Property A and coarse embeddability for fuzzy metric spaces
aa r X i v : . [ m a t h . GN ] F e b PROPERTY A AND COARSE EMBEDDABILITY FORFUZZY METRIC SPACES
YEONG CHYUAN CHUNG
Abstract.
We define property A for fuzzy metric spaces in the sense ofGeorge and Veeramani, show that it is an invariant in the coarse categoryof fuzzy metric spaces, and provide characterizations of it for uniformlylocally finite fuzzy metric spaces. We also show that uniformly locallyfinite fuzzy metric spaces with property A are coarsely embeddable intoHilbert space. Introduction
A form of probabilistic geometry was introduced by Menger [12, 15, 14, 13]who axiomatized the idea that the distance between two points is proba-bilistic rather than deterministic by replacing the numerical-valued distancebetween two points by a cumulative distribution function, reflecting in away the uncertainty inherent in measurements. In fact, his definition of an“ensemble flou” in [13] is identical to the definition of a fuzzy set later givenby Zadeh [29], thereby anticipating the theory of fuzzy sets by more than adecade.Menger’s idea was picked up and developed by others, notably Schweizerand Sklar [23, 24]. Motivated by these earlier works, Kramosil and Michálekintroduced fuzzy metric spaces in [10]. George and Veeramani gave a mod-ified definition in [4] such that the induced topology is Hausdorff, and thisis the definition we will use in this paper. Besides theoretical interest in thesubject, such as topological and fixed point properties, fuzzy metrics havefound applications in engineering problems, notably in image processing fil-ters that produce better quality and sharpness than classical ones built usingmetrics [6, 11, 19].Coarse geometry can be briefly described as the study of geometric objectsviewed from afar. Coarse geometric ideas were already present in Mostow’srigidity theorem [16] and work on growth of groups [7]. It has also formedconnections with Banach space theory [17], topology and index theory [20,27], as well as operator algebras and noncommutative geometry [26, 28].Most of these involve the coarse geometry of metric spaces.Zarichnyi wrote a short note [30] on the coarse geometry of fuzzy metricspaces, and Grzegrzolka studied the asymptotic dimension of fuzzy metricspaces in [8]. In this paper, we take the coarse geometric study of fuzzy
Date : February 23, 2021. metric spaces a step further by studying property A and coarse embeddabil-ity for fuzzy metric spaces. Property A was introduced for metric spacesin [28] as a sufficient condition for coarse embeddability into Hilbert space,and it is also true that metric spaces with finite asymptotic dimension haveproperty A [9, Lemma 4.3]. We show that both remain true for fuzzy metricspaces. We also note that property A was defined for general coarse spacesin [22].We end this introduction with an outline of the rest of the paper. In sec-tion 2, we recall basic definitions and properties pertaining to fuzzy metricspaces in the sense of George and Veeramani. In section 3, we define prop-erty A for fuzzy metric spaces, and show that fuzzy metric spaces with finiteasymptotic dimension have property A. In section 4, we show that propertyA is an invariant in the coarse category of fuzzy metric spaces, and is in factinherited by fuzzy metric subspaces. In section 5, we provide characteriza-tions of property A for uniformly locally finite fuzzy metric spaces. Finally,in section 6, we show that uniformly locally finite fuzzy metric spaces withproperty A coarsely embed into Hilbert space.2.
Fuzzy metric spaces
In this section, we record basic definitions and properties pertaining tofuzzy metric spaces as defined in [4].
Definition 2.1.
A binary operation ∗ : [0 , × [0 , → [0 , is called a continuous t -norm if(i) ∗ is associative and commutative,(ii) ∗ is continuous,(iii) a ∗ a for all a ∈ [0 , ,(iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for a, b, c, d ∈ [0 , .We will write a ∗ ( m ) for a ∗ a ∗ · · · ∗ a ( m times). Definition 2.2.
A triple ( X, M, ∗ ) is a fuzzy metric space if X is a set, ∗ is a continuous t -norm, and M : X × X × (0 , ∞ ) → [0 , is a functionsatisfying the following conditions for all x, y, z ∈ X and all s, t > :(i) M ( x, y, t ) > ,(ii) M ( x, y, t ) = 1 if and only if x = y ,(iii) M ( x, y, t ) = M ( y, x, t ) ,(iv) M ( x, y, t ) ∗ M ( y, z, s ) ≤ M ( x, z, t + s ) ,(v) M ( x, y, · ) : (0 , ∞ ) → [0 , is continuous. It was shown in [5, Lemma 4] that M ( x, y, · ) is non-decreasing for x, y ∈ X . ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 3
Example 2.3.
Let ( X, d ) be a metric space, and let a ∗ b = ab for a, b ∈ [0 , .Define M ( x, y, t ) = tt + d ( x, y ) . Then ( X, M, ∗ ) is a fuzzy metric space, which we call the standard fuzzymetric space induced by the metric d . Example 2.4. [4, Example 2.11]
Let X = N , define a ∗ b = ab , and define M ( x, y, t ) = ( x/y if x ≤ yy/x if y ≤ x for all t > . Then ( X, M, ∗ ) is a fuzzy metric space. Moreover, M is notthe standard fuzzy metric corresponding to any metric on X . Definition 2.5.
Let ( X, M, ∗ ) be a fuzzy metric space. The ball B ( x, r, t ) with center x ∈ X , r ∈ (0 , , and t > is defined by B ( x, r, t ) = { y ∈ X : M ( x, y, t ) > − r } . If (
X, d ) is a metric space, then we may compare balls in (
X, d ) with ballsin the corresponding standard fuzzy metric space.
Lemma 2.6.
Let ( X, d ) be a metric space, and let ( X, M, ∗ ) be the corre-sponding standard fuzzy metric space. For R > , let B d ( x, R ) = { y ∈ X : d ( x, y ) < R } . Then B d ( x, R ) = B (cid:18) x, Rt + R , t (cid:19) = B (cid:18) x, r, R (1 − r ) r (cid:19) for each x ∈ X , t > , r ∈ (0 , , and R > . In particular, B d ( x, R ) = B ( x, , R ) for each x ∈ X and R > .Proof. The proof is a straightforward computation using the definition ofthe standard fuzzy metric space. (cid:3)
Definition 2.7.
Let ( X, M, ∗ ) be a fuzzy metric space. A subset A of X isbounded if there exist r ∈ (0 , and t > such that M ( x, y, t ) > − r for all x, y ∈ A . Throughout this paper, we will follow [8] in assuming that a ∗ b = 0whenever a = 0 and b = 0, i.e., there are no zero divisors with respect to ∗ .This ensures that the union of two bounded sets is bounded [8, Proposition2.8]. For example, the Łukasiewicz t -norm given by a ∗ b = max( a + b − , E of X × X for whichthere exist r ∈ (0 ,
1) and t > M ( x, y, t ) > − r for all ( x, y ) ∈ E forms a coarse structure on X in the sense of [21, Definition 2.3]. YEONG CHYUAN CHUNG Property A for fuzzy metric spaces
In this section, we define property A for fuzzy metric spaces, and showthat fuzzy metric spaces with finite asymptotic dimension have property A.
Definition 3.1.
A fuzzy metric space ( X, M, ∗ ) is said to have property Aif for every ε > , t > , and r ∈ (0 , , there exist t ′ > , r ′ ∈ (0 , , and afamily { A x } x ∈ X of non-empty finite subsets of X × N such that(i) A x ⊂ B ( x, r ′ , t ′ ) × N for each x ∈ X ,(ii) the symmetric difference A x ∆ A y satisfes | A x ∆ A y | < ε | A x ∩ A y | when M ( x, y, t ) > − r . This definition is inspired by the original definition of property A formetric spaces in [28], which we now recall.
Definition 3.2.
A metric space ( X, d ) is said to have property A if for every ε > and R > , there exist S > and a family { A x } x ∈ X of non-emptyfinite subsets of X × N such that(i) A x ⊂ B d ( x, S ) × N for each x ∈ X ,(ii) the symmetric difference A x ∆ A y satisfes | A x ∆ A y | < ε | A x ∩ A y | when d ( x, y ) < R . Proposition 3.3.
Let ( X, d ) be a metric space, and let ( X, M, ∗ ) be thecorresponding standard fuzzy metric space. Then ( X, d ) has property A ifand only if ( X, M, ∗ ) has property A.Proof. We will use Lemma 2.6 to compare balls in (
X, d ) with balls in(
X, M, ∗ ).Suppose ( X, d ) has property A. Given ε > t >
0, and r ∈ (0 , R = tr − r . There exist S > { A x } x ∈ X of non-empty finitesubsets of X × N such that A x ⊂ B d ( x, S ) × N ⊆ B ( x, St + S , t ) × N for each x ∈ X , and | A x ∆ A y | < ε | A x ∩ A y | whenever y ∈ B d ( x, R ) and thus whenever y ∈ B ( x, r, t ). Hence ( X, M, ∗ ) has property A.Conversely, suppose ( X, M, ∗ ) has property A. Given R > ε >
0, let t = R and r = . There exist t ′ > r ′ ∈ (0 , { A x } x ∈ X of non-empty finite subsets of X × N such that A x ⊂ B ( x, r ′ , t ′ ) × N ⊆ B d ( x, t ′ r ′ − r ′ ) × N for each x ∈ X , and | A x ∆ A y | < ε | A x ∩ A y | whenever M ( x, y, t ) > − r = and thus whenever d ( x, y ) < R . (cid:3) Our definition is also compatible with Sako’s definition of property A forgeneral coarse spaces in [22].We now proceed to show that fuzzy metric spaces with finite asymptoticdimension have property A, and we begin by recalling the definition of as-ymptotic dimension for fuzzy metric spaces from [8].
Definition 3.4.
A family U of subsets of a fuzzy metric space ( X, M, ∗ ) isuniformly bounded if there exist r ∈ (0 , and t > such that M ( x, y, t ) > − r for all x, y ∈ U and U ∈ U . ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 5
Definition 3.5.
Two subsets U and U ′ of a fuzzy metric space ( X, M, ∗ ) are ( r, t ) -disjoint for some r ∈ (0 , and t > if M ( U, U ′ , t ) < − r , where M ( U, U ′ , t ) = sup { M ( x, y, t ) : x ∈ U, y ∈ U ′ } .A family U of subsets of ( X, M, ∗ ) is said to be ( r, t ) -disjoint if any twodistinct members of U are ( r, t ) -disjoint. Definition 3.6.
Let ( X, M, ∗ ) be a fuzzy metric space. We say that theasymptotic dimension of X does not exceed n , and write asdim ( X ) ≤ n , iffor every r ∈ (0 , and t > , there exist ( r, t ) -disjoint families U , . . . , U n of subsets of X such that S i U i is a uniformly bounded cover of ( X, M, ∗ ) .If no such n exists, we say that asdim ( X ) = ∞ . If asdim ( X ) ≤ n but asdim ( X ) (cid:2) n − , then we say that asdim ( X ) = n . Definition 3.7.
A Lebesgue pair of a cover U of a fuzzy metric space ( X, M, ∗ ) is a pair of numbers r ∈ (0 , and t > such that for any x ∈ X , B ( x, r, t ) ⊆ U for some U ∈ U . Lemma 3.8. [8, Theorem 4.5]
Let ( X, M, ∗ ) be a fuzzy metric space. If asdim ( X ) ≤ n , then for every r ∈ (0 , and t > , there exists a uniformlybounded cover U of X with Lebesgue pair ( r, t ) and multiplicity at most n +1 . The main result of this section is analogous to [2, Theorem 2.1], and showsthat fuzzy metric spaces with finite asymptotic dimension have property A.
Theorem 3.9.
Let ( X, M, ∗ ) be a fuzzy metric space, and n ≥ . If asdim ( X ) ≤ n , then for each ε > , t > , and r ∈ (0 , , there exist t ′ > , r ′ ∈ (0 , , and non-empty finite subsets A x ⊂ B ( x, r ′ , t ′ ) × N for x ∈ X suchthat | A x ∆ A y | < ε | A x ∩ A y | when M ( x, y, t ) > − r , and the projection of A x onto X contains at most n + 1 elements for each x ∈ X . In particular,fuzzy metric spaces with finite asymptotic dimension have property A.Proof. Assume asdim ( X ) ≤ n , and fix ε > t >
0, and r ∈ (0 , U be a uniformly bounded cover of X with multiplicity at most n + 1 andwith Lebesgue pair ( R, T ), where R = 1 − (1 − r ) ∗ ( ⌊ n +1) /ε ⌋ +1) ∈ (0 , T = (2 + n +1 ε ) t . Here, ⌊ n +1 ε ⌋ denotes the integer part of 2 + n +1 ε .Note that 1 − R ≤ − r and t < T so B ( x, r, t ) ⊆ B ( x, R, T ) for each x ∈ X .Since U is uniformly bounded, there exist r ′ ∈ (0 ,
1) and t ′ > M ( x, y, t ′ ) > − r ′ for all x, y ∈ U and U ∈ U . Pick an element a U ∈ U foreach U ∈ U . We call a finite sequence x , . . . , x n of points in X an ( r, t )-chain from x to x n if M ( x i , x i − , t ) > − r for i = 1 , . . . , n . For x ∈ X and U ∈ U , let l U ( x ) denote the length of the shortest ( r, t )-chain from x toa point outside U . If there is no such chain, set l U ( x ) equal to ⌊ n +1 ε ⌋ .Now let A x = [ U ∋ x { a U } × { , . . . , l U ( x ) } . This is a finite set for each x ∈ X since the cover U has finite multiplicity.If x ∈ U , then M ( x, a U , t ′ ) > − r ′ so A x ⊂ B ( x, r ′ , t ′ ) × N for each x ∈ X . YEONG CHYUAN CHUNG
Since U has multiplicity at most n + 1, the projection of A x onto X containsat most n + 1 elements for each x ∈ X .Assume M ( x, y, t ) > − r for the rest of the proof. If y, y , . . . , y k forms an( r, t )-chain, then so does x, y, y , . . . , y k . If there is no ( r, t )-chain from x to X \ U , then there is also no ( r, t )-chain from y to X \ U . Thus | l U ( x ) − l U ( y ) | ≤
1. Since U has Lebesgue pair ( R, T ) and multiplicity at most n + 1, there areat most 2 n + 1 elements of U containing either x or y . Moreover, if x ∈ U and y / ∈ U , then l U ( x ) = 1 since x, y form an ( r, t )-chain. Thus | A x ∆ A y | ≤ n + 1 . By definition of a Lebesgue pair, there exists U ∈ U such that B ( x, R, T ) ⊂ U . If x, x , . . . , x m forms an ( r, t )-chain, and x m ∈ X \ U , then M ( x, x m , T ) ≤ − R . Now if m < ⌊ n +1 ε ⌋ , then1 − R ≥ M ( x, x m , T ) = M ( x, x m , (2 + 2 n + 1 ε ) t ) ≥ M ( x, x m , mt ) ≥ M ( x, x , t ) ∗ M ( x , x , t ) ∗ · · · ∗ M ( x m − , x m , t ) ≥ (1 − r ) ∗ ( m ) . Since 1 − R = (1 − r ) ∗ ( ⌊ n +1) /ε ⌋ +1) , we get ⌊ n +1 ε ⌋ +1 ≤ m < ⌊ n +1 ε ⌋ ,which is a contradiction. Hence m ≥ ⌊ n +1 ε ⌋ .Thus { a U } × { , . . . , ⌊ n +1 ε ⌋ − } ⊆ A x ∩ A y , and | A x ∩ A y | ≥ ⌊ n + 1 ε ⌋ − > n + 1 ε . Consequently, | A x ∆ A y || A x ∩ A y | < n + 1(2 n + 1) /ε = ε. (cid:3) The fuzzy metric space in the next example is not the standard fuzzymetric space corresponding to any metric, and it was shown to have asymp-totic dimension zero in [8, Example 6.3]. It has property A by the theoremabove but we can also show this directly using the definition.
Example 3.10.
Let X = N , define a ∗ b = ab , and define M ( x, y, t ) = ( if x = y /xy if x = y for all t > . Given ε > , r ∈ (0 , , and t > , choose N ∈ N such that N +1 < − r . For n ∈ N , set A n = ( { N } × { } if n ≤ N { n } × { } if n > N For n < N , we have M ( n, N, t ) = nN > N , and it follows that A n ⊂ B ( n, − N , t ) × N for all n ∈ N . If M ( n, m, t ) > − r and m = n , ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 7 then mn > − r > N +1 so A n = A m = { N } × { } , and it follows that | A n ∆ A m | < ε | A n ∩ A m | whenever M ( n, m, t ) > − r . Hence ( X, M, ∗ ) hasproperty A. Dranishnikov [3, Section 4] gave examples of metric spaces with bothinfinite asymptotic dimension and property A so the standard fuzzy metricspaces corresponding to these metric spaces also have both properties.4.
Coarse invariance
In this section, we show that property A is an invariant in the coarsecategory of fuzzy metric spaces introduced in [8, Section 5]. In fact, it is aproperty that is inherited by fuzzy metric subspaces.
Definition 4.1.
Let ( X, M , ∗ ) and ( Y, M , ∗ ) be fuzzy metric spaces. Let f : X → Y be a map.(i) f is said to be uniformly expansive if for all A > and t > ,there exist B ∈ (0 , and t ′ > such that M ( f ( x ) , f ( x ′ ) , t ′ ) ≥ B whenever M ( x, x ′ , t ) ≥ A for x, x ′ ∈ X .(ii) f is said to be effectively proper if for all C > and t > , thereexist D ∈ (0 , and t ′ > such that M ( x, x ′ , t ′ ) ≥ D whenever M ( f ( x ) , f ( x ′ ) , t ) ≥ C for x, x ′ ∈ X .(iii) f is a coarse embedding if it is both uniformly expansive and effec-tively proper.(iv) f is a coarse equivalence if it is a coarse embedding and it is coarselyonto in the sense that there exist r ∈ (0 , and t > such that foreach y ∈ Y there exists x ∈ X satisfying M ( f ( x ) , y, t ) > − r . Definition 4.2.
Let X be a set, and let ( Y, M, ∗ ) be a fuzzy metric space.Let f : X → Y and g : X → Y be functions. Then f is close to g , denoted f ∼ g , if there exist r ∈ (0 , and t > such that M ( f ( x ) , g ( x ) , t ) > − r for all x ∈ X . Proposition 4.3. [8, Proposition 5.4]
Let ( X, M , ∗ ) and ( Y, M , ∗ ) befuzzy metric spaces. Let f : X → Y be a function. Then f is a coarse equiv-alence if and only if f is uniformly expansive and there exists a uniformlyexpansive g : Y → X such that the compositions f ◦ g and g ◦ f are close tothe identity maps of Y and X respectively.The function g is called a coarse inverse of f . Theorem 4.4.
Let ( X, M , ∗ ) and ( Y, M , ∗ ) be fuzzy metric spaces. Let f : X → Y be a coarse equivalence. Then ( X, M , ∗ ) has property A if andonly if ( Y, M , ∗ ) does.Proof. Assume that f : X → Y is a coarse equivalence, and g : Y → X is acoarse inverse. Assume that ( X, M , ∗ ) has property A, and let ε > t > r ∈ (0 ,
1) be given. Since g is uniformly expansive, there exist B ∈ (0 , t > M ( g ( y ) , g ( y ′ ) , t ) ≥ B whenever M ( y, y ′ , t ) ≥ − r .From the definition of property A, there exist t > r >
0, and a family
YEONG CHYUAN CHUNG { A x } x ∈ X of non-empty finite subsets of X × N such that A x ⊂ B ( x, r , t ) × N for each x ∈ X , and | A x ∆ A x ′ | < ε | A x ∩ A x ′ | whenever M ( x, x ′ , t ) > B . Fix y ∈ Y . For each y ∈ Y , let n y = | ( f − ( y ) × N ) ∩ A g ( y ) | , and define B y = [ y ∈ Y { ( y, , ( y, , . . . , ( y, n y ) } . The sets ( f − ( y ) × N ) ∩ A g ( y ) partition A g ( y ) so | B y | = | A g ( y ) | < ∞ . Thisproduces a family { B y } y ∈ Y of non-empty finite subsets of Y × N .If ( y ′ , n ) ∈ B y , then there exists ( x, m ) ∈ ( f − ( y ′ ) × N ) ∩ A g ( y ) . Now M ( g ( y ) , x, t ) > − r so M ( f ( g ( y )) , y ′ , t ) ≥ C for some t > C ∈ (0 , f ◦ g is close to the identity map on Y , there exist r ∈ (0 , t > M ( y, f ( g ( y )) , t ) > − r so M ( y, y ′ , t + t ) ≥ M ( y, f ( g ( y )) , t ) ∗ M ( f ( g ( y )) , y ′ , t ) ≥ C ∗ (1 − r ) > − r for some r ∈ (0 , x, m ) ∈ A g ( y ) for some m , then ( f ( x ) , ∈ B y , and it follows that | A g ( y ) ∩ A g ( y ′ ) | ≤ | B y ∩ B y ′ | and | A g ( y ) ∆ A g ( y ′ ) | ≥ | B y ∆ B y ′ | for y, y ′ ∈ Y .Hence | B y ∆ B y ′ | < ε | B y ∩ B y ′ | whenever M ( y, y ′ , t ) > − r . Therefore( Y, M , ∗ ) has property A. (cid:3) Remark . The same proof as above shows that if g : Y → X is a coarseembedding, and ( X, M , ∗ ) has property A, then so does ( Y, M , ∗ ). Thisis because g : ( Y, M , ∗ ) → ( g ( Y ) , M | g ( Y ) × g ( Y ) × (0 , ∞ ) , ∗ ) is a coarse equiv-alence and we may consider a coarse inverse f : g ( Y ) → Y . Therefore,property A is inherited by fuzzy metric subspaces.5. Characterizations of property A
In this section, we provide characterizations of property A for uniformlylocally finite fuzzy metric spaces in terms of positive definite kernels, mapsinto Hilbert spaces, and operators on Hilbert space. We begin with defini-tions of terms that will appear in the result.
Definition 5.1.
A fuzzy metric space ( X, M, ∗ ) is said to be uniformly lo-cally finite if for every r ∈ (0 , and t > , there exists N r,t such that | B ( x, r, t ) | < N r,t for all x ∈ X . If a fuzzy metric space (
X, M, ∗ ) is uniformly locally finite, then X isnecessarily countable. Indeed, given x ∈ X , if y = x and n ∈ N , then M ( x, y, n ) = 1 − r for some r ∈ (0 , m ∈ N such that m > r ≥ m +1 , we get M ( x, y, n ) > − m , i.e., y ∈ B ( x, m , n ). Therefore, for a fixed x ∈ X , we have X = S n,m ∈ N B ( x, m , n ) so X is countable since each ballis finite. Also, for each x ∈ X , t >
0, and m ∈ N , there exists R ∈ (0 , m )such that B ( x, R, t ) = { x } . Indeed, let B ( x, m , t ) = { x, x , . . . , x n } , and let R = 1 − max i M ( x, x i , t ) ∈ (0 , m ). Then M ( x, x i , t ) ≤ − R for i = 1 , . . . , n .If y ∈ X \ B ( x, m , t ), then M ( x, y, t ) ≤ − m < − R . Hence B ( x, R, t ) = { x } . ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 9
Definition 5.2.
Let X be a set. A map k : X × X → C (resp. R ) iscalled a positive definite kernel if for all finite sequences x , . . . , x n in X and λ , . . . , λ n ∈ C (resp. R ), we have P ni =1 λ i λ j k ( x i , x j ) ≥ . Definition 5.3.
Let ( X, M, ∗ ) be a uniformly locally finite fuzzy metricspace, and write B ( ℓ ( X )) for the set of all bounded linear operators on ℓ ( X ) . An operator S ∈ B ( ℓ ( X )) has finite propagation if there exist t > and r ∈ (0 , such that h Sδ y , δ x i = 0 whenever M ( x, y, t ) < − r .The uniform Roe algebra C ∗ u ( X ) is the operator norm closure in B ( ℓ ( X )) of the set of all bounded linear operators on ℓ ( X ) with finite propagation. Note that the composition S ◦ S has finite propagation whenever S and S do. Indeed, if h S δ y , δ x i = 0 whenever M ( x, y, t ) < − r , and h S δ y , δ x i = 0 whenever M ( x, y, t ) < − r , then h S ◦ S δ y , δ x i = 0 when-ever M ( x, y, t + t ) < (1 − r ) ∗ (1 − r ).The following follows from [22, Theorem 3.1] on general uniformly locallyfinite coarse spaces but we shall rewrite the proof in terms of fuzzy met-ric spaces. The reader may refer to [25, Theorem 1.1] for the analogousstatements for uniformly locally finite metric spaces. Theorem 5.4.
The following are equivalent for a uniformly locally finitefuzzy metric space ( X, M, ∗ ) :(i) X has property A.(ii) For every ε > , r ∈ (0 , , and t > , there exists a map η : X → ℓ ( X ) such that • || η x || = 1 for all x ∈ X , • || η x − η y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that the support of η x satisfies supp( η x ) ⊆ B ( x, R, T ) for all x ∈ X .(iii) For every ε > , r ∈ (0 , , and t > , there exists a map η : X → ℓ ( X ) such that • || η x || = 1 for all x ∈ X , • || η x − η y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that the support of η x satisfies supp( η x ) ⊆ B ( x, R, T ) for all x ∈ X .(iv) For every ε > , r ∈ (0 , , and t > , there exist a Hilbert space H and a map η : X → H such that • || η x || = 1 for all x ∈ X , • || η x − η y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that h η x , η y i = 0 whenever M ( x, y, T ) < − R .(v) For every ε > , r ∈ (0 , , and t > , there exist R ∈ (0 , , T > ,and a positive definite kernel k : X × X → R such that • | − k ( x, y ) | < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that k ( x, y ) = 0 whenever M ( x, y, T ) < − R . (vi) For every ε > , r ∈ (0 , , and t > , there exist R ∈ (0 , , T > ,and a positive definite kernel k : X × X → C such that • | − k ( x, y ) | < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that k ( x, y ) = 0 whenever M ( x, y, T ) < − R , • convolution with k defines a bounded linear operator S k belong-ing to the uniform Roe algebra C ∗ u ( X ) .Proof. (i) ⇒ (ii): Suppose X has property A. Fix ε > r ∈ (0 , t >
0. There exist r ′ ∈ (0 , t ′ >
0, and a family { A x } x ∈ X of non-emptyfinite subsets of X × N such that A x ⊂ B ( x, r ′ , t ′ ) × N for each x ∈ X , and | A x ∆ A y | < ε | A x ∩ A y | whenever M ( x, y, t ) > − r .Define A x ( y ) ⊂ N by A x ( y ) = { n ∈ N : ( y, n ) ∈ A x } . For x, y ∈ X , define ζ x ( y ) = | A x ( y ) | and η x = ζ x / || ζ x || . Then η x is a unitvector in ℓ ( X ) for each x ∈ X .If M ( x, y, t ) > − r , then || η x − η y || = || || ζ y || ζ x − || ζ x || ζ y || || ζ x || || ζ y || ≤ | || ζ y || − || ζ x || | · || ζ x || + || ζ x || · || ζ x − ζ y || || ζ x || || ζ y || ≤ || ζ x − ζ y || || ζ y || ≤ | A x ∆ A y || A y | < ε. Moreover, η x ( y ) = 0 if and only if ζ x ( y ) = 0, and in this case ( y, n ) ∈ A x forsome n ∈ N so y ∈ B ( x, r ′ , t ′ ).(ii) ⇒ (iii): Assuming (ii), fix ε > r ∈ (0 , t >
0. Then there isa map ξ : X → ℓ ( X ) such that • || ξ x || = 1 for all x ∈ X , • || ξ x − ξ y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 ,
1) and
T > ξ x satisfiessupp( ξ x ) ⊆ B ( x, R, T ) for all x ∈ X .Replacing ξ x by the map y
7→ | ξ x ( y ) | , we may assume ξ x takes nonnegativevalues while retaining the properties above. For each x ∈ X , define a unitvector η x in ℓ ( X ) by η x ( y ) = p ξ x ( y ). If M ( x, y, t ) > − r , then || η x − η y || ≤ q || ξ x − ξ y || < ε. Moreover, supp( η x ) = supp( ξ x ) ⊆ B ( x, R, T ) for all x ∈ X .(iii) ⇒ (iv): Assuming (iii), if h η x , η y i 6 = 0, then there exists z ∈ supp( η x ) ∩ supp( η y ) so M ( x, z, T ) > − R and M ( z, y, T ) = M ( y, z, T ) > − R . Hence ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 11 M ( x, y, T ) ≥ M ( x, z, T ) ∗ M ( z, y, T ) ≥ (1 − R ) ∗ (1 − R ) = 1 − R ′ for some R ′ ∈ (0 , ⇒ (v): Assuming (iv), we get a positive definite kernel k : X × X → R by k ( x, y ) = Re ( h η x , η y i ). Now || η x − η y || = || η x || + || η y || − Re ( h η x , η y i ) = 2 − Re ( h η x , η y i ) so if M ( x, y, t ) > − r , then | − k ( x, y ) | = | − Re ( h η x , η y i ) | = || η x − η y || / < ε /
2. Also, k ( x, y ) = 0 whenever M ( x, y, T ) < − R .(v) ⇒ (vi): Taking k, R, T from (v), since X is uniformly locally finite,there exists N R,T such that | B ( x, R, T ) | < N R,T for all x ∈ X so for each x ∈ X , we have |{ y ∈ X : k ( x, y ) = 0 }| ≤ N R,T . Also, | k ( x, y ) | ≤ p | k ( x, x ) k ( y, y ) | = 1 by Cauchy-Schwarz. Convolution with k defines alinear map S k : ℓ ( X ) → ℓ ( X ) given by( S k ξ ) x = X y ∈ X k ( x, y ) ξ y . Now || S k ξ || = X x ∈ X | X y ∈ B ( x,R,T ) k ( x, y ) ξ y | ≤ X x ∈ X ( X y ∈ B ( x,R,T ) | ξ y | ) ≤ X y ∈ X N R,T X x ∈ B ( y,R,T ) | ξ y | ≤ X y ∈ X N R,T | ξ y | = N R,T || ξ || . Hence S k is a bounded operator on ℓ ( X ). Moreover, h S k δ y , δ x i = k ( x, y ) so S k has finite propagation.(vi) ⇒ (iii): Assuming (vi), note that S k is a positive operator by seeingthat h S k f, f i ≥ f in the dense subset of all finitely supportedfunctions in ℓ ( X ). Thus it has a unique positve square root S l , and thereis an operator S m with finite propagation such that || S l − S m || < min( ε, ε || S l || + ε ) ) . By passing to ( S m + S ∗ m ) /
2, we may assume that S m is self-adjoint. Theoperator S m corresponds to a kernel m , and there exist R ′ ∈ (0 ,
1) and T ′ > m ( x, y ) = 0 whenever M ( x, y, T ′ ) < − R ′ . Also, || S k − S m || = || S l − S m || ≤ || S l |||| S l − S m || + || S l − S m |||| S m || < ε. For each x ∈ X , define θ x ∈ ℓ ( X ) by θ x ( y ) = m ( x, y ). Note that h θ x , θ y i = X z ∈ X m ( x, z ) m ( y, z ) = X z ∈ X m ( x, z ) m ( z, y )so |h θ x , θ y i − k ( x, y ) | < ε . If M ( x, y, t ) > − r , then | − h θ x , θ y i| ≤ |h θ x , θ y i − k ( x, y ) | + | − k ( x, y ) | < ε so || θ x − θ y || = h θ x , θ x i + h θ y , θ y i − Re ( h θ x , θ y i ) < ε . Also, || θ x || > − ε so setting η x = θ x / || θ x || , we have || η x − η y || ≤ | || θ y || − || θ x || | · || θ x || + || θ x || · || θ x − θ y || || θ x || || θ y || ≤ || θ x − θ y || || θ y || < s ε − ε . Moreover, since m ( x, y ) = 0 whenever M ( x, y, T ′ ) < − R ′ , we havesupp( η x ) ⊆ B ( x, R ′ , T ′ ).(iii) ⇒ (i): Assuming (iii), fix ε > t >
0, and r ∈ (0 , { η x } x ∈ X in ℓ ( X ) such that || η x − η y || < ε whenever M ( x, y, t ) > − r , and there exist R ∈ (0 ,
1) and
T > η x ) ⊆ B ( x, R, T )for all x ∈ X . Since X is uniformly locally finite, there exists N R,T suchthat | supp( η x ) | < N R,T for all x ∈ X .Define unit vectors ξ x ∈ ℓ ( X ) by ξ x ( y ) = | η x ( y ) | . If M ( x, y, t ) > − r ,then || ξ x − ξ y || ≤ || η x + η y || || η x − η y || < ε . Choose N ∈ N such that N > N
R,T /ε , and define ζ x ( y ) = j/N , where j ∈ { , . . . , N } and j −
Hence | A x ∆ A x || A x ∩ A x | < N (1 + ε )(1 − ε ) − N (1 − ε − ε ) N (1 − ε − ε )= 8 ε − ε − ε , which tends to zero as ε tends to 0, showing that X has property A. (cid:3) Remark . In the proof above, the assumption of uniform local finitenesswas used only in the implications (v) ⇒ (vi) and (iii) ⇒ (i).Using (ii) in the previous theorem, we will refine Theorem 3.9 to show thatif the multiplicities of certain uniformly bounded covers of a fuzzy metricspace grow at a subexponential rate, then the space has property A.For the next result, we will write mult( U ) for the multiplicity of a cover U , and we will write L ( U ) ≥ ( r, t ) to mean that U has a Lebesgue pairwith values at least r and t . Using Lemma 2.6 and [8, Proposition 3.4], onerecovers [18, Theorem 1] when our result is applied to the standard fuzzymetric space corresponding to a metric space. Theorem 5.6.
Let ( X, M, ∗ ) be a uniformly locally finite fuzzy metric space.Suppose there exists r ∈ (0 , such that the function ad X ( t ) = min { mult( U ) : U uniformly bounded cover of X, L ( U ) ≥ ( r ′ , t ) }− has subexponential growth, where r ′ = 1 − (1 − r ) ∗ (2) . Then ( X, M, ∗ ) hasproperty A.Proof. Given ε > r ∈ (0 , t >
0, we will construct a map η : X → ℓ ( X ) such that • || η x || = 1 for all x ∈ X , • || η x − η y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 ,
1) and
T > η x ) ⊆ B ( x, R, T )for all x ∈ X .Let U = { U i } i ∈ I be a uniformly bounded cover of X with Lebesgue pairat least ( r ′ , n ) and multiplicity ad X (6 n ) + 1. Uniform boundedness gives R ∈ (0 ,
1) and
T > M ( x, y, T ) > − R for all x, y ∈ U i and i ∈ I .For each i ∈ I , choose x i ∈ U i and let J : ℓ ( I ) → ℓ ( X ) be the contractionsending δ i to δ x i . For each x ∈ X and k ∈ N , set S x ( r, k ) = { i ∈ I : B ( x, r, k ) ⊂ U i } . For x, y ∈ X and t ∈ (0 , k ), if M ( x, y, t ) > − r and M ( y, z, k ) > − r ,then M ( x, z, k + t ) ≥ (1 − r ) ∗ (1 − r ) > − r ′ . Also, if M ( x, y, t ) > − r and M ( x, z, k − t ) > − r , then M ( y, z, k ) ≥ M ( x, y, t ) ∗ M ( x, z, k − t ) ≥ (1 − r ) ∗ (1 − r ) > − r ′ so we have S x ( r ′ , k + t ) ⊆ S x ( r, k ) ∩ S y ( r, k ) ⊆ S x ( r, k ) ∪ S y ( r, k ) ⊆ S x ( r ′ , k − t ) For each non-empty finite set S , set ξ S = | S | − χ S , where χ S denotes thecharacteristic function of S . Then for any non-empty finite sets S and T , || ξ S − ξ T || = 2 (cid:18) − | S ∩ T | max( | S | , | T | ) (cid:19) . Define ζ nx = 1 n n X k = n +1 ξ S x ( r,k ) ∈ ℓ ( I ) and η nx = J ( ζ nx ) ∈ ℓ ( X ) . Note that || η nx || = 1 for all x ∈ X . If x i ∈ supp( η nx ), then i ∈ S x ( r, k ) forsome k . Thus supp( η nx ) ⊆ B ( x, R, T ) for all x ∈ X .Now assume M ( x, y, t ) > − r with t ∈ { , . . . , k − } , and n ≥ t . Then || ζ nx − ζ ny || ≤ n n X k = n +1 || ξ S x ( r,k ) − ξ S y ( r,k ) || ≤ n n X k = n +1 (cid:18) − | S x ( r ′ , k + t ) || S x ( r ′ , k − t ) | (cid:19) . On the other hand,1 n n X k = n +1 | S x ( r ′ , k + t ) || S x ( r ′ , k − t ) | ≥ n Y k = n +1 | S x ( r ′ , k + t ) || S x ( r ′ , k − t ) | /n = Q n + tk =2 n − t +1 | S x ( r ′ , k ) | Q n + tk = n − t +1 | S x ( r ′ , k ) | ! /n ≥ mult( U ) − t/n since 1 ≤ | S x ( r ′ , k ) | ≤ mult( U ) for each k ∈ { n − t + 1 , . . . , n + t } . Hence || η nx − η ny || ≤ || ζ nx − ζ ny || ≤ − mult( U ) − t/n ) . Since mult( U ) = ad X (6 n ) + 1 grows subexponentially in n , one sees that || η nx − η ny || < ε for all sufficiently large n . (cid:3) Coarse embeddability into Hilbert space
The following lemma shows that coarse embeddability of a fuzzy metricspace into the standard fuzzy metric space corresponding to a metric can beverified using the metric instead of the standard fuzzy metric. Using it, wewill show that uniformly locally finite fuzzy metric spaces with property Aare coarsely embeddable into Hilbert space, which is in line with the originalmotivation for the introduction of property A for metric spaces as shown in[28, Theorem 2.2].We shall omit the proof of the lemma as it is straightforward using thedefinitions and Lemma 2.6.
Lemma 6.1.
Let ( X, M , ∗ ) be a fuzzy metric space, let ( Y, d ) be a metricspace, and let ( Y, M , ∗ ) be the corresponding standard fuzzy metric space. ROPERTY A AND COARSE EMBEDDABILITY FOR FUZZY METRIC SPACES 15 (i) A map f : ( X, M , ∗ ) → ( Y, M , ∗ ) is uniformly expansive if andonly if for all A > and t > , there exists S > such that d ( f ( x ) , f ( x ′ )) ≤ S whenever M ( x, x ′ , t ) ≥ A .(ii) A map f : ( X, M , ∗ ) → ( Y, M , ∗ ) is effectively proper if andonly if for all R > there exists D ∈ (0 , and t > such that M ( x, x ′ , t ) ≥ D whenever d ( f ( x ) , f ( x ′ )) ≤ R .Remark . One can reverse the roles of X and Y above to get analogousstatements when X is a metric space instead. Consequently, one sees thatthere is a coarse embedding between two metric spaces if and only if there is acoarse embedding between the corresponding standard fuzzy metric spaces,and similarly for coarse equivalences. Lemma 6.3.
Let ( X, M, ∗ ) be a uniformly locally finite fuzzy metric spacewith property A. For every ε > , r ∈ (0 , , and t > , there exists a map ξ : X → ℓ ( X × N ) such that • || ξ x || = 1 for all x ∈ X , • || ξ x − ξ y || < ε if M ( x, y, t ) > − r , • there exist R ∈ (0 , and T > such that the support of ξ x satisfies supp( ξ x ) ⊆ B ( x, R, T ) × N for all x ∈ X .Proof. Given ε > r ∈ (0 , t >
0, there exist R ∈ (0 , T >
0, and a family { A x } x ∈ X of non-empty finite subsets of X × N such that A x ⊂ B ( x, R, T ) × N for each x ∈ X , and | A x ∆ A y | < ε | A x ∩ A y | whenever M ( x, y, t ) > − r .Write χ A x for the characteristic function of A x , and set ξ x = | A x | − / χ A x for each x ∈ X . If M ( x, y, t ) > − r , then | A x | + | A y | = 2 | A x ∩ A y | + | A x ∆ A y | < (2 + ε ) | A x ∩ A y | so h ξ x , ξ y i = | A x ∩ A y | q | A x || A y | ≥ | A x ∩ A y || A x | + | A y | >
22 + ε , and || ξ x − ξ y || < ε ε < ε . (cid:3) Theorem 6.4.
A uniformly locally finite fuzzy metric space with propertyA coarsely embeds into Hilbert space.Proof.
For each n ∈ N , choose ξ n as in Lemma 6.3 corresponding to ε = 2 − n , t = n , and r n ∈ (0 ,
1) with r n increasing to 1. Let R n ∈ (0 ,
1) and T n > ξ nx ) ⊆ B ( x, R n , T n ) × N . Then || ξ nx − ξ ny || = √ M ( x, y, T n ) < (1 − R n ) ∗ (1 − R n ).Fix z ∈ X , and define a map F : X → L ∞ n =1 ℓ ( X × N ) by F ( x ) = ∞ M n =1 ( ξ nx − ξ nz ) . Suppose x = y , and let M = sup n M ( x, y, n ). There exists N such that r n > − M for all n > N . There exists N ′ such that M ( x, y, n ) > M for all n > N ′ . Hence 1 − r n < M < M ( x, y, n ) for all n > N ′′ = max( N, N ′ ) so || ξ nx − ξ ny || < − n for all n > N ′′ . It follows that || F ( x ) − F ( y ) || = ∞ X n =1 || ξ nx − ξ ny || ≤ N ′′ X n =1 || ξ nx − ξ ny || + ∞ X n = N ′′ +1 − n < N ′′ + 1 . In the case where y = z , the calculation above shows that F ( x ) indeedbelongs to L ∞ n =1 ℓ ( X × N ).Given R >
0, if || F ( x ) − F ( y ) || ≤ R , then the set { n ∈ N : M ( x, y, T n ) < (1 − R n ) ∗ (1 − R n ) } has at most R / N = max { n ∈ N : M ( x, y, T n ) < (1 − R n ) ∗ (1 − R n ) } , we have M ( x, y, T N +1 ) ≥ (1 − R N +1 ) ∗ (1 − R N +1 ) so F is effectively proper.On the other hand, let A > t >
0. If A ≥ M ( x, y, t ) ≥ A ,then x = y so assume A ∈ (0 , n ≥ t such that r n ≥ − A . If M ( x, y, t ) ≥ A , then M ( x, y, n ) ≥ M ( x, y, t ) ≥ A ≥ − r n so || F ( x ) − F ( y ) || ≤ − n , and F is uniformly expansive. (cid:3) Examples of uniformly locally finite metric spaces without property A butare coarsely embeddable into Hilbert space were constructed in [1] so thestandard fuzzy metric spaces corresponding to these metric spaces also donot have property A but are coarsely embeddable into Hilbert space.
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