Algebraic characterization of quasi-isometric spaces via the Higson compactification
aa r X i v : . [ m a t h . M G ] N ov Algebraic characterization of quasi-isometric spaces viathe Higson compactification
Jes´us A. ´Alvarez L´opez a,1 , Alberto Candel b, ∗ a Departamento de Xeometr´ıa e Topolox´ıa, Facultade de Matem´aticas, Universidade deSantiago de Compostela, Campus Universitario Sur, 15706 Santiago de Compostela, Spain b Department of Mathematics, California State University at Northridge, Northridge, CA91330, U.S.A.
Abstract
The purpose of this article is to characterize the quasi-isometry type of a propermetric space via the Banach algebra of Higson functions on it.
Keywords:
Metric space, Coarse space, Quasi-isometry, Coarse maps, Higsoncompactification, Higson algebra
1. Introduction
The Gelfand representation is a contravariant functor from the categorywhose objects are commutative Banach algebras with unit and whose mor-phisms are unitary homomorphisms into the category whose objects are compactHausdorff spaces and whose morphisms are continuous mappings. This functorassociates to a unitary Banach algebra its set of unitary characters (they areautomatically continuous), topologized by the weak* topology.That functor (the Gelfand representation) is right adjoint to the functor thatto a compact Hausdorff space S assigns the algebra C ( S ) of continuous complexvalued functions on S normed by the supremum norm. If A is a commutativeBanach algebra without radical, then A is isomorphic to an algebra of continuouscomplex valued functions on the space of unitary characters of A .The Gelfand representation is the base for an algebraic characterization ofcompactifications of topological spaces. A compactification of a topologicalspace, X , is a pair ( X κ , e ) consisting of a compact Hausdorff topological space X κ and an embedding e : X → X κ with open dense image. (Thus, onlylocally compact spaces admit compactifications in this sense.) The complement ∗ Corresponding Author
Email addresses: [email protected] (Jes´us A. ´Alvarez L´opez), [email protected] (Alberto Candel) Research of first author supported by DGICYT and MICINN, Grants PB95-0850 andMTM2008-02640.
Preprint submitted to Elsevier May 28, 2018 κ \ e ( X ) is called the corona (or growth) of the compactification ( X κ , e ), anddenoted by κX . Usually X is identified with its image e ( X ) and thus regardedas a subspace of X κ . In this case, the closure X = X κ and the boundary ∂X = κX .Let X be a topological space and let ( X κ , e ) be a compactification of X .Then C b ( X κ ) = C ( X κ ) is a Banach algebra and the embedding e : X → X κ induces an algebraic isomorphism of C b ( X κ ) into the Banach algebra C b ( X ) viacomposition with the embedding e . The image of C b ( X κ ) in C b ( X ) consistsprecisely of all the bounded continuous functions on X that admit a continuousextension to X κ (via e ). It therefore contains the constant functions on X andgenerates the topology of X in the sense that if E is a compact subset of X and x ∈ X \ E , then there is a function in e ∗ C b ( X κ ) that takes on the value0 at x and is identically 1 on E . Conversely, if A is a Banach subalgebra of C b ( X ) that contains the constant functions on X and generates the topology of X , then A is isomorphic to the algebra of (bounded) continuous functions on acompactification of X .For example, C b ( X ), the algebra of bounded continuous functions on X ,corresponds to the Stone- ˇCech compactification of X , and C + C ( X ), the sub-algebra of C b ( X ) generated by the constants and the continuous functions thatvanish at infinity, corresponds to the one-point compactification of X . Theorem 1.1 (Gelfand) . Two locally compact Hausdorff spaces, X and Y ,are homeomorphic if and only if the Banach algebras C b ( X ) and C b ( Y ) arealgebraically isomorphic. In fact, an algebraic isomorphism C b ( Y ) → C b ( X ) induces a homeomor-phism X β → Y β that maps X onto Y .The present paper is motivated by algebraic characterizations of topologicalstructures for which the above theorem is a milestone. Refinements of thismilestone that motivated the present paper include the work of Nakai [15] onthe algebraic characterization of the holomorphic and quasi conformal structuresof Riemann surfaces.Let R be a Riemann surface and M ( R ) the algebra of bounded complexvalued functions on R which are absolutely continuous on lines and have fi-nite Dirichlet integral D ( f ) = Z R df ∧ ⋆ df . Endowed with the norm k f k =sup x ∈ R | f ( x ) | + [ D ( f )] / , M ( R ) is a commutative Banach algebra. It containsthe constant functions and it also contains the compactly supported smoothfunctions, so it generates the topology of R . Therefore M ( R ) is the algebra ofcontinuous complex valued functions on a compactification of R , the so calledRoyden compactification.Work on the Royden compactification culminated in the following theoremof Nakai [15]: Theorem 1.2 (Nakai) . Two Riemann surfaces R and R ′ are quasi-conformallyequivalent if and only if the corresponding algebras M ( R ) and M ( R ′ ) are alge-braically isomorphic. wo Riemann surfaces R and R ′ are conformally equivalent if and only ifthere is a norm preserving isomorphism between M ( R ) and M ( R ′ ) . The Royden algebra can be defined on any locally compact metric space,(
X, d ), endowed with a Borel measure µ . If f is a complex valued functionon X , then its gradient norm is the function |∇ f | on X given by |∇ f | ( x ) =lim sup z → x | f ( x ) − f ( z ) | d ( x, z ) . A function f on X is a Royden function if it is bounded,continuous, and satisfies R X |∇ f | · µ < ∞ . The family of Royden functionson X form a subalgebra of the algebra of bounded continuous functions whichcontains the constant functions and the compactly supported functions. Its com-pletion with respect to the norm given by k f k = sup x ∈ X | f ( x ) | + ( Z X |∇ f | µ ) / isa Banach algebra and it gives rise to a compactification of X , called the Roydencompactification of X . If f n is a Cauchy sequence of Royden functions (withrespect to the Royden norm), then f n converges uniformly to a continuous func-tion on X . Therefore, to each element in the Royden algebra, there correspondsa bounded continuous function on X . This correspondence M ( X ) → C b ( X ) isa norm decreasing, injective, algebraic homomorphism.Nakai and others have studied and extended Nakai’s Theorem on the Roy-den algebra and Royden compactification of Riemann surfaces to other metricspaces: Riemannian manifolds (Nakai [16], Lelong-Ferrand [6]), and domains inEuclidean spaces (Lewis [14]). A generalization of Nakai’s Theorem involvingRoyden p -compactifications was also given in [17]. The following theorem is arepresentative result of those works. Theorem 1.3.
Let R , R ′ be Riemannian manifolds of dimension dim R =dim R ′ > , endowed with the induced path metric structure and Riemannianmeasure. Then R and R ′ are quasi-isometrically homeomorphic if and only if R and R ′ have homeomorphic Royden compactifications. In the present paper we prove an analogous theorem for metric spaces andtheir coarse quasi-isometries in the sense of Gromov. The algebra of functionsthat characterizes the coarse quasi-isometry type is the Higson algebra. A Hig-son function ( cf.
Definition 4.1 below) on a locally compact metric space, (
M, d ),is a bounded Borel function, f , on M such that, for each r >
0, its r -expansion ∇ r f ( x ) = sup {| f ( x ) − f ( y ) | | d ( x, y ) ≤ r } is in B ( M ), the algebra of boundedBorel functions on M that vanish at ∞ . The Higson functions on M form aBanach algebra, denoted by B ν ( M ), and the subalgebra of continuous Higsonfunctions, C ν ( M ), determine the so called Higson compactification M ν of M .The Higson corona (or growth) is the complement νM = M ν \ M . Some topo-logical properties of this Higson compactification were studied in [4], [5] and[13]. Theorem 1.4.
Two proper metric spaces ( M, d ) and ( M ′ d ′ ) are coarsely equiv-alent if and only if there is an algebraic homomorphism of Higson algebras B ν ( M ′ ) → B ν ( M ) that induces an isomorphism C ( νM ′ ) → C ( νM ) , where νM nd νM ′ are the coronas of the respective Higson compactifications of M and M ′ . The “only if” part of Theorem 1.4 has no version with continuous Higsonfunctions, which justifies the use of Borel ones. For instance, Z and R arecoarsely equivalent, but any continuous map R → Z is constant, and thereforeno homomorphism C ν ( Z ) → C ν ( R ) induces an isomorphism C ( ν Z ) → C ( ν R ).Other geometric properties of metric spaces have been shown to have apurely algebraic characterization; one example of such properties is illustratedby recent work of Bourdon [3]. To each metric space he associates an algebraof functions based on a Besov norm, and then he proves that two metric spacesare homeomorphic via a quasi-Moebius homeomorphism if and only if thosealgebras are isomorphic.It appears of interest to analyze what other geometric structures on a metricspace can be inferred from naturally defined Banach algebras of functions on it.We thank the referee for valuable comments that helped correct the presen-tation of this paper.
2. Coarse quasi-isometries
Let (
M, d ) and ( M ′ , d ′ ) be arbitrary metric spaces. A mapping f : M → M ′ is said to be Lipschitz if there is some
C > d ′ ( f ( x ) , f ( y )) ≤ C d ( x, y )for all x, y ∈ M . Any such constant C is called a Lipschitz distortion of f . Themap f is said to be bi-Lipschitz when there is some C ≥ C d ( x, y ) ≤ d ′ ( f ( x ) , f ( y )) ≤ C d ( x, y )for all x, y ∈ M . In this case, the constant C will be called a bi-Lipschitzdistortion of f .A family of Lipschitz maps is called equi-Lipschitz when all the maps in ithave some common Lipschitz distortion. A family of bi-Lipschitz maps is saidto be equi-bi-Lipschitz when all of its maps have some common bi-Lipschitzdistortion, which is called an equi-bi-Lipschitz distortion .A net in a metric space ( M, d ) is a subset A ⊂ M such that d ( x, A ) ≤ K for some K > x ∈ M . On the other hand, a subset A of M is saidto be separated when there is some δ > d ( x, y ) > δ for every pairof different points x, y ∈ A . The terms K -net and δ -separated net will be alsoused. Lemma 2.1.
Let
K > . There is some K -separated K -net in M . Moreoverany K -net of M contains a K -separated K -net of M .Proof. Let S be the family of K -separated subsets of M . By using Zorn’slemma, it follows that there exists a maximal element A ∈ S . If d ( x, A ) > K x ∈ M , then A ∪ { x } ∈ S , contradicting the maximality of A . Hence A is a K -net in M .Let A be a K -net for M . The above shows that there is a K -separated K -net B for the metric space A . It easily follows that B is a 2 K -net for M .The concept of coarse quasi-isometry was introduced by M. Gromov [8] asfollows. A coarse quasi-isometry between metric spaces ( M, d ) and ( M ′ , d ′ ) isa bi-Lipschitz bijection between some nets A ⊂ M and A ′ ⊂ M ′ ; in this case, M and M ′ are said to have the same coarse quasi-isometry type or to be coarselyquasi-isometric . A coarse quasi-isometry between M and itself will be calleda coarse quasi-isometric transformation of M . For some K > C ≥ K, C ) is said to be a coarsely quasi-isometric distortion of a coarsequasi-isometry if it is a bi-Lipschitz bijection between K -nets with bi-Lipschitzdistortion C . A family of equi-coarse quasi-isometries is a collection of coarsequasi-isometries that have a common coarse distortion.Two coarse quasi-isometries between ( M, d ) and ( M ′ , d ′ ), say f : A → A ′ and g : B → B ′ , are close if there are some r, s > d ( x, B ) ≤ r , d ( y, A ) ≤ r ,d ( x, y ) ≤ r = ⇒ d ′ ( f ( x ) , g ( y )) ≤ s , for all x ∈ A and y ∈ B . (Such coarse quasi-isometries f and g are said to be( r, s ) -close .)It is well known that “being coarsely quasi-isometric” is an equivalence re-lation on metric spaces. Indeed, this is a consequence of the fact that the“composite” of coarse quasi-isometries makes sense up to closeness [1].The equivalence classes of the closeness relation on coarse quasi-isometriesbetween metric spaces form a category of isomorphisms. This is indeed thesubcategory of isomorphisms of the following larger category. For any set S anda metric space M , with metric d , two maps f, g : S → M are said to be close when there is some R > d ( f ( x ) , g ( x )) ≤ R for all x ∈ S ; it maybe also said that these maps are R -close . If ( M ′ , d ′ ) is another metric space, a(not necessarily continuous) map f : M → M ′ is said to be large scale Lipschitz [8] if there are constants λ ≥ c > d ′ ( f ( x ) , f ( y )) ≤ λ d ( x, y ) + c for all x, y ∈ M ; in this case, the pair ( λ, c ) will be called a large scale Lipschitzdistortion of f . The map f is said to be large scale bi-Lipschitz if there areconstants λ ≥ c > λ d ( x, y ) − c ≤ d ′ ( f ( x ) , f ( y )) ≤ λ d ( x, y ) + c It is also called rough isometry in the context of potential theory [12] This term is used in [10]. Other terms used to indicate the same property are coarselyequivalent [19], parallel [8], bornotopic [18], and uniformly close [2]. x, y ∈ M ; in this case, the pair ( λ, c ) will be called a large scale bi-Lipschitzdistortion of f . The map f will be called a large scale Lipschitz equivalence if it is large scale Lipschitz and if there is another large scale Lipschitz map g : M ′ → M so that g ◦ f and f ◦ g are close to the identity maps on M and M ′ , respectively. In this case, if ( λ, c ) is a large scale Lipschitz distortion of f and g , and g ◦ f and f ◦ g are R -close to the identity maps for some R > λ, c, R ) will be called a large scale Lipschitz equivalence distortion of f .A large scale Lipschitz equivalence is easily seen to be large scale bi-Lipschitz.It is well known that two metric spaces are coarsely quasi-isometric if andonly if they are isomorphic in the category of metric spaces and closeness equiv-alence classes of large scale Lipschitz maps; this is part of the content of thefollowing two results, where the constants involved are specially analyzed. Proposition 2.2.
Let f : A → A ′ be any coarse quasi-isometry between metricspaces ( M, d ) and ( M ′ , d ′ ) with coarse distortion ( K, C ) . Then f is inducedby a large scale Lipschitz equivalence ϕ : M → M ′ with large scale Lipschitzequivalence distortion ( C, CK, K ) .Proof. For each x ∈ M and x ′ ∈ M ′ , choose points h ( x ) ∈ A and h ′ ( x ) ∈ A ′ suchthat d ( x, h ( x )) ≤ K and d ′ ( x ′ , h ′ ( x ′ )) ≤ K . Moreover assume that h ( x ) = x forall x ∈ A , and that h ′ ( x ′ ) = x ′ for all x ′ ∈ A ′ . Then f and f − are respectivelyinduced by the maps ϕ = f ◦ h : M → M ′ and ψ = f − ◦ h ′ : M ′ → M . For all x, y ∈ M , d ′ ( ϕ ( x ) , ϕ ( y )) ≤ C d ( h ( x ) , h ( y )) ≤ C d ( h ( x ) , x ) + C d ( x, y ) + C d ( y, h ( y )) ≤ C d ( x, y ) + 2 CK ;furthermore d ( x, ψ ◦ ϕ ( x )) = d ( x, h ( x )) ≤ K .
Similarly, d ( ψ ( x ′ ) , ψ ( y ′ )) ≤ C d ′ ( x ′ , y ′ ) + 2 CK ,d ( x ′ , ϕ ◦ ψ ( x ′ )) ≤ K , for all x ′ , y ′ ∈ M ′ , and the result follows. Proposition 2.3.
Let ϕ : M → M ′ be a large scale Lipschitz equivalence withlarge scale Lipschitz equivalence distortion ( λ, c, R ) . Then, for each ε > , themap ϕ induces a coarse quasi-isometry f : A → A ′ between M and M ′ withcoarse distortion ( R + 2 λR + λc + λε + c , λ (1 + 2 R + cε )) . Proof.
Let ψ : M ′ → M be a large scale Lipschitz map with large scale Lipschitzdistortion ( λ, c ) such that ψ ◦ ϕ and ϕ ◦ ψ are R -close to the identity maps on M and M ′ , respectively. 6y Lemma 2.1, there is a (2 R + c + ε )-separated (2 R + c + ε )-net A of M .Let A ′ = ϕ ( A ), and let f : A → A ′ denote the restriction of ϕ . For all x, y ∈ M ,the inequality d ( x, y ) ≤ d ( x, ψ ◦ ϕ ( x )) + d ( ψ ◦ ϕ ( x ) , ψ ◦ ϕ ( y )) + d ( ψ ◦ ϕ ( y ) , y )implies d ( x, y ) ≤ λ d ′ ( ϕ ( x ) , ϕ ( y )) + 2 R + c . (2.1)In particular, if ϕ ( x ) = ϕ ( y ), then d ( x, y ) ≤ R + c . Therefore f is bijectivebecause A is (2 R + c )-separated.For any x ′ ∈ M ′ , there is some x ∈ A such that d ( x, ψ ( x ′ )) ≤ R + c + ε .Then d ′ ( x ′ , ϕ ( x )) ≤ d ′ ( x ′ , ϕ ◦ ψ ( x ′ )) + d ′ ( ϕ ◦ ψ ( x ′ ) , ϕ ( x )) ≤ R + λ d ( ψ ( x ′ ) , x ) + c ≤ R + 2 λR + λc + λε + c . So A ′ is a ( R + 2 λR + λc + λε + c )-net of M ′ .Because A is (2 R + c + ε )-separated, if x, y ∈ A are distinct, then d ′ ( f ( x ) , f ( y )) ≤ λ d ( x, y ) + c ≤ ( λ + c R + c + ε ) d ( x, y ) . By the same reason and (2.1), it follows that d ′ ( f ( x ) , f ( y )) > ε/λ . Hence d ( x, y ) ≤ λ d ′ ( f ( x ) , f ( y )) + 2 R + c ≤ λ (1 + 2 R + cε ) d ′ ( f ( x ) , f ( y ))again by (2.1), which finishes the proof.
3. Coarse structures
The concept of coarse structure was introduced in Roe [19], and furtherdeveloped in Higson-Roe [10], as a generalization of the concept of the closenessrelation on maps from a set into a metric space. The basic definitions and resultspertaining to coarse structures are recalled presently.
Definition 3.1. A coarse structure on a set X is a correspondence that assignsto each set S an equivalence relation (called “being close ”) on the set of maps S → X such that the following compatibility conditions are satisfied:(i) if p, q : S → X are close and h : S ′ → S is any map, then p ◦ h and q ◦ h are close;(ii) if S = S ′ ∪ S ′′ and if p, q : S → X are maps whose restrictions to both S ′ and S ′′ are close, then p and q are close; and(iii) all constant maps S → X are close to each other.7 set endowed with a coarse structure is called a coarse space . Definition 3.2.
Let X be a coarse space. A subset E ⊂ X × X is called controlled if the restrictions to E of the two factor projections X × X → X areclose.The coarse structure of a coarse space X is determined by its controlled sets:two maps p, q : S → X are close if and only if the image of ( p, q ) : S → X × X iscontrolled. Thus a coarse structure can be also defined in terms of its controlledsets [19], [10].A subset B ⊂ X is called bounded if B × B is controlled, equivalently, if theinclusion mapping B ֒ → X is close to a constant mapping. More generally, acollection U of subsets of X is said to be uniformly bounded if S U ∈U U × U iscontrolled. The coarse space X is called separable if it has a countable uniformlybounded cover. Definition 3.3.
A mapping f : X → X ′ between coarse spaces is called a coarse map if(i) whenever p, q : S → X are close maps, the composites f ◦ p, f ◦ q : S → X ′ are close maps; and(ii) if B is a bounded subset of X ′ , then f − ( B ) is bounded in X .Two coarse spaces, X and X ′ , are coarsely equivalent if there are coarsemappings f : X → X ′ and g : X ′ → X such that f ◦ g is close to the identityof X ′ and g ◦ f is close to the identity of X . In this case, f (and g ) are calledcoarse equivalences. The coarse category is the category whose objects are coarsespaces and whose morphisms are equivalence classes of coarse mappings, twomappings being equivalent if they are close. Definition 3.4.
Let X , X ′ be coarse spaces. A mapping ϕ : X → X ′ is uniformly coarse if(i) for every controlled set E ⊂ X × X , the image ( ϕ × ϕ )( E ) ⊂ X ′ × X ′ iscontrolled, and(ii) for every controlled set F ⊂ X ′ × X ′ , the preimage ( ϕ × ϕ ) − ( F ) ⊂ X × X is controlled. Proposition 3.5.
Let X and X ′ be coarse spaces, and let ϕ : X → X ′ and ψ : X ′ → X be mappings satisfying (i) of Definition 3.3, and such that ψ ◦ ϕ isclose to the identity of X and ϕ ◦ ψ is close to the identity of X ′ . Then ϕ and ψ are uniformly coarse ( and consequently X and X ′ are coarsely equivalent ) . Controlled sets are called entourages in Roe [19]. roof. It is plain that (i) of Definition 3.4 is equivalent to (i) of Definition 3.3,and that (ii) of Definition 3.4 implies (ii) of Definition 3.3.Let p and p denote the projection mappings X × X → X . If F ⊂ X ′ × X ′ is controlled, then ( ψ × ψ )( F ) ⊂ X × X is controlled, and so the mappings p ◦ ( ψ × ψ ) and p ◦ ( ψ × ψ ) of F into X are close. Therefore, the mappings p ◦ ( ψ × ψ ) ◦ ( ϕ × ϕ ) = p ◦ ( ψ ◦ ϕ × ψ ◦ ϕ ) and p ◦ ( ψ × ψ ) ◦ ( ϕ × ϕ ) = p ◦ ( ψ ◦ ϕ × ψ ◦ ϕ )from ( ϕ × ϕ ) − ( F ) into X are also close. Since ψ ◦ ϕ is close to the identity on X , the mappings p and p from ( ϕ × ϕ ) − ( F ) into X are also close, establishingproperty (ii) of Definition 3.4 for ϕ . Definition 3.6.
A coarse structure on a set X is said to be a proper coarsestructure if(i) X is equipped with a locally compact Hausdorff topology;(ii) X has a uniformly bounded open cover; and(iii) every bounded subset of X has compact closure.A set equipped with a proper coarse structure will be called a proper coarsespace . Note that bounded subsets of a proper coarse space are those subsetswith compact closure.A metric space, ( M, d ), has a natural coarse structure, that is defined bydeclaring two maps f, g : S → M (where S is any set) to be close whensup { d ( f ( s ) , g ( s )) | s ∈ S } < ∞ . This closeness relation defines a coarse struc-ture on M , which is called its metric coarse structure. The terms metric close-ness and metric controlled set can be used in this case. This coarse structure isproper if and only if the metric space M is proper in the sense that its closedballs are compact. In the case of metric coarse structures, the above abstractcoarse notions have their usual meanings for metric spaces.More generally, following Hurder [11], a coarse distance (or coarse metric ) ona set X is a symmetric map d : X × X → [0 , ∞ ) satisfying the triangle inequality;in this case, ( X, d ) is called a coarse metric space . Any coarse distance defines acoarse structure in the same way as a metric does, and will be also called a metric coarse structure. In this section and the above one, all notions and propertiesare given for metric spaces for simplicity, but they have obvious versions forcoarse metric spaces.
Definition 3.7.
If (
M, d ) and ( M ′ , d ′ ) are metric spaces, the two conditions ofDefinition 3.3 on a map f : M → M ′ to be coarse can be written as follows:(i) ( Uniform expansiveness. ) For each R >
S > d ( x, z ) ≤ R = ⇒ d ′ ( f ( x ) , f ( z )) ≤ S for all x, z ∈ M . This term is taken from [10]. It is also called bounded coarse structure in [19] This name comes from Roe [19]. Other terms used to denote the same property are uniformly bornologous
Roe [18] and coarsely Lipschitz
Block-Weinberger [2].
Metric properness. ) For each bounded subset B ⊂ M ′ , the inverse image f − ( B ) is bounded in M .The last property admits a uniform version: a map f : M → M ′ is said tobe uniformly metrically proper if for each R >
S > d ′ ( f ( x ) , f ( z )) ≤ R = ⇒ d ( x, z ) ≤ S for all x, z in M . By using uniform metric properness instead of metric proper-ness, we get what is called the rough category . More precisely, a map betweenmetric spaces, f : M → M ′ , is called a rough map if it is uniformly expansiveand uniformly metrically proper; if moreover there is a rough map g : X ′ → X so that the compositions g ◦ f and f ◦ g are respectively close to the identitymaps on X and X ′ , then f is called a rough equivalence ; in this case, X and X ′ are said to be roughly equivalent . Thus rough equivalences are the maps thatinduce isomorphisms in the rough category. There are interesting differencesbetween the rough category and the coarse category of metric spaces Roe [19],but the following result shows that they have the same isomorphisms. Proposition 3.8.
Any coarse equivalence between metric spaces is uniformlymetrically proper. Moreover the definition of uniform metric properness is sat-isfied with constants that depend only on the constants involved in the definitionof coarse equivalence.Proof.
Let f : M → M ′ and g : M ′ → M be coarse maps so that g ◦ f and f ◦ g are r -close to the identity maps on M and M ′ for some r >
0. Then, because g is uniformly expansive, for any R >
S > d ′ ( x ′ , z ′ ) ≤ R = ⇒ d ( g ( x ′ ) , g ( z ′ )) ≤ S for all x ′ , z ′ ∈ M ′ . It follows that when x, z ∈ M are such that d ′ ( f ( x ) , f ( z )) ≤ R , then d ( x, z ) ≤ d ( x, g ◦ f ( x )) + d ( g ◦ f ( x ) , g ◦ f ( z ) + d ( g ◦ f ( z ) , z ) ≤ S + 2 r , which establishes that f is uniformly metrically proper.It is not possible to define “equi-coarse maps” or “equi-coarse equivalences”between arbitrary coarse spaces, but in the case of metric coarse structures thefollowing related concepts can be defined. A family of maps, f i : X i → X ′ i , i ∈ Λ, is said to be a family of: • equi-uniformly expansive maps if they satisfy the condition of uniformexpansiveness involving the same constants; This term is used in Roe [19] This term is used in [19]. Another term used to denote the same property is effectivelyproper [2]. The term uniform closeness is used in [2] to indicate this equivalence between metricspaces equi-uniformly metrically proper maps if they satisfy the condition of uni-form metric properness involving the same constants; • equi-rough maps if they are equi-uniformly expansive and equi-uniformlymetrically proper; and • equi-rough equivalences if they are equi-rough, and there is another col-lection of equi-rough maps g i : X ′ i → X i , i ∈ Λ, and there is some r > g i ◦ f i and f i ◦ g i are r -close to the identity mapson X i and X ′ i , respectively, for all i ∈ Λ.According to Proposition 3.8, a collection of equi-rough equivalences can bealso properly called a family of equi-coarse equivalences .Gromov [9, Theorem 1.8.i] characterizes complete path metric spaces (thatis, metric spaces where the distance between any two points equals the infimumof the lengths of all paths joining those two points) as those complete metricspaces, (
X, d ), that satisfy the following property: for all points x, y in X andevery ε >
0, there is some point z such that max { d ( x, z ) , d ( y, z ) } < d ( x, y )+ ε .This condition can be called “ approximate convexity :” a subset of R n satisfiesthis property (with respect to the induced metric) if and only if it has convexclosure. Gromov [9, Theorem 1.8.i] establishes that a complete, locally compactmetric space is approximately convex if and only if it is geodesic: the distancebetween any two points equals the length of some curve joining those two points.The following definition is a coarsely quasi-isometric version of the aboveapproximate convexity property. Definition 3.9.
A metric space, (
M, d ), is said to be coarsely quasi-convex ifthere are constants a, b, c > x, y ∈ M , there is some finitesequence of points x = x , . . . , x n = y in M such that d ( x k − , x k ) ≤ c for all k ∈ { , . . . , n } , and n X k =1 d ( x k − , x k ) ≤ a d ( x, y ) + b . A family of metric spaces is said to be equi-coarsely quasi-convex if all of themsatisfy the condition of being coarsely quasi-convex with the same constants a , b , and c . Remark . Definition 3.9 can be compared with the concept of monogeniccoarse space [20]. In the case of a metric coarse structure, the condition of beingmonogenic is obtained by removing the constants a, b and the last inequalityfrom Definition 3.9.A typical example of a coarsely quasi-convex space that is not approximatelyconvex is the set V of vertices of a connected graph G with the metric d V induced by G . This V satisfies the condition of being coarsely quasi-convexwith constants a = b = c = 1. This metric on V is the restriction of a metric on G that can be defined as follows. Choose any metric d e on each edge e of G so11hat e is isometric to the unit interval. Then the distance between two points x, y ∈ G is the minimum of the sums of the form d e ( x, v ) + d V ( v, w ) + d f ( w, y ) , where x, y lie in edges e, f , and v, w are vertices of e, f , respectively. Observethat G is geodesic and V is a 1 / G . More generally, any net of a geodesicmetric space is coarsely quasi-convex. This is a particular case of the followingresult. Theorem 3.11.
A metric space, ( M, d ) , is coarsely quasi-convex if and onlyif there exists a coarse quasi-isometry f : A → A ′ between ( M, d ) and somegeodesic metric space ( M ′ , d ′ ) . In this case, the coarsely quasi-isometric distor-tion of f depends only on the constants involved in the condition coarse quasi-convexity satisfied by M , and conversely; equivalently, a family of metric spacesis equi-coarsely quasi-convex if and only if they are equi-coarsely quasi-isometricto geodesic metric spaces.Proof. Suppose that there is a coarse quasi-isometry f : A → A ′ , with coarsedistortion ( K, C ), between (
M, d ) and a geodesic metric space ( M ′ , d ′ ). For all x, y ∈ M , there are some ¯ x, ¯ y ∈ A with d ( x, ¯ x ) , d ( y, ¯ y ) ≤ K . Then there is somefinite sequence f (¯ x ) = x ′ , . . . , x ′ n = f (¯ y ) in M ′ such that d ′ ( x ′ k − , x ′ k ) < k ∈ { , . . . , n } and n X k =1 d ′ ( x ′ k − , x ′ k ) = d ′ ( f (¯ x ) , f (¯ y )) . Moreover, we can assume that this is one of the shortest sequences satisfyingthis condition. If d ′ ( x ′ k − , x ′ k ) < / d ′ ( x ′ k , x ′ k +1 ) < / k , thenthe term x ′ k could be removed from the sequence, contradicting its minimality.It follows that d ′ ( x ′ k − , x ′ k ) ≥ / n/
2] indexes k . So( n − / ≤ [ n/ / ≤ n X k =1 d ′ ( x ′ k − , x ′ k ) = d ′ ( f (¯ x ) , f (¯ y )) , which implies n ≤ d ′ ( f (¯ x ) , f (¯ y )) + 1 . (3.1)For each k ∈ { , . . . , n } , there is some ¯ x ′ k ∈ A ′ with d ′ (¯ x ′ k , x ′ k ) ≤ K , and let¯ x k = f − (¯ x ′ k ); for simplicity, take ¯ x ′ = x ′ and ¯ x ′ n = x ′ n , and thus ¯ x = ¯ x and¯ x n = ¯ y . To simplify the notation, let also x = x , x n = y , and x k = ¯ x k for k ∈ { , . . . , n − } . Then d ( x k − , x k ) ≤ d ( x k − , ¯ x k − ) + d (¯ x k − , ¯ x k ) + d (¯ x k , x k ) ≤ K + C d ′ (¯ x ′ k − , ¯ x ′ k ) ≤ K + C ( d ′ (¯ x ′ k − , x ′ k − ) + d ′ ( x ′ k − , x ′ k ) + d ′ ( x ′ k , ¯ x ′ k )) ≤ K + 2 CK + C k ∈ { , . . . , n } , and n X k =1 d ( x k − , x k ) ≤ d ( x, ¯ x ) + n X k =1 d (¯ x k − , ¯ x k ) + d (¯ y, y ) ≤ K + C n X k =1 d ′ (¯ x ′ k − , ¯ x ′ k ) ≤ K + C n X k =1 ( d ′ (¯ x ′ k − , x ′ k − ) + d ′ ( x ′ k − , x ′ k ) + d ′ ( x ′ k , ¯ x ′ k )) ≤ K + 2 CKn + C n X k =1 d ′ ( x ′ k − , x ′ k ) ≤ K + 2 CK (4 d ′ ( f (¯ x ) , f (¯ y )) + 1)) + C d ′ ( f (¯ x ) , f (¯ y )) ≤ K + 2 CK + (8 CK + C ) C d (¯ x, ¯ y ) ≤ K + 2 CK + (8 CK + C ) C ( d (¯ x, x ) + d ( x, y ) + d ( y, ¯ y )) ≤ K + 2 CK + 2(8 CK + C ) CK + (8 CK + C ) C d ( x, y ) , where (3.1) was used in the fifth inequality. Thus the condition of Definition 3.9is satisfied with a , b and c depending only on K and C , as desired.Assume now that ( M, d ) satisfies the coarsely quasi-convex condition (Defi-nition 3.9) with constants a , b and c . By Lemma 2.1, there is a c -separated c -net A in M . By attaching an edge to any pair of points x, y ∈ A with d ( x, y ) ≤ c ,there results a graph M ′ whose set of vertices is A . For any x, y ∈ A , thereis a finite sequence x = x, x , . . . , x n = y in M with d ( x k − , x k ) ≤ c for all k ∈ { , . . . , n } , and n X k =1 d ( x k − , x k ) ≤ a d ( x, y ) + b . For each k , take some ¯ x k ∈ A with d ( x k , ¯ x k ) ≤ c ; in particular, take ¯ x = x and¯ x n = y . Then there is an edge between each ¯ x k − and ¯ x k because d (¯ x k − , ¯ x k ) ≤ d (¯ x k − , x k − ) + d ( x k − , x k ) + d ( x k , ¯ x k ) ≤ c . Therefore M ′ is a connected graph. Let d ′ denote the geodesic metric on M ′ ,defined as above, with each edge having a metric that makes it isometric to theunit interval. Since A is a 1-net in M ′ , it only remains to check that the identitymap ( A, d ) → ( A, d ′ ) is bi-Lipschitz with bi-Lipschitz distortion depending onlyon a , b and c . Fix any pair of different points x, y ∈ A , and take a sequence x = ¯ x , . . . , ¯ x n = y as above; after removing some points of this sequence, ifnecessary, it may be assumed that ¯ x k − = ¯ x k for all k . Since there is an edgebetween each ¯ x k − and ¯ x k , it follows that d ′ ( x, y ) ≤ n . Since A is c -separated, cn ≤ n X k =1 d (¯ x k − , ¯ x k ) ≤ a d ( x, y ) + b ≤ ac + bc d ( x, y ) , d ′ ( x, y ) ≤ ac + bc d ( x, y ) . On the other hand, if d ′ ( x, y ) = m , then there is a sequence x = y , . . . , y m = y in A with the property that each pair y k − , y k is joined by an edge; thus d ( y k − , y k ) ≤ c for each k , and so d ( x, y ) ≤ m X k =1 d ( y k − , y k ) ≤ cm = 3 c d ′ ( x, y ) . Remark . Theorem 3.11 is a coarsely quasi-isometric version of [20, Propo-sition 2.57], which asserts that the monogenic coarse structures are those thatare coarsely equivalent to geodesic metric spaces.
Proposition 3.13.
The following properties hold true: (i)
Any large scale Lipschitz map between metric spaces is uniformly expan-sive; moreover, a family of equi-large scale Lipschitz maps between metricspaces is equi-uniformly expansive. (ii)
Any large scale Lipschitz equivalence is a rough equivalence; moreover, afamily of equi-large scale Lipschitz equivalences between metric spaces is afamily of equi-rough equivalences.Proof.
Let (
M, d ) and ( M ′ , d ′ ) be metric spaces, and let f : M → M ′ be a largescale Lipschitz map. If ( λ, c ) is a large scale Lipschitz distortion of f , then f obviously satisfies the definition of uniform expansiveness with S = λR + c foreach R >
0. This proves property (i) because S depends only on R , λ and c .For (ii), suppose that f is a large scale Lipschitz equivalence. Then there isa large scale Lipschitz map g : M ′ → M , whose large scale Lipschitz distortioncan be assumed to be also ( λ, c ), such that g ◦ f and f ◦ g are r -close to theidentity maps on M and M ′ , for some r >
0. Then d ( x, y ) ≤ d ( x, g ◦ f ( x )) + d ( g ◦ f ( x ) , g ◦ f ( y )) + d ( g ◦ f ( y ) , y ) ≤ λ d ′ ( f ( x ) , f ( y )) + 2 r . Hence f satisfies the definition of uniform metric properness with S = λR + 2 r ,for each R >
0. This proves property (ii) because S depends only on R , λ and r . Example . Let N = { n | n ∈ N } and N = { n | n ∈ N } with therestriction of the Euclidean metric on R . Suppose that N and N are largescale Lipschitz equivalent; i.e. , there are large scale Lipschitz maps f : N → N and g : N → N with large scale Lipschitz distortion ( λ, c ) such that g ◦ f and f ◦ g are close to identity maps on N and N . Let σ, τ : N → N be the mapsdefined by f ( n ) = σ ( n ) and g ( n ) = τ ( n ) . Since ( n + 1) − n → ∞ and( n + 1) − n → ∞ as n → ∞ , there is some a ∈ N such that g ◦ f ( n ) = n and14 ◦ g ( n ) = n for all n ≥ a , and so τ ◦ σ ( n ) = σ ◦ τ ( n ) = n for n ≥ a . Assumefor a while that there is some integer b ≥ a such that τ ( n ) ≥ n + a + 2 for all n ≥ b . Thus τ ( { b, b + 1 , . . . } ) ⊂ { b + a + 2 , b + a + 3 , . . . } and therefore { a, a + 1 , . . . , b + a + 1 } ⊂ τ ( { , , . . . , b − } )because { a, a + 1 , . . . } ⊂ τ ( N ). So b + 1 = { a, a + 1 , . . . , b + a + 1 } ≤ τ ( { , . . . , b − } ) ≤ b , which is a contradiction. Hence there is some sequence n k ↑ ∞ in N such that τ ( n k ) ≤ n k + a + 1 for all k . So | τ ( n k ) − n k | ≤ ( τ ( n k ) + n k ) ≤ (2 n k + a + 1) . We can assume that n k ≥ a for all k . Then n k = f ◦ g ( n k ) − f ◦ g ( n ) + n ≤ λ | g ( n k ) − g ( n ) | + c + n = λ | τ ( n k ) − τ ( n ) | + c + n ≤ λ ( | τ ( n k ) − n k | + n k − n + | n − τ ( n ) | ) + c + n ≤ λ ((2 n k + a + 1) + n k − n + (2 n + a + 1) ) + c + n = 5 λn k + 4 λ ( a + 1) n k + 3 λn + 4 λ ( a + 1) n + 2 λ ( a + 1) + c + n , which is a contradiction because n k → ∞ as k → ∞ . Therefore there is nolarge scale Lipschitz equivalence between N and N , and thus these spacesare not coarsely quasi-isometric. But they are coarsely equivalent; indeed, themap n n is distance decreasing, and the map n n is coarse: if 0 < | n − m | < R , then n + m < R also, so | n − m | < S with S = R .Example 3.14 shows that the converse of Proposition 3.13-(2) does not holdin general. Nevertheless, the following proposition shows that coarse equiva-lences coincide with large scale Lipschitz equivalences for metric spaces that arecoarsely quasi-convex. Proposition 3.15.
Any uniformly expansive map of a coarsely quasi-convexmetric space to another metric space is large scale Lipschitz; moreover, a familyof equi-uniformly expansive maps between metric spaces, whose domains areequi-coarsely quasi-convex, is a family of equi-large scale Lipschitz maps.Proof.
Let (
M, d ) and ( M ′ , d ′ ) be metric spaces, and let f : M → M ′ bea uniformly expansive map. Suppose that M satisfies the condition of beingcoarsely quasi-convex with constants a , b , and c . Fix points x, y ∈ M , and let15 = x , . . . , x n = y be a sequence of smallest length such that d ( x k − , x k ) ≤ c ,for k = 1 , · · · , n , and n X k =1 d ( x k − , x k ) ≤ a d ( x, y ) + b . If both d ( x k − , x k ) < c/ d ( x k , x k +1 ) < c/ k ∈ { , . . . , n − } , then d ( x k − , x k +1 ) < c , and thus x k could be removed from the sequence x , x , · · · , x n , contradicting that this was a sequence of smallest length. Hencethere are at least ( n − / k ∈ { , · · · , n } such that d ( x k − , x k ) ≥ c/ a d ( x, y ) + b ≥ n X k =1 d ( x k − , x k ) ≥ ( n − c , or n ≤ ac d ( x, y ) + 4 bc + 1 . (3.2)Since f is uniformly expansive, there is some S > d ′ ( f ( z ) , f ( z ′ )) ≤ S for all z, z ′ ∈ M with d ( z, z ′ ) ≤ c . So, by (3.2), d ′ ( f ( x ) , f ( y )) ≤ n X k =1 d ′ ( f ( x k − ) , f ( x k )) ≤ nS ≤ aSc d ( x, y ) + 4 bSc + S, which establishes that f is large scale Lipschitz with large scale Lipschitz dis-tortion depending only on S , a , b and c . Corollary 3.16.
Any coarse equivalence between coarsely quasi-convex metricspaces is a large scale Lipschitz equivalence; moreover, a family of equi-coarseequivalences between equi-coarsely quasi-convex spaces is a family of equi-largescale Lipschitz equivalences.Proof.
This is elementary by Proposition 3.15.
Corollary 3.17.
Two coarsely quasi-convex metric spaces are coarsely quasi-isometric if and only if they are coarsely equivalent; moreover, if M i and M ′ i , i ∈ Λ , are families of equi-coarsely quasi-convex metric spaces, then all pairs M i and M ′ i are equi-coarsely quasi-isometric if and only they are equi-coarselyequivalent.Proof. This follows from Propositions 2.2, 2.3 and 3.15, and Corollary 3.16.
4. The Higson Compactification
A significant example of coarse structure is induced by any compactifica-tion X κ = X of a topological space X , with corona ∂X = κX = X κ \ X . Only metrizable compactifications are considered in [18], but this kind of coarse structurecan be defined for arbitrary compactifications [10, 19] X is defined by declaring a subset E ⊂ X × X to becontrolled when E ∩ ∂ ( X × X ) ⊂ ∆ ∂X in X × X , where ∂ ( X × X ) = (cid:0) ∂X × X (cid:1) ∪ (cid:0) X × ∂X (cid:1) . This is called the topological coarse structure associated to the given compact-ification; it is proper if X is metrizable [20], [21].A compactification X of a proper coarse space X is said to be a coarsecompactification , with coarse corona ∂X = X \ X , if the identity map from X with its given coarse structure to X endowed with the topological coarsestructure arising from X is a coarse map. Intuitively, the slices of any controlledsubset of X × X become small when approaching the boundary ∂X ; in particular,this holds for the sets of any uniformly bounded family in X .The structure of coarse compactifications of a proper coarse space X canbe described algebraically as follows. Let B ( X ) be the Banach algebra of allbounded functions X → C with the supremum norm, and let B ( X ) be theBanach subalgebra of all functions f ∈ B ( X ) that vanish at infinity; i.e. , suchthat, for any ε >
0, there is some compact subset K ⊂ X so that | f ( x ) | < ε forall x ∈ X \ K . For any f ∈ B ( X ) and every controlled subset E ⊂ X × X , letthe E -expansion of f be the function ∇ E f ∈ B ( X ) defined by ∇ E f ( x ) = sup {| f ( x ) − f ( y ) | | ( x, y ) ∈ E } . Definition 4.1.
A function f ∈ B ( X ) is called a Higson function if ∇ E f ∈B ( X ) for all controlled subsets E ⊂ X × X .The set B ν ( X ) of all Higson functions on X is a Banach subalgebra of B ( X )[19], [10], [20]. If only bounded continuous functions are considered, then thenotation C b ( X ), C ( X ) and C ν ( X ) will be used instead of B ( X ), B ( X ) and B ν ( X ), respectively.The terms X -close maps, X -controlled sets and X -coarse compactificationwill be used in the case of the topological coarse structure induced by a com-pactification X of a locally compact space X .The following lemma shows that Higson functions are preserved by coarsemaps. Lemma 4.2.
Let X be a compactification of a locally compact space X withboundary ∂X . The following conditions are equivalent for any subset E ⊂ X × X : (i) E is X -controlled. (ii) ∇ E f ∈ B ( X ) for every f ∈ B ( X ) having an extension ¯ f : X → C that iscontinuous on the points of ∂X . This term is used in [10]. It is also called continuously controlled coarse structure in [19] ∇ E f ∈ C ( X ) for every f ∈ C b ( X ) having a continuous extension to X .Proof. To prove that property (i) implies property (ii), suppose that E is X -controlled, and assume that some f ∈ B ( X ) has an extension ¯ f : X → C thatis continuous on the points of ∂X . Since the function ( x, y ) (cid:12)(cid:12) ¯ f ( x ) − ¯ f ( y ) (cid:12)(cid:12) on X × X vanishes on ∆ ∂X and is continuous on the points of ∂X × ∂X , there issome open neighborhood Ω of ∆ ∂X in X × X such that (cid:12)(cid:12) ¯ f ( x ) − ¯ f ( y ) (cid:12)(cid:12) < ε forall ( x, y ) ∈ Ω. On the other hand, since E is X -controlled, there is some openneighborhood U of ∂X in X such that E ∩ ( U × X ) ⊂ Ω . If K is the compact set K = X \ U , then ∇ E f ( x ) < ε for all x ∈ X \ K = X ∩ U ,and so ∇ E f ∈ B ( X ).Property (iii) is a particular case of property (ii).To prove that property (iii) implies property (i), assume that ∇ E f ∈ C ( X )for all f ∈ C b ( X ) that admit a continuous extension to X . If E were not X -controlled, there would be a pair of different points, x ∈ ∂X and y ∈ X , suchthat either ( x, y ) or ( y, x ) is in E . Since the family of controlled sets is invariantby transposition [19], [10], [20], it may be assumed that ( x, y ) ∈ E . Then, forany continuous function ¯ f : X → C with ¯ f ( x ) = ¯ f ( y ), the restriction f = ¯ f | X would satisfy lim inf z → x ∇ E f ( z ) ≥ (cid:12)(cid:12) ¯ f ( x ) − ¯ f ( y ) (cid:12)(cid:12) > , which would be a contradiction. Therefore E is X -controlled.The following is a direct consequence of Lemma 4.2, which is contained in[20, Proposition 2.39]. Corollary 4.3.
Let X be a compactification of a proper coarse space X withboundary ∂X . Then the following conditions are equivalent: (i) X is a coarse compactification of X . (ii) B ν ( X ) contains every function in B ( X ) that admits an extension to X that is continuous on the points of ∂X . (iii) C ν ( X ) contains every continuous function X → C that extends continu-ously to X . Proposition 4.4.
Let X and X ′ be compactifications of locally compact spaces X and X ′ with boundaries ∂X and ∂X ′ , respectively. Then the following prop-erties hold: (i) A map ϕ : X → X ′ is coarse if it has an extension ¯ ϕ : X → X ′ that iscontinuous on the points of ∂X and such that ¯ ϕ ( ∂X ) ⊂ ∂X ′ . (ii) Let ϕ, ψ : X → X ′ be maps with extensions ¯ ϕ, ¯ ψ : X → X ′ satisfying theconditions of property (i) . Then ϕ and ψ are X ′ -close if and only if ¯ ϕ = ¯ ψ on ∂X . roof. Let ϕ : X → X ′ be a map satisfying the conditions of property (i).If B is a bounded subset of X ′ , then B has compact closure in X ′ , and thus B ∩ ∂X ′ = ∅ . So¯ ϕ ( ϕ − ( B ) ∩ ∂X ) ⊂ ¯ ϕ ( ϕ − ( B )) ∩ ∂X ′ ⊂ B ∩ ∂X ′ = ∅ because ¯ ϕ is continuous on the points of ∂X . It follows that ϕ − ( B ) ∩ ∂X = ∅ ,and thus ϕ − ( B ) has compact closure in X ; that is, ϕ − ( B ) is bounded in X .Let E be a controlled subset of X × X , and let f : X ′ → C be a boundedfunction that admits an extension ¯ f to X ′ that is continuous on the points of ∂X ′ . Lemma 4.2 implies that ∇ ( ϕ × ϕ )( E ) f = ∇ E ( f ◦ ϕ ) ∈ B ( X ) because ¯ f ◦ ¯ ϕ is an extension of the function f ◦ ϕ that is continuous at the points of ∂X .It follows that ( ϕ × ϕ )( E ) is a controlled subset of X ′ × X ′ by Lemma 4.2.Therefore ϕ is a coarse map, which establishes property (i).Let ϕ, ψ : X → X ′ be maps with extensions ¯ ϕ, ¯ ψ : X → X ′ satisfyingthe conditions of property (i). Suppose first that ¯ ϕ = ¯ ψ on ∂X , and let E = { ( ϕ ( x ) , ψ ( x )) | x ∈ X } . Fix any point ( x ′ , y ′ ) ∈ E ∩ ∆ ∂ ( X ′ × X ′ ) . Thus, foreach neighborhood Ω of ( x ′ , y ′ ) in E , there is some point x Ω ∈ X so that( ϕ ( x Ω ) , ψ ( x Ω )) ∈ Ω; such points x Ω form a net ( x Ω ) in X . Suppose e.g. that x ′ ∈ ∂X ′ . Then the net ( ϕ ( x Ω )) is unbounded in X ′ , and thus the net ( x Ω ) isunbounded in X because ϕ is a coarse map according to property (i). So thereis an accumulation point x of ( x Ω ) in ∂X . Since ¯ ϕ and ¯ ψ are continuous at x ,it follows that (cid:0) ¯ ϕ ( x ) , ¯ ψ ( x ) (cid:1) is an accumulation point of the net ( ϕ ( x Ω ) , ψ ( x Ω )),which converges to ( x ′ , y ′ ). Hence ( x ′ , y ′ ) = (cid:0) ¯ ϕ ( x ) , ¯ ψ ( x ) (cid:1) ∈ ∆ ∂X ′ because ¯ ϕ = ¯ ψ on ∂X . This shows that E is X ′ -controlled, and thus ϕ is X ′ -close to ψ .Assume now that ϕ is X ′ -close to ψ ; i.e. , the set E = { ( ϕ ( x ) , ψ ( x )) | x ∈ X } is X ′ -controlled. The conditions on ¯ ϕ and ¯ ψ imply that (cid:0) ¯ ϕ × ¯ ψ (cid:1) (∆ ∂X ) = (cid:0) ¯ ϕ × ¯ ψ (cid:1) (cid:0) ∆ X ∩ ( ∂X × ∂X ) (cid:1) ⊂ ( ϕ × ψ )(∆ X ) ∩ ( ∂X ′ × ∂X ′ )= E ∩ ( ∂X ′ × ∂X ′ ) ⊂ ∆ ∂X ′ , which establishes that ¯ ϕ = ¯ ψ on ∂X , and completes the proof of property (ii). Remark . The above result can be compared with the “if” part of [20, Propo-sition 2.33]. Continuity on X is not needed in Proposition 4.3, only the con-tinuity on ∂X is used, and properness is replaced by the condition to preservethe boundary of the compactifications. The reciprocal of property (i) holdswhen the compactifications are first countable [20, Proposition 2.33], and thisassumption is necessary for the reciprocal [20, Example 2.34]. Second countability is required in [20, Proposition 2.33], but only first countability is usedin the proof. X ν ,which is the maximal ideal space of C ν ( X ); it is called the Higson compactifica-tion of X , and its boundary νX is called the Higson corona . Since each Higsonfunction on X has a unique extension to X ν that is continuous on the points of νX , there are canonical isomorphisms C ( νX ) ∼ = C ν ( X ) /C ( X ) ∼ = B ν ( X ) / B ( X ) . (4.1)This isomorphism can be used to define the Higson boundary νX for any coarsespace X [20].For subsets A of X or of X × X , the notation A ν will be used to indicatethe closure of A in X ν or in X ν × X ν , respectively. The notation ν ( X × X ) =( νX × X ) ∪ ( X × νX ) will be also used.The following lemma is contained in the proof of [20, Proposition 2.41]. Lemma 4.6.
Let X and X ′ be proper coarse spaces and let ϕ : X → X ′ be acoarse map. Then: (i) f ◦ ϕ ∈ B ( X ) for all f ∈ B ( X ′ ) , and (ii) f ◦ ϕ ∈ B ν ( X ) for all f ∈ B ν ( X ′ ) . Proposition 4.7.
Let X and X ′ be proper coarse spaces. Any coarse map ϕ : X → X ′ has a unique extension ¯ ϕ : X ν → X ′ ν that is continuous on thepoints of νX and such that ¯ ϕ ( νX ) ⊂ νX ′ .Proof. According to Lemma 4.6, ϕ induces a homomorphism ϕ ∗ : B ν ( X ′ ) →B ν ( X ) defined by ϕ ∗ ( f ) = f ◦ ϕ , which maps B ( X ′ ) to B ( X ). By (4.1), ϕ ∗ induces a homomorphism C ( νX ′ ) → C ( νX ). Then, by considering maximalideal spaces, we get a map ¯ ϕ : X ν → X ′ ν , which extends ϕ and maps νX into νX ′ . The continuity of ¯ ϕ on the points of νX is a consequence of the factthat any Higson function has a unique extension to the Higson compactificationwhich is continuous on the Higson corona. Remark . The above result is slightly stronger than [20, Proposition 2.41],which only shows the continuity of the restriction ¯ ϕ : νX → νX ′ .Sometimes the Higson compactification can be easily determined, as shownby the following result [20, Proposition 2.48]. Proposition 4.9.
Let X be a proper coarse space with the topological coarsestructure induced by a first-countable compactification X of X with boundary ∂X . Then X and X ν are equivalent compactifications of X , and thus ∂X ishomeomorphic to νX . The statement of [20, Proposition 2.48] requires second countability but, indeed, its proofonly uses first countability.
Proposition 4.10.
Let ( M, d ) be a proper metric space, and let M ν be itsHigson compactification. A point p in M ν is in M if and only if the set { p } isa G δ -set.Proof. The “only if” part is elementary. To prove the “if” part, let p ∈ M ν besuch that { p } is a G δ set. Then there is a sequence ( x n ) in M that converges to p . Suppose that p M ; i.e., p ∈ νM . Passing to a subsequence if necessary, itmay be assumed that there is a sequence of positive real numbers r n ↑ ∞ suchthat the metric balls B ( x n , r n ) are mutually disjoint. Let f : M → R be thefunction given by f ( x ) = ( − n r n − d ( x, x n ) r n x in B ( x n , r n )0 otherwise.Then f extends to a continuous function ¯ f on M ν , and so lim n →∞ ¯ f ( x n ) = ¯ f ( p ).But the definition of f implies that ¯ f ( x n ) = ( − n , so the limit lim n →∞ ¯ f ( x n ) doesnot exist. Remark . (i) The argument of [20, Example 2.53] can also be used toshow that the point p (in proof above) is in M .(ii) G δ properties are common in the study of the structure of the Stone-Cˇechcompactification of spaces (e.g., Walker [22]). The property brought tolight here also plays a role in Nakai’s work on the Royden compactificationof Riemann surfaces [15]. Proposition 4.12.
Let ( M, d ) be a non-compact proper metric space. Let W ⊂ M be a subset that contains metric balls of arbitrarily large radius. Then theclosure of W in M ν is a neighborhood of a point in νM .Proof. If W ⊂ M contains ball of arbitrarily large radius, then, because M is not compact, there is a sequence, ( x n ), of points in W without limit pointin M , and a sequence of positive real numbers r n ↑ ∞ such that the metricballs B ( x n , r n ) are mutually disjoint and contained in W . If f is the functionconstructed in Proposition 4.10, then g = | f | admits a continuous extension, ¯ g ,to M ν that satisfies ¯ g ( p ) = 1 for any p ∈ νM that is an accumulation point ofthe sequence ( x n ). Therefore ¯ g − (0 ,
1] is an open neighborhood of p containedin the closure of W in M ν .The Higson compactification of a proper coarse space is defined as the max-imal ideal space of the algebra of Higson functions on the space. The questionarises whether it is possible to construct the Higson compactification directly21orm the topological structure of the space, or whether the Higson compactifi-cation is a Wallman-Frink compactification. A Wallman-Frink compactificationcan be defined using H -ultrafilters, where H is the ring of zero sets of Higsonfunctions, topologized in a appropriate manner. The resulting space may notbe Hausdorff and has the Higson compactification as a quotient space. Under-standing the precise relationship between the two compactifications will lead toan intrinsic characterization of H -set, toward which Proposition 4.12 is a minorcontribution.Even if the statement of Proposition 4.9 was not true when the first-countabilityaxiom is removed, the following result is always true by the maximality of theHigson compactification among all coarse compactifications. Proposition 4.13.
Proper topological coarse structures are induced by theirHigson compactifications.
The following is a direct consequence of Propositions 4.4, 4.7 and 4.13.
Corollary 4.14.
Let X and X ′ be proper topological coarse spaces. Then thefollowing properties hold: (i) A map ϕ : X → X ′ is coarse if and only if it has an extension ϕ ν : X ν → X ′ ν that is continuous on the points of νX and such that ϕ ν ( νX ) ⊂ νX ′ . (ii) Two coarse maps ϕ, ψ : X → X ′ are close if and only if the extensions ϕ ν and ψ ν , given by property (i) , are equal on νX . The following result shows that proper metric coarse structures are particularcases of the topological ones (Roe [20, Proposition 2.47]).
Proposition 4.15.
The metric coarse structure of a proper metric space isequal to the topological coarse structure induced by its Higson compactification.
Proposition 4.15 and Corollary 4.14 have the following consequences.
Theorem 4.16.
Let X and X ′ be proper metric spaces. Then a map ϕ : X → X ′ is a coarse equivalence if and only if it has an extension ϕ ν : X ν → X ′ ν suchthat ϕ ν ( νX ) ⊂ νX ′ , ϕ ν is continuous on the points of νX and the restriction ϕ ν : νX → νX ′ is a bijection.Proof. The “if” part follows from Corollary 4.14. To prove the “only if” part,assume that ϕ : X → X ′ is coarse and admits an extension ϕ ν : X ν → X ′ ν thatis continuous on the points of νX and takes νX bijectively onto νX ′ (hence ϕ ν induces a homeomorphism of νX onto νX ′ because νX is compact andHausdorff).The hypotheses imply that ϕ is uniformly metrically proper. Indeed, if thatwas not the case, there would be a positive number R > x n ) and ( z n ) in X such that d ′ ( ϕ ( x n ) , ϕ ( z n )) ≤ R but d ( x n , z n ) ≥ n for all n . Because ϕ : X → X ′ is coarse (metric proper and uniformly expansive),it may be assumed, after passing to subsequences if needed, that neither of22he sequences ( x n ) and ( z n ) has accumulation points in X , and that neither( ϕ ( x n )) nor ( ϕ ( x n )) have accumulation points in X ′ . Because d ( x n , z n ) ≥ n ,an application of Proposition 4.12 shows that the set of accumulation points ofthe sequence ( x n ), say P , and of the sequence ( z n ), say Q , are disjoint closedsubsets of νX . Being continuous on νX , the mapping ϕ ν takes P and Q to theset of accumulation points of the corresponding sequences ( ϕ ( x n )) and ( ϕ ( x n )),respectively. But, since d ′ ( ϕ ( x n ) , ϕ ( z n )) ≤ R , it follows that ϕ ν ( P ) = ϕ ν ( Q ).This contradicts that ϕ ν induces a homeomorphism of the compact Hausdorffspace νX onto νX ′ .It is also true that there is an N > ϕ ( X ) is N -densein X ′ . For if not there would be a sequence ( x ′ n ) in X ′ such that the union, W ,of the metric balls B ( x ′ n , n ) is disjoint from the image ϕ ( X ). By Corollary 4.12,the closure of W in X ′ ν is a neighborhood of a point p in νX ′ . This clearlycontradicts the hypothesis that the mapping ϕ admits an extension to X ′ thattakes νX onto νX ′ and is continuous at the points of νX ′ .Thus, by the above, there is some number N such that ϕ ( X ) is N -dense in X ′ , and so a mapping ψ : X ′ → X can be defined, by choosing, for each x ′ in X ′ a point ψ ( x ′ ) in X such that ϕ ( ψ ( x ′ )) is in B ( x ′ , N ).The map ψ is uniformly expansive (Definition 3.7 (i)). Let R > x ′ and z ′ in X ′ be such that d ′ ( x ′ , z ′ ) ≤ R . Then, by the definition of ψ ,the points ψ ( x ′ ) and ψ ( z ′ ) satisfy d ′ ( ϕ ( ψ ( x ′ )) , ϕ ( ψ ( z ′ ))) ≤ d ′ ( x ′ , z ′ ) + 2 N ≤ R + 2 N . Because ϕ is uniformly metrically proper, given R + 2 N >
0, there is S = S ( R + 2 N ) > d ′ ( ϕ ( x ) , ϕ ( z )) ≤ R + 2 N , then d ( x, z ) ≤ S .This applies in particular to x = ψ ( x ′ ) and z = ψ ( z ′ ).The map ψ is metrically proper (Definition 3.7 (ii)). Let B ⊂ X be boundedand suppose that ψ − ( B ) ⊂ X ′ is not bounded. Then there is a sequence x ′ , x ′ , · · · in ψ − ( B ) such that d ′ ( x ′ n +1 , x ′ ) ≥ n for all n . The points x n = ψ ( x ′ n )are all in B , and satisfy d ′ ( ϕ ( x n ) , x ′ n ) ≤ N for all n , by the construction of ψ .Therefore d ′ ( ϕ ( x ) , ϕ ( x n )) ≥ d ( x ′ , x ′ n ) − N ≥ n − N , so that the sequence ϕ ( x n ) is unbounded in X ′ . Since all the x n are in the bounded set B , thiscontradicts the uniform expansiveness of ϕ .The composite mapping ϕ ◦ ψ : X ′ → X ′ is close to the identity of X ′ becausefor any x ′ in X ′ , the point ψ ( x ′ ) is such that d ′ ( ϕ ( ψ ( x ′ )) , x ′ ) ≤ N .The composite mapping ψ ◦ ϕ : X → X is close to the identity on X .Indeed, by the definition of ψ , for any x in X , the point ψ ( ϕ ( x )) is such that d ′ ( ϕ ( ψ ( ϕ ( x ))) , ϕ ( x )) ≤ N . Because ϕ is uniformly metrically proper, there isan S = S ( N ) such that d ( ψ ( ϕ ( x )) , x ) ≤ S for all x in X .According to Corollary 3.17, in the case of coarsely quasi-convex metricspaces, the property “coarse equivalence” in this statement can be replaced bythe property “coarse quasi-isometry.” Theorem 4.17.
Let ( M, d ) and ( M ′ , d ′ ) be proper metric spaces, and supposethat ϕ : C ν ( M ′ ) → C ν ( M ) is an algebraic isomorphism of their associatedHigson algebras. Then M and M ′ are coarsely equivalent. Furthermore, if M and M ′ are coarsely quasi-convex, then ϕ induces a coarse quasi-isometrybetween M and M ′ . roof. The algebra C ν ( M ) has trivial radical because it is a Banach subalgebraof C b ( M ). Therefore, by Gelfand et al. [7, Theorem 2 of § ϕ : C ν ( M ′ ) → C ν ( M ) is automatically continuous and induces ahomeomorphism of Higson compactifications F : M ν → M ′ ν .That homeomorphism F must send M homeomorphically onto M ′ because M is first countable but no point in the Higson corona of M is a G δ -set (Propo-sition 4.10). Then the induced map F : M → M ′ is a coarse equivalence byTheorem 4.16. The last part of the statement now follows by invoking Propo-sition 3.15, which shows that a coarse mapping between coarsely quasi-convexspaces is a coarse quasi-isometry.We now prove a slightly strengthened version of Theorem 1.4 stated in theintroduction. Theorem 4.18.
Two proper coarse metric spaces, ( M, d ) and ( M ′ , d ′ ) , arecoarsely equivalent if and only if there is an algebraic isomorphism C ( νM ′ ) → C ( νM ) induced by a homomorphism B ν ( M ′ ) → B ν ( M ) . Furthermore, if M and M ′ are defined by coarsely quasi-convex metrics ( or coarse metrics ) d and d ′ ,then the above condition is equivalent to the existence of a coarse quasi-isometrybetween ( M, d ) and ( M ′ , d ′ ) .Proof. Let ϕ : B ν ( M ′ ) → B ν ( M ) be an algebraic homomorphism inducing analgebraic isomorphism C ( νM ′ ) → C ( νM ).Fix K > K -separated K -nets A ⊂ M and A ′ ⊂ M ′ . The inclusion mapping A → M induces a norm-decreasing algebraichomomorphism B ν ( M ) → C ν ( A ).There is a Borel partition of M ′ of the form { F x | x ∈ A ′ } with x ∈ F x ⊂ B ( x, K ) for each x ∈ A ′ . Such a partition can be constructed by induction on n for an enumeration ( x n ) of the points of A ′ : take F x = B ( x , K ), and F x n +1 = { x n +1 } ∪ ( B ( x n +1 , K ) \ ( F x ∪ · · · ∪ F x n ))if F x , . . . , F x n are constructed. Let χ x denote the characteristic function of F x for each x ∈ A ′ . Given a function f on A ′ , P f = P x ∈ A ′ f ( x ) χ x is a Higsonfunction on M ′ by the argument of Roe in [18, Proposition (5.10)]. This definesa homomorphism of algebras P : C ν ( A ′ ) → B ν ( M ′ ) because the sets F x form apartition. Moreover the composition of P with the restriction homomorphism B ν ( M ′ ) → C ν ( A ′ ) is the identity on C ν ( A ′ ) because x ∈ F x for all x ∈ A ′ .It follows from the above that there is a homomorphism of algebras C ν ( A ′ ) → C ν ( A ) that induces the original isomorphism C ( νA ′ ) = C ( νM ′ ) → C ( νA ) = C ( νM ). Since C ( νA ) = C ν ( A ) /C ( A ) and C ( νA ) = C ν ( A ) /C ( A ), this homo-morphism of algebras induces a continuous mapping ϕ ν : A ν → A ′ ν that sends νA into νA ′ homeomorphically, and such that the restriction ϕ = ϕ ν | A sends A into A ′ . It thus follows from Theorem 4.16 that ϕ induces a coarse equivalence M → M ′ .If the metrics d and d ′ are coarsely quasi-convex, then ϕ can be improved toa coarse quasi-isometry, because of Corollary 3.17.24 xample . This result implies that a coarse equivalence between two locallycompact metric spaces induces a homeomorphism between their respective coro-nas. The converse is not true in general. It follows from Example 3.14 that theHigson compactifications of the subspaces N and N of the natural numbersare the same as their Stone- ˇCech compactifications (which are homeomorphicto the Stone- ˇCech compactification of the natural numbers).If the Continuum Hypothesis is accepted, then the Stone- ˇCech corona ofthe natural numbers has 2 c automorphisms (Walker [22]). On the other hand,there are at most c maps of N into N . Therefore, many homeomorphisms ofthe Higson corona of N into that of N are not induced by a map of N into N . References [1] Jes´us A. ´Alvarez L´opez and Alberto Candel,
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