof. Since φ is a quantum function and dist( P, Q ) ≥ r implies dist( φ ( P ) , φ ( Q )) ≥ ˜ ω φ ( r ) , we have φ (cid:0) ( P ) r (cid:1) = φ (cid:0) Id M − _ (cid:8) Q ∈ M | dist( P, Q ) ≥ r (cid:9)(cid:1) = Id N − _ (cid:8) φ ( Q ) | dist( P, Q ) ≥ r (cid:9) ≥ Id N − _ (cid:8) φ ( Q ) | dist( φ ( P ) , φ ( Q )) ≥ ˜ ω φ ( r ) (cid:9) ≥ Id N − _ (cid:8) R ∈ N | dist( φ ( P ) , R ) ≥ ˜ ω φ ( r ) (cid:9) = (cid:0) φ ( P ) (cid:1) ˜ ω φ ( r ) . (cid:3) The next theorem follows immediately. Note that a quantum function isunital and so it maps covers to covers.
Theorem 4.6.
Let φ : M → N be a quantum coarse embedding betweenquantum metric spaces. Then asdim( N ) ≤ asdim( M ) . As a consequence of Theorem 4.6, asymptotic dimension plays well withthe quotient [KW12, Definition 2.35] and direct sum [KW12, Definition 2.32(b)] constructions for quantum metric spaces. Compare this to the corre-sponding results on subspaces and (disjoint) unions of classical metric spacesfound in [BD11, Proposition 2.2.6] and [BD11, Corollary 2.3.3]. Note alsothat these results are related to some of the conditions required of an abstractdimension theory for a class of C ∗ -algebras from [Thi13]. Corollary 4.7.
Let M and N be quantum metric spaces.(a) If N is a metric quotient of M , then asdim( N ) ≤ asdim( M ) .(b) asdim( M ⊕ N ) = max { asdim( M ) , asdim( N ) } .Proof. (a): This follows immediately from Theorem 4.6 and Remark 3.6 ofthis paper and [KW12, Corollary 2.37]. (b): Since each of M and N is a metric quotient of M ⊕ N , we have asdim(
M ⊕ N ) ≥ max { asdim( M ) , asdim( N ) } from part (a). Now let n =max { asdim( M ) , asdim( N ) } and take any r > . By Definition 4.2, thereexist uniformly bounded, r -disjoint families of projections P , P , . . . , P n such that S ni =0 P i is a cover for M , and there also exist uniformly bounded, r -disjoint families of projections Q , Q , . . . , Q n such that S ni =0 Q i is a coverfor N . For ≤ j ≤ n , define R j = { P ⊕ | P ∈ P j } ∪ { ⊕ Q | Q ∈ Q j } . Itis clear that each R j is uniformly bounded, and moreover the union S ni =0 R i is a cover for M ⊕ N . Additionally, each family R j is r -disjoint, since ( P ⊕ r = ( P ) r ⊕ and (0 ⊕ Q ) r = 0 ⊕ ( Q ) r , by [KW12, Proposition 2.34].Therefore, asdim( M ⊕ N ) ≤ n . (cid:3) An analog of the result concerning asymptotic dimension of possibly nondis-joint unions of classical metric spaces [BD11, Corollary 2.3.3] can also beestablished, at least for reflexive quantum metric spaces. In a reflexive quan-tum metric space M , diameters of projections in M may be computed usingonly projections in M . Lemma 4.8.
Let ( M , {V t } t ∈ [0 , ∞ ) ) be a reflexive quantum metric space andlet P be a nonzero projection in M . Then diam( P ) = sup { dist( Q, R ) | QP AP R = 0 for some A ∈ B ( H ) } = sup { dist( Q, R ) | QP, RP = 0 } Proof.
It is clear from Definition 2.3 that diam( P ) ≥ sup { dist( Q, R ) | QP AP R = 0 for some A ∈ B ( H ) } . The result is then trivial when diam( P ) = 0 . So suppose diam( P ) = 0 andtake any < ε < diam( P ) . Let Q, R be any projections in
M ⊗ B ( H ) such that dist( Q, R ) > diam( P ) − ε while Q ( P AP ⊗ Id) R = 0 for some A ∈ B ( H ) . By [KW12, Proposition 2.10], P AP / ∈ V diam( P ) − ε . But thenby [KW12, Proposition 2.24], there exist projections Q ′ , R ′ ∈ M such that dist( Q ′ , R ′ ) ≥ diam( P ) − ε while QP AP R = 0 . As ε > was arbitrary, itfollows that diam( P ) ≤ sup { dist( Q, R ) | QP AP R = 0 for some A ∈ B ( H ) } . (cid:3) Lemma 4.9.
Let M be a reflexive quantum metric space, let P, Q, R beprojections in M , and take any r, s > . Then:(a) ( P ) r Q = 0 ⇐⇒ dist( P, Q ) < r .(b) dist( Q, R ) ≤ dist( Q, P ) + dist(
R, P ) + diam( P ) .(c) diam (cid:0) ( P ) r (cid:1) ≤ diam( P ) + 2 r .(d) diam( P ∨ Q ) ≤ dist( P, Q ) + diam( P ) + diam( Q ) .(e) ( P ) r ( Q ) s = 0 = ⇒ diam( P ∨ Q ) ≤ diam( P ) + diam( Q ) + 2( r + s ) .Proof. (a) The implication = ⇒ follows immediately from Definition 2.4.So suppose Q is such that dist( P, Q ) < r and furthermore, that ( P ) r Q = 0 .Then Q ≤ Id − ( P ) r = _ { Q ′ ∈ M | dist( P, Q ′ ) ≥ r } UANTUM ASYMPTOTIC DIMENSION 11 and thus r > dist(
P, Q ) ≥ dist( P, Id − ( P ) r ) = inf { dist( P, Q ′ ) | dist( P, Q ′ ) ≥ r } ≥ r, a contradiction. Therefore ( P ) r Q = 0 if dist( P, Q ) < r .(b) By the remark after Definition 2.2, with S ranging over projections in M , dist( Q, R ) ≤ dist( Q, P ) + sup { dist( S, R ) | P S = 0 }≤ dist( Q, P ) + dist(
R, P ) + sup { dist( S, S ′ ) | P S = 0 , P S ′ = 0 }≤ dist( Q, P ) + dist(
R, P ) + diam( P ) , where the last inequality follows from the fact that SP AP S ′ = 0 for some A ∈ B ( H ) whenever SP, P S ′ = 0 .(c) Suppose S, S ′ are projections in M such that ( P ) r S = 0 and ( P ) r S ′ =0 . By parts (a) and (b), this means dist( S, S ′ ) ≤ diam( P ) + 2 r . As S, S ′ were arbitrary, Lemma 4.8 implies that diam(( P ) r ) ≤ diam( P ) + 2 r .(d) Suppose S, S ′ are projections in M such that S ( P ∨ Q ) = 0 and S ′ ( P ∨ Q ) = 0 . If SP = 0 and S ′ P = 0 , then dist( S, S ′ ) ≤ diam( P ) . If SQ = 0 and S ′ Q = 0 , then dist( S, S ′ ) ≤ diam( Q ) . Finally, if SP = 0 and S ′ Q = 0 , then by part (b), dist( S, S ′ ) ≤ dist( S, Q ) + dist(
Q, S ′ ) + diam( Q ) ≤ dist( S, P ) + dist(
P, Q ) + diam( P ) + diam( Q )= dist( P, Q ) + diam( P ) + diam( Q ) . The same inequality holds if SQ = 0 and S ′ P = 0 . As S, S ′ were arbitrary,Lemma 4.8 implies that diam( P ∨ Q ) ≤ dist( P, Q ) + diam( P ) + diam( Q ) .(e) By parts (c) and (d) diam( P ∨ Q ) ≤ diam(( P ) r ∨ ( Q ) s ) ≤ dist(( P ) r , ( Q ) s ) + diam(( P ) r ) + diam(( Q ) s ) ≤ diam( P ) + diam( Q ) + 2( r + s ) . (cid:3) The following is a direct adaptation of [BD11, Prop. 2.3.1] for reflexivequantum metric spaces.
Proposition 4.10.
Let M be a reflexive quantum metric space, and let P , Q be families of projections in M . Fix r > and for each Q ∈ Q , let P Q = { P ∈ P | ( P ) r ( Q ) r = 0 } and P Q = _ P ∈ P Q P. Suppose that P is r -disjoint and R -bounded with R > r , and Q is R -disjointand D -bounded. Then Q ∪ r P is r -disjoint and ( D +2( R + D +4 r )) -bounded,where Q ∪ r P = { Q ∨ P Q | Q ∈ Q } ∪ (cid:8) P ∈ P | ( P ) r ( Q ) r = 0 for all Q ∈ Q (cid:9) . Proof.
Fix Q ∈ Q . By Lemmas 4.8 and 4.9 (b) and (e), diam( P Q ) = sup { dist( S, S ′ ) | SP Q , S ′ P Q = 0 }≤ sup { dist( S, Q ) + dist( S ′ , Q ) | SP Q , S ′ P Q = 0 } + diam( Q ) ≤ P ∈ P Q { diam( P ∨ Q ) } + diam( Q ) ≤ sup P ∈ P Q { diam( P ) } + diam( Q ) + 4 r ! + diam( Q ) ≤ R + D + 4 r ) + D. Thus, the bound on the diameter of Q ∨ P Q and hence the entire family P ∪ r Q is shown.We now show that Q ∪ r P is r -disjoint. If we take two elements of Q ∪ r P coming from P , then they are r -disjoint by assumption. Using thefact that ( R ∨ S ) r = ( R ) r ∨ ( S ) r for all projections R and S , it is also clearthat any two elements such that one is of the form Q ∨ P Q and the otheris in (cid:8) P ∈ P | ( P ) r ( Q ) r = 0 for all Q ∈ Q (cid:9) will be r -disjoint. The onlyremaining case is to consider two elements of the form Q ∨ P Q and Q ′ ∨ P Q ′ ,where Q, Q ′ ∈ Q are distinct. Note that in this case ( Q ) r ( Q ′ ) r = 0 . Consider P, P ′ such that P ∈ P Q and P ′ ∈ P Q ′ . If ( P ) r ( Q ′ ) r = 0 , then by Lemma4.9 (a), (b), and (c), dist( Q, Q ′ ) ≤ dist( Q, ( P ) r ) + dist(( P ) r , Q ′ ) + diam(( P ) r ) < r + diam( P ) + 2 r < R. By Lemma 4.9 (a), this implies ( Q ) R Q ′ = 0 , a contradiction. Thus ( P ) r ( Q ′ ) r =0 and similarly ( P ′ ) r ( Q ) r = 0 . And if ( P ) r ( P ′ ) r = 0 , then by Lemma 4.9(a), (b), (c), and (d), dist( Q, ( Q ′ ) r ) ≤ dist( Q, ( P ) r ∨ ( P ′ ) r ) + diam(( P ) r ∨ ( P ′ ) r ) ≤ r + diam( P ) + 2 r + diam( P ′ ) + 2 r < R. By Lemma 4.9 (a), this implies ( Q ) R ( Q ′ ) R = 0 , a contradiction. Thus ( P ) r ( P ′ ) r = 0 . As P , P ′ were arbitrary, it follows that ( Q ∨ P Q ) r ( Q ′ ∨ P Q ′ ) r =0 . Therefore Q ∪ r P is r -disjoint. (cid:3) We can now prove the following theorem which provides a bound on the as-ymptotic dimension of “nondisjoint unions” of quantum metric spaces. Com-pare this to [BD11, Corollary 2.3.3].
Theorem 4.11.
Let M be a reflexive quantum metric space. Suppose that N and N are metric quotients of M , corresponding to central projections R and R , respectively. If R ∨ R = Id , then asdim( M ) ≤ max { asdim( N ) , asdim( N ) } . (Note that, in particular, this includes the case M = N ⊕ N ). UANTUM ASYMPTOTIC DIMENSION 13
Proof.
Let n = max { asdim( N ) , asdim( N ) } and fix r > . Take n + 1 uniformly bounded, r -disjoint families of projections P , P , . . . , P n in N such that S ni =0 P i is a cover for N and let R > r be a uniform diameterbound for S ni =0 P i . Now take n + 1 uniformly bounded, R -disjoint familiesof projections Q , Q , . . . , Q n in N such that S ni =0 Q i is a cover for N and let D > be a uniform diameter bound for S ni =0 Q i . By viewing aprojection P in N as the projection P ⊕ ∈ R M ⊕ (Id − R ) M ∼ = M anda projection Q ∈ N as the projection Q ⊕ ∈ R M ⊕ (Id − R ) M ∼ = M ,it follows from [KW12, Proposition 2.34] that the families P , P , . . . , P n and Q , Q , . . . , Q n have the same bounds and disjointedness when viewedas families of projections in M . Thus, for each ≤ j ≤ n , the families R j = Q j ∪ r P j in M are r -disjoint and uniformly bounded by Proposition4.10. And since R ∨ R = Id , it follows that S ni =0 R i is a cover for M .Therefore asdim( M ) ≤ n . (cid:3) Remark 4.12:
In order to prove the general quantum analog of [BD11,Corollary 2.3.3] found in Theorem 4.11, we had to make the assumptionthat the quantum metric space is reflexive. Our proof follows [BD11] ratherclosely and relies on the ability to place an upper bound on the diameter ofa neighborhood of a projection in terms of the diameter of the projectionitself. This bound is found in Part (c) of Lemma 4.9, which is the first placewe use the reflexivity assumption. We do not know whether the reflexivityassumption can be dropped in the statement of Theorem 4.11, but if it can,we expect a method different from that found in [BD11] would be needed toprove it. 5.
Asymptotic dimension and quantum expanders
In this section, we will show that a quantum metric space equi-coarselycontaining a sequence of classical expanders (or more generally, a sequenceof reflexive quantum expanders) has infinite asymptotic dimension. Thisis a generalization of [DS12, Sec. 2.3] which shows that even for generalquantum metric spaces, information about their large-scale structure canbe inferred from their bounded metric subspaces. While this statement isquite believable in light of [DS12] and Theorem 4.6, it is not obvious atall that a quantum metric space should coarsely contain a classical metricspace of infinite asymptotic dimension even though it equi-coarsely containsa sequence of expander graphs! To prove the statement, we establish aquantum version of a vertex-isoperimetric inequality for expanders from aknown edge-isoperimetric inequality.In what follows, we denote the space of n × n matrices with complexentries by M n . The n × n identity matrix will be denoted by I n . TheHilbert-Schmidt norm for matrices will be denoted by k · k HS , the tracenorm will be denoted by k · k , and the operator norm will be denoted by k · k ∞ . We will use the initialization CPTP for a map Φ : M n → M n to indicate that Φ is completely positive and trace-preserving. Given a com-pletely positive map Φ : M n → M n , there exist by Choi’s theorem [Cho75]matrices K , K , . . . , K N ∈ M n such that Φ( X ) = P Nj =1 K j XK ∗ j for allmatrices X ∈ M n . If Φ is additionally trace-preserving, it may be shownalso that P Ni =1 K ∗ i K i = I n . It is then possible to define a quantum met-ric V = {V t } t ∈ [0 , ∞ ) on M n by V = C · I n , V = span { K ∗ j K i } ≤ i,j ≤ N , and V t = V ⌊ t ⌋ for t > [KW12, Sec. 3.2]. There are good information-theoreticalreasons [DSW13, Wea15] and metric reasons [KW12] for regarding a CPTPmap Φ (or rather, the operator system V ) as a quantum analog of a com-binatorial graph and the quantum metric V a quantum analog of a graphmetric. By an abuse of language, the terminology “quantum graph” will beused for any of Φ , V , and ( M n , V ) . Definition 5.1:
Given δ, ε, t > and n ∈ N , a quantum metric on M n issaid to satisfy a ( δ, ε, t ) -isoperimetric inequality if rank (cid:0) ( P ) δ (cid:1) ≥ (1 + ε ) rank( P ) for all projections P ∈ M n such that diam( P ) ≤ t . Remark 5.2:
By Lemma 4.9 (c), if a reflexive quantum metric on M n sat-isfies a ( δ, ε, t ) -isoperimetric inequality, it follows from repeated applicationsof it that, given any m ∈ N , rank (cid:0) ( P ) mδ (cid:1) ≥ (1 + ε ) m rank( P ) for all projections P ∈ M n such that diam( P ) + 2 mδ ≤ t . Definition 5.3:
Given a family of quantum coarse embeddings { φ α : M →N α } , we say that the family is equi-coarse if there exist functions f, g satisfy-ing lim t →∞ f ( t ) = ∞ and g ( t ) < ∞ for all t ≥ such that for for each t ≥ and each α , f ( t ) ≤ ˜ ω φ α ( t ) and ˜ ρ φ α ( t ) ≤ g ( t ) . By analogy with the classicalsetting, in this case we say that M equi-coarsely contains the family {N α } .The strategy of proof in the next Proposition is based on [DS12, Thm.2.9]. Proposition 5.4.
Let M be a quantum metric space, and fix δ, ε > .Suppose that { ( M n t , V t ) } t> is a family of reflexive quantum metric spacesand { φ t : M → M n t } t> is an equi-coarse family of quantum coarse embed-dings. If M n t satisfies a ( δ, ε, t ) -isoperimetric inequality for every t > , then asdim( M ) = ∞ .Proof. Suppose M has finite asymptotic dimension n , and take any m ∈ N such that (1+ ε ) m − > n . Let f, g be functions satisfying lim r →∞ f ( r ) = ∞ and g ( r ) < ∞ for all r ≥ such that f ( r ) ≤ ˜ ω φ t ( r ) and ˜ ρ φ t ( r ) ≤ g ( r ) forall t, r > ; and pick r > such that f ( r ) > mδ . Let P , P , . . . , P n beuniformly bounded r -disjoint families of projections in M such that S nj =0 P j UANTUM ASYMPTOTIC DIMENSION 15 is a cover for M and let d be such that diam( P ) ≤ d for every P ∈ S nj =0 P j .Finally, let t = 2 mδ + g ( d ) . It follows from Lemmas 4.4 and 4.5 that foreach ≤ j ≤ n , the families Q j = { φ t ( P ) | P ∈ P j } are f ( r ) -disjoint(and therefore mδ -disjoint) and uniformly bounded by g ( d ) , and S nj =0 Q j isa cover for M n t since S nj =0 P j is a cover for M . Thus, the ( δ, ε, mδ + g ( d )) -isoperimetric inequality for M n t implies by Lemma 4.9 (c) and Remark 5.2that for each ≤ j ≤ n , n t = rank( I n t ) ≥ X Q ∈ Q j rank(( Q ) mδ ) ≥ (1 + ε ) m X Q ∈ Q j rank( Q ) , and adding over j yields ( n + 1) · n t ≥ (1 + ε ) m n X j =0 X Q ∈ Q j rank( Q ) ≥ (1 + ε ) m · n t , where the last inequality follows from the fact that S nj =0 Q j is a cover for M n t . This implies that n ≥ (1 + ε ) m − > n , a contradiction. Therefore asdim( M ) = ∞ . (cid:3) We will show that a sequence of reflexive quantum expanders (which in-cludes the case of classical expanders) satisfies the isoperimetric inequalitycondition found in Proposition 5.4. We first recall the definition of quantumexpander sequence and an associated Cheeger-type inequality below.
Definition 5.5 ([Pis14]) : Given < ε < and n ∈ N , a CPTP map Φ : M n → M n is said to have an ε -spectral gap if (cid:13)(cid:13) Φ( X ) − n tr( X ) I n (cid:13)(cid:13) HS ≤ (1 − ε ) (cid:13)(cid:13) X − n tr( X ) I n (cid:13)(cid:13) HS for all X ∈ M n . Definition 5.6 ([Pis14]) : A CPTP map
Φ : M n → M n is called a d -regular ε -quantum expander if Φ has an ε -spectral gap and there exist unitaries U , . . . , U d ∈ M n such that Φ( X ) = d P dj =1 U j XU ∗ j for each X ∈ M n . Asequence of CPTP maps { Φ m : M n m → M n m } is called a sequence of d -regular ε -quantum expanders if Φ m is a d -regular ε -quantum expander foreach m ∈ N and n m → ∞ as m → ∞ .The following is just a restatement of [TKR +
10, Lemma 20], which canbe described as a quantum Cheeger inequality.
Lemma 5.7.
Let
Φ : M n → M n be a CPTP unital map with an ε -spectralgap. Then tr (cid:0) ( I n − P )Φ ∗ Φ( P ) (cid:1) tr( P ) ≥ (1 − ε ) / for all projections P ∈ M n such that < rank( P ) ≤ n/ . Remark 5.8:
The expression appearing in the preceding lemma can berewritten in terms of the inner product associated to the Hilbert-Schmidtnorm. Indeed, tr (cid:0) ( I n − P )Φ ∗ Φ( P ) (cid:1) = tr (cid:0) ( I n − P ) ∗ Φ ∗ Φ( P ) (cid:1) = h Φ ∗ Φ( P ) , I n − P i HS = h Φ( P ) , Φ( I n − P ) i HS . Thus, Lemma 5.7 says that Φ maps orthogonal pairs P, I n − P to nonorthog-onal pairs in a uniform way.The next result says that if the rank of a projection inside an expander issmall, then any large neighborhood of the projection has strictly larger rankthan the projection itself. Proposition 5.9.
Let
Φ : M n → M n be a d -regular ε -quantum expander.Then for any δ > , rank (cid:0) ( P ) δ (cid:1) ≥ (1 + ε ′ ) rank( P ) whenever P ∈ M n is a projection such that rank( P ) ≤ n/ , where the quan-tum metric on M n is the one induced by Φ and ε ′ = (1 − ε ) / .Proof. Let P ∈ M n be a projection such that rank( P ) ≤ n/ . We will showthat if Q ∈ M n is a projection such that dist( P, Q ) ≥ δ , then rank( Q ) ≤ n − (1 + ε ′ ) rank( P ) . The result will then follow from Definition 2.4.Let U , . . . , U d ∈ M n be unitaries such that Φ( X ) = d P dj =1 U j XU ∗ j foreach X ∈ M n . If dist( P, Q ) ≥ δ > , it follows from the definition of thequantum metric induced by Φ that P U ∗ j U i Q = 0 for all ≤ i, j ≤ d . Thus, h Φ( P ) , Φ( Q ) i HS = 1 d d X i,j =1 h U j P U ∗ j , U i QU ∗ i i HS = 1 d d X i,j =1 tr (cid:0) U i QU ∗ i U j P U ∗ j (cid:1) = 1 d d X i,j =1 tr (cid:0) U ∗ i U j P U ∗ j U i Q (cid:1) = 0 . Therefore, by Lemma 5.7 (Definition 5.6 implies that Φ is unital) and Remark5.8,(5.1) ε ′ tr( P ) ≤ h Φ( P ) , Φ( I n − P ) i HS = h Φ( P ) , Φ( I n − P − Q ) i HS . Now, by the trace duality between the trace and operator norms on M n , h Φ( P ) , Φ( R ) i HS = 1 d d X i,j =1 h U j P U ∗ j , U i RU ∗ i i HS = 1 d d X i,j =1 tr (cid:0) U i RU ∗ i U j P U ∗ j (cid:1) = 1 d d X i,j =1 tr (cid:0) RU ∗ i U j P U ∗ j U i (cid:1) ≤ d d X i,j =1 k R k (cid:13)(cid:13) U ∗ i U j P U ∗ j U i (cid:13)(cid:13) ∞ ≤ k R k UANTUM ASYMPTOTIC DIMENSION 17 for all projections R ∈ M n . Therefore, using the fact that for projectionsthe rank, the trace, and the trace norm coincide, it follows from (5.1) that ε ′ rank( P ) ≤ k I n − P − Q k = tr( I n − P − Q )= tr( I n ) − tr( P ) − tr( Q ) = n − rank( P ) − rank( Q ) , which yields the desired inequality. (cid:3) We would like to use Proposition 5.9 to establish that quantum expanderssatisfy a ( δ, ε, t ) -isoperimetric inequality. To do this, we have to first estab-lish a relationship between the rank and the diameter of a projection insidean expander. We do this more generally for projections inside any connectedquantum graph and then show that expanders are connected. The impor-tance of the connectedness assumption is that it implies that every projectionhas finite diameter. A quantum graph (that is, an operator system) S ⊆ M n is said to be connected if there is m ∈ N such that S m = M n [CDS19,Definition 3.1]. See [CDS19] for more information about connected quantumgraphs. Proposition 5.10.
Let S be a connected quantum graph, and R ∈ M n aprojection. If k ∈ N is such that diam( R ) ≤ k , then R S k R = RM n R .Proof. Suppose to the contrary that R S k R ( RM n R , and pick any A ∈ RM n R \ R S k R . Then by [Wea12, Lemma 2.8], there exist projections P, Q ∈ RM n R ⊗B ( ℓ ) such that P ( RAR ⊗ Id) Q = 0 , while P ( RBR ⊗ Id) Q = 0 forall B ∈ S k . Let ˜ P , ˜ Q be the range projections of ( R ⊗ Id) P and ( R ⊗ Id) Q ,respectively. The above implies that ˜ P ( RAR ⊗ Id) ˜ Q = 0 , while ˜ P ( B ⊗ Id) ˜ Q =0 for all B ∈ S k . By Definition 2.2 and Definition 2.3, this means that diam( R ) > k . This is a contradiction, and so RS k R = RM n R . (cid:3) Lemma 5.11.
Let
Φ : M n → M n be CPTP map and let K , K , . . . , K N ∈ M n be such that Φ( X ) = P Nj =1 K j XK ∗ j for all matrices X ∈ M n . If thequantum graph associated to Φ is connected, then for every projection R ∈ M n , rank( R ) ≤ N diam( R ) , where the diameter is taken with respect to the quantum graph metric asso-ciated to Φ .Proof. Let k = diam( R ) and let S = span { K ∗ j K i | ≤ i, j ≤ N } be theassociated quantum graph. It follows from Proposition 5.10 that RM n R = R S k R and therefore rank( R ) = dim( RM n R ) = dim( R S k R ) ≤ dim( S k ) ≤ ( N ) k , which yields the desired inequality. (cid:3) Proposition 5.12.
Let
Φ : M n → M n be a CPTP unital map with an ε -spectral gap. Then the associated quantum graph is connected. In particular,every d -regular ε -quantum expander is connected. Proof.
Let K , . . . , K N ∈ M n be matrices such that Φ( X ) = P Nj =1 K j XK ∗ j for each X ∈ M n . Suppose that the quantum graph S = span { K ∗ j K i | ≤ i, j ≤ N } is disconnected. By [CDS19, Theorem 3.3], there exists a nontrivialprojection P ∈ M n such that P S ( I n − P ) = 0 , and without loss of generality,we may assume < rank( P ) ≤ n/ . In particular, P K ∗ j K i ( I n − P ) = 0 forall ≤ i, j ≤ N . Therefore h Φ( P ) , Φ( I n − P ) i HS = N X i,j =1 h K j P K ∗ j , K i ( I n − P ) K ∗ i i HS = N X i,j =1 tr (cid:0) K i ( I n − P ) K ∗ i K j P K ∗ j (cid:1) = N X i,j =1 tr (cid:0) K ∗ i K j P K ∗ j K i ( I n − P ) (cid:1) = 0 . This contradicts Lemma 5.7, and so the quantum graph associated to Φ isconnected. (cid:3) Propositions 5.4 and 5.9 and Lemma 5.11 together yield our main theorem.
Theorem 5.13.
If a quantum metric space M is equi-coarsely embeddableinto a sequence of reflexive d -regular ε -quantum expanders, then asdim( M ) = ∞ . In particular, this holds whenever M admits a sequence of reflexive d -regular ε -quantum expanders as metric quotients. We point out that, in particular, Theorem 5.13 covers the case when M equi-coarsely contains a sequence of d -regular ε -classical expanders, thanksto the following proposition. Proposition 5.14.
The canonical quantum metric associated to a classicalmetric is always reflexive.Proof.
Let ( X, d ) be a classical metric. Let t ≥ . By [KW12, Prop. 2.5],the canonical quantum metric on ℓ ∞ ( X ) associated to d is given by(5.2) V t = (cid:8) A ∈ B ( ℓ ( X )) | d ( x, y ) > t ⇒ h Ae y , e x i = 0 (cid:9) . Let us now show that V t is reflexive. To that end, let B ∈ B ( ℓ ( X )) be suchthat for any projections P, Q ∈ B ( ℓ ( X )) such that P V t Q = { } , it followsthat P BQ = 0 ; we need to show that B belongs to V t . Recall that V xy denotes the mapping g
7→ h g, e y i e x . Let x, y ∈ X satisfy d ( x, y ) > t . Notethat V xx and V yy are projections, and it follows from (5.2) that V xx V t V yy = { } . Therefore, V xx BV yy = 0 . But this implies h Be y , e x i = 0 , and thus B ∈ V t by apppealing to (5.2) again. (cid:3) One final remark is in order regarding Theorem 5.13 and Proposition5.14. We have shown that equi-coarse containment of reflexive quantumexpanders implies infinite asymptotic dimension and we have also shownthat quantum expanders induced by classical expanders are reflexive. Whilethis is enough to provide a generalization of [DS12, Sec. 2.3] to the realmof quantum metric spaces, what we have not shown is the existence of a
UANTUM ASYMPTOTIC DIMENSION 19 nontrivial reflexive quantum expander. That is, we do not actually knowwhether every reflexive quantum expander is induced by a classical expander.It would be interesting to know the answer to this question, but it would bemore interesting still to know whether the reflexivity assumption in Theorem5.13 (or more generally Proposition 5.4) can be dropped. As with the proofof Theorem 4.11, it was very important to be able to place an upper boundon the diameter of a neighborhood of a projection in terms of the diameter ofthe projection (Part (c) of Lemma 4.9). This is what allowed us to repeatedlyapply the isoperimetric inequality to derive the inequality found in Remark5.2.
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E-mail address : [email protected] Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA
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