A Banach-Stone Theorem for Uniformly Continuous Functions

Monatsh. Math. 131, 189±192 (2000)

A Banach±Stone Theorem for Uniformly Continuous Functions By

M. Isabel Garrido1 and JesuÂs A. Jaramillo2 1

Universidad de Extremadura, Badajoz, Spain Universidad Complutense de Madrid, Spain

2

(Received 3 March 2000; in revised form 29 June 2000) Abstract. In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U…X† of all uniformly continuous real functions on X. 2000 Mathematics Subject Classi®cation: 54C30, 54E35, 54D35 Key words: Uniformly continuous real functions, lattice homomorphisms, Banach±Stone theorems

There exists a variety of results in the literature linking the topological structure of a topological space X with algebraic or topological-algebraic structures of C…X†, the set of all continuous real functions on X. We ®rst recall the classical BanachStone theorem asserting that, for a compact space X, the linear metric structure of C…X† (endowed with the sup-norm) determines the topology of X. Further results along this line were obtained by Hewitt [6] and Shirota [9]. They proved respectively that, for a realcompact space X, the topology of X is determined by the ring structure of C…X† and by the lattice structure of C…X†. Moreover, Shirota proved in [9] that the lattices U…X† and U  …X† determine the topology of a complete metric space X, where U…X† denotes the family of all uniformly continuous real functions on X, and U  …X† denotes the subfamily of all bounded functions in U…X†. In the last few years there has been interest in the connection between the uniformity of a metric space X and some further structures over U…X† and U  …X†. Thus, in [1] Araujo and Font, using some results by Lacruz and Llavona [8], proved that the metric linear structure of U  …X† endowed with the sup-norm determines the uniformity of X, in the case that X is the unit ball of a Banach space. This result has been extended to any complete metric space X by HernaÂndez [5]. In this note, considering U…X† and U  …X† as unital vector lattices, we provide a direct and simple proof of the fact that the uniformity of a complete metric space X is indeed characterized not only by U  …X† but also by U…X†. In this way, we extend the above mentioned results.

Partially supported by DGES grants PB96/1262 and PB96/0607

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1. The Structure Space Let X be a topological space, and denote by C…X† the set of all continuous real functions on X. Let L be a unital vector sublattice of C…X† that separates points and closed subsets of X, that is, if F is a closed subset of X and x 2 X n F, there exists some f 2 L so that f …x† 2 = f …F†. We say that  : L ! R is a lattice homomorphism whenever it satis®es: …i† …f ‡ g† ˆ … f † ‡ …g†, for all f , g 2 L and ,  2 R. …ii† …j f j† ˆ j… f †j, for all f 2 L. …iii† …1† ˆ 1. Note that in this case  is monotone, i.e., … f † 5 0 when f 5 0. We de®ne the structure space of L as the set H…L† of all lattice homomorphisms on L, considered as a topological subspace of the product RL . This construction is standard (see, e.g. Isbell [7]), but we give some details for completeness. It is easy to check that H…L† is closed in RL , and therefore H…L† is a realcompact space. In addition, since L separates points and closed sets, X can be embedded as a topological subspace of H…L† identifying each x 2 X with the point evaluation homomorphism x given by x … f † ˆ f …x†, for every f 2 L. We are going to see that X is dense in H…L†. Indeed, given  2 H…L†, f1 ; . . . ; fn 2 L, and " > 0, there exists some x P 2 X such that jx … fi † ÿ … fi †j < ", for i ˆ 1; . . . ; n. Otherwise the function g ˆ niˆ1 j fi ÿ … fi †j 2 L would satisfy g 5 " and …g† ˆ 0, and this is impossible since  is monotone. It follows that H…L† is a realcompacti®cation of X that will be, in fact, a compacti®cation in the case that every function in L is bounded. Moreover, every f 2 L can be continuously extended to H…L† by means of prf , the corresponding projection map de®ned on RL . 2. The Results Now let …X; d† be a metric space and let L  C…X† be a unital vector lattice. We say that L is uniformly separating if for every pair of subsets A and B of X with d…A; B† > 0 there exists some f 2 L such that f …A† \ f …B† ˆ ;. Typical examples are L ˆ U…X†, the lattice of all uniformly continuous real functions on X, as well as L ˆ U  …X† the lattice of bounded elements in U…X†. Indeed, if A and B are subsets of X with d…A; B† ˆ  > 0, then the function f ˆ inf fd…
; A†; g 2 U  …X† satis®es f ˆ 0 on A and f ˆ  on B. We need the following simple Lemma. Lemma 1. Let …X; d† be a complete metric space and let L  C…X† be a uniformly separating unital vector lattice. Then  2 H…L† has a countable neighborhood basis in H…L† if, and only if,  2 X. Proof. Suppose ®rst that  2 H…L† n X has a countable neighborhood basis. Since X is dense in H…L†, there exists a sequence …xn † in X converging to . The completeness of X implies that …xn † has no d-Cauchy subsequence, and therefore there are some " > 0 and some subsequence …xnk † such that d…xnk ; xnj † 5 " for

A Banach±Stone Theorem for Uniformly Continuous Functions

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k 6ˆ j. Now the sets A ˆ fxnk : k eveng and B ˆ fxnk : k oddg satisfy d…A; B† 5 ", and then there is a function f 2 L with f …A† \ f …B† ˆ ;. But this is a contradiction since f extends continuously to H…L† and  is in the closure of both A and B. Conversely, if  2 X, let Bn be the open ball in X of center  and radius 1=n, and let clH…L† Bn be its closure in H…L†. Then the family fclH…L† Bn g is easily seen to be a countable neighborhood basis as required. & Theorem 2. Let …X; dX † and …Y; dY † be complete metric spaces. Let LX  U…X† and LY  U…Y† be unital vector lattices which are uniformly separating. If LX is isomorphic to LY as unital vector lattices, then X is uniformly homeomorphic to Y. Proof. Suppose that  : LY ! LX is an isomorphism of unital vector lattices. We de®ne h : H…LX † ! H…LY † by h…† ˆ   . Then h is a bijection, and in fact h is a homeomorphism since prg  h ˆ pr…g† , for each g 2 LY (recall that pr denotes the corresponding projection map). Applying the previous Lemma to LX and LY we obtain that h…X† ˆ Y. We are going to see that hjX : X ! Y is uniformly continuous. First note that dX …A; B† ˆ 0 ) dY …hjX …A†; hjX …B†† ˆ 0: Otherwise, there exist A; B  X such that dX …A; B† ˆ 0 and dY …hjX …A†; hjX …B†† > 0. Since LY is uniformly separating, there exists g 2 LY with g…hjX …A†† \ g…hjX …B†† ˆ ;. But this is impossible because g  hjX ˆ …g† 2 LX is a uniformly continuous function, and dX …A; B† ˆ 0. Now applying a classical result due to Efremovich [2] (see also Engelking [3], p. 573), it follows that hjX is uniformly continuous. The same is true for …hjX †ÿ1 . & Note that the above Theorem shows that the existence of a homeomorphism between the realcompacti®cations H…LX † and H…LY † implies the existence of a uniform homeomorphism between the complete metric spaces X and Y. Next, as a direct consequence of Theorem 2, we obtain a Banach-Stone type result for uniformly continuous functions. Corollary 3. Let X and Y be complete metric spaces. The following are equivalent: …i† U…X† is isomorphic to U…Y† as unital vector lattices. …ii† U  …X† is isomorphic to U  …Y† as unital vector lattices. …iii† X is uniformly homeomorphic to Y. e denotes its completion, then both Remarks. (1) If X is a metric space and X metric spaces have the same uniformly continuous real functions. Therefore, completeness of spaces cannot be avoided in the above results. (2) On the other hand, note that Theorem 2 and Corollary 3 do not extend to the general class of uniform spaces. Indeed, let …X; d† be an uncountable discrete metric space, and let  be the weak uniformity on X generated by the set of all real functions on X. It is easy to see that  is not metrizable. On the other hand, if X has non-measurable cardinal then …X; † is complete (see e.g. Gillman-Jerison [4]). In

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this case, …X; d† and …X; † are complete uniform spaces which are not uniformly homeomorphic, but nevertheless they have the same uniformly continuous real functions. Acknowledgements. It is a pleasure to thank Professors Salvador HernaÂndez and JesuÂs Araujo for several interesting conversations concerning this note. We are also grateful to the referee for his/her suggestions.

References [1] Araujo J, Font JJ (2000) Linear isometries on subalgebras of uniformly continuous functions. Proc Edinburgh Math Soc 43: 139±147 [2] Efremovich VA (1951) The geometry of proximity I. Math Sbor 31: 189±200 [3] Engelking R (1977) General Topology. Warsaw: PWN-Polish Scienti®c [4] Gillman L, Jerison M (1976) Rings of continuous functions. New York: Springer [5] HernaÂndez S (1999) Uniformly continuous mappings de®ned by isometries of spaces of bounded uniformly continuous functions. Topology Atlas No 394 [6] Hewitt E (1948) Rings of real-valued continuous functions I. Trans Amer Math Soc 64: 54±99 [7] Isbell JR (1958) Algebras of uniformly continuous functions. Ann of Math 68: 96±125 [8] Lacruz M, Llavona JG (1997) Composition operators between algebras of uniformly continuous functions. Arch Math 69: 52±56 [9] Shirota T (1952) A generalization of a theorem of I. Kaplansky. Osaka Math J 4: 121±132 Authors' addresses: M. Isabel Garrido, Departamento de MatemaÂticas, Universidad de Extremadura, 06071 Badajoz, Spain, e-mail: [email protected]; JesuÂs A. Jaramillo, Departamento de AnaÂlisis MatemaÂtico, Universidad Complutense de Madrid, 28040 Madrid, Spain, e-mail: jaramil@eucmax. sim.ucm.es