Isometric shifts between spaces of continuous functions ∗ Jes´ us Araujo†

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Introduction

We present an account of old and new results on isometric shifts between spaces of K-valued continuous functions (being K = R or C). The notion of isometric shift is just a natural generalization to Banach spaces of that of shift operator in `2 and other sequence spaces, (x1 , x2 , . . .) 7→ (0, x1 , x2 , . . .) (see [6, 14]). We say that a linear operator T : E → E is an isometric shift if 1. T is an isometry, 2. The codimension of T (E) in E is 1, T n 3. ∞ n=1 T (E) = {0}. Two basic spaces that can be seen both as spaces of sequences and spaces of continuous functions are `∞ and c, consisting of all K-valued bounded and convergent sequences, respectively. `∞ and c are easily seen to be isometrically isomorphic to C(βN) and C(N ∪ {∞}), respectively (here we consider spaces C(X) endowed with the supremum norm k·k∞ ). Obviously, in both cases the usual shift defined in the sequence spaces is also an isometric shift in the above sense when considering them as function spaces. ∗

2010 Mathematics Subject Classification. Primary 47B38; Secondary 46E15, 47B33, 47B37, 54D65, 54H20. † Research partially supported by the Spanish Ministry of Science and Education (Grant number MTM2006-14786).

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We focus on some questions that have made it possible to develop the theory of isometric shifts between spaces of continuous functions. Suppose that X is a compact Hausdorff space and there exists a shift operator T : C(X) → C(X), where C(X) denotes the space of all continuous functions from X to K endowed with the supremum norm k·k∞ . Question 1 Can X have an infinite connected component? Question 2 Must X be separable? Question 3 Can X be the Cantor set K or a related space? Of course there is still a very natural question that we have not included above but that lies behind all the study: Question 4 Can we describe T ? Sometimes the answer to these and other questions depend on the scalar field. We will denote by CR (X) and CC (X) the spaces of real-valued and complex-valued continuous functions on X, respectively. Some other papers have recently studied operators related to isometric shifts (also defined on other spaces of functions). Among them, we will mention for instance [3, 5, 8, 15, 16, 18, 19, 20], and [22] (see also references therein).

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Question 1

The original statement of Question 1 was not exactly as given above. It was posed in the following terms by Holub in [14]. Conjecture If X has at least one infinite connected component, there is no shift or backward shift on CR (X). We do not treat here backward shifts, and just mention that the answer to the corresponding part of the question was completely solved in [19]. As for the part on isometric shifts, in fact that author gave a first partial answer ([14, Theorem 2.1]). Theorem 2.1 If X has a finite number of components, then CR (X) admits no shift. 2

Obviously, we immediately deduce that CR [a, b] admits no shift. But, even if we are just dealing with the case of isometric shifts between spaces C(X), it is worth to see that something else can be said for other spaces on intervals (see [14, Theorem 2.3]). Theorem 2.2 Let E be a Banach space consisting of real-valued continuous functions on an interval [a, b] for which 1. k·kE = k·k∞ + p(·), where p is a seminorm on E; 2. 1 ∈ E and p(1) = 0; 3. Given any (nontrivial) interval I ⊂ [a, b], there exists an infinitedimensional subspace of E whose members have support in I. Then there is no isometric shift on E. When appropiate norms are given in the spaces, an immediate corollary is the following. Corollary 2.3 There is no shift on the space C n [a, b] of all real valued functions on [a, b] having n continuous derivatives there (n ≥ 1). It must be said that the techniques used to prove Theorem 2.1 do not carry over to the complex case. In fact, some other results were given before the next theorem, which represents a large generalization of Theorem 2.1, could be proved (see [7, Theorem 6.1]). Theorem 2.4 Let M be any compact manifold with or without boundary. Then C(M ) does not admit an isometric shift. In fact, the authors prove more: they show that C(M ) does not admit even a codimension 1 linear isometry. Notice that in general a codimension 1 linear isometry need not to be an isometric shift, as the following example shows (see [7, Example 3.2]). Example 2.5 Identify as usual the spaces C (N ∪ {∞}) and c, and define T : C (N ∪ {∞}) → C (N ∪ {∞}) by T (x1 , x2 , x3 , . . .) := (x1 , 0, x2 , x3 , . . .) for each (xn ) ∈ c.

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Nevertheless, Question 1 was settled in the negative in [10]. There is indeed a space X with an infinite connected component and such that C(X) admits a shift. The example given turns out to be a compactification of integers. We finally give a generalization of Theorem 2.1 for the infinite case and K = R, C, as it appears in [10, Theorem 2.7]. Theorem 2.6 Suppose that X has a countably infinite number of components, all of whom are infinite. Then C(X) does not admit an isometric shift. In view of all the above, the following question appears to be very natural, assuming that there exists a shift operator on C(X): Question 1b Can X be infinite and connected?

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Question 4 and new questions

The new results that changed dramatically the way to look at isometric shifts, allowing in particular to obtain Theorem 2.4, are those concerning the representation of isometric shifts. Using a result of Holszty´ nski ([13]), Gutek, Hart, Jamison, and Rajagopalan classified isometric shifts into two types, called type I and type II (see [10]). First, If T : C(X) → C(X) is an isometric shift, then there exist a closed subset Y ⊂ X, a continuous and surjective map φ : Y → X, and a function a ∈ C(Y ), |a| ≡ 1, such that (T f )(x) = a(x) · f (φ(x)) for all x ∈ Y and all f ∈ C(X). Then the classification is as follows: • T is said to be of type I if Y can be taken to be equal to X \ {p}, where p ∈ X is an isolated point. • T and is said to be of type II if Y can be taken equal to X. Moreover, if T is of type I, then the map φ : X \ {p} → X is indeed a homeomorphism. In fact, that classification is not mutually exclusive, as there can be examples of isometric shifts that are of both types I and II, as the authors show. 4

The description given above helps the authors to provide, in the same paper, a first partial answer to Question 2. Theorem 3.1 If C(X) admits an isometric shift of type II, then X is separable. Corollary 3.2 If C(X) admits an isometric shift and X has no isolated points, then X is separable. Can the same be said with respect to shift operators of type I? Suppose then that T is of type I, and denote 1 := p, 2 := φ−1 (1), and in general n := φ−1 (n − 1) = φ1−n (1) for each n ≥ 2. Since p is isolated, all points in the set N := {1, 2, 3, ...} are also isolated, and consequently X contains a copy of N. We write T = T [a, φ, ∆] to describe an isometric shift T : C(X) → C(X), where X is compact and contains N . It means that φ : X \ {1} → X is a homeomorphism, satisfying in particular φ(n + 1) = n for all n ∈ N. It also means that a ∈ C(X \ {1}), |a| ≡ 1, and that ∆ is a continuous linear functional on C(X) with k∆k ≤ 1. The description of T we have is (T f )(x) = a(x)f (φ(x)), when x 6= 1, and (T f )(1) = ∆(f ), for every f ∈ C(X). Taking into account the above and that Question 2 remains unsolved only for type I isometric shifts, we can ask a new question, very much related but much simpler. Question 2a If C(X) admits an isometric shift of type I, must N be dense in X? Examples of isometric shifts for which clX (N ) = X are called primitive. In fact the answer to Question 2a is negative and was given first by Farid and Varadarajan in [7]. They proved the following. Theorem 3.3 Let n ∈ N. For X := N ∪ {∞}, there exists an isometric shift T : C(X) → C(X) such that X \ clX (N ) is a set of n isolated points. We compare it with the next two results, given later in [11], were the sharp distinction between the real and complex cases is made clear. Theorem 3.4 For X := N ∪ {∞}, if T : CR (X) → CR (X) is a type I isometric shift, then X \ N is finite. 5

Theorem 3.5 For X := N∪{∞}, there exists an isometric shift T : CC (X) → CC (X) such that X \ N is infinite. All examples given so far of spaces X for which C(X) admits type I isometric shifts are compactifications of integers, that is, N ∪ {∞}, βN, and the counterexample to Question 1 are compactifications of integers. It is interesting to see that in some cases the denseness of isolated points allows defining isometric shifts (see [11, Theorem 2.1]). Theorem 3.6 Let X be an infinite metric space and let D be a dense set of isolated points. If X \D is connected, then C(X) admits a primitive isometric shift. Question 2b If C(X) admits an isometric shift of type I, must X have a dense set of (countably many) isolated points? The answer to this question is again ”no”, although up to 2001 all known examples were compactifications of integers. Two very different spaces were given in [4] and [21]. The example in [21] involves the Cantor set but still is totally disconnected. As for that in [4] satisfies X \ clX (N ) = T (the unit circle in C), thus giving a new negative answer to Question 1 as well. Looking at the above examples, assuming that C(X) admits an isometric shift of type I, we can ask, as Gutek and Norden did in [11], Question 2c How big can X \ clX (N ) be? This will be studied by considering how complex φ can be. Related to this, we can ask also Question 2d How complex can X be if a is simple?

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Questions 2c and 1 again

How to measure the complexity of T , f 7→ a · f ◦ φ? We can do it in two ways, namely: • Forcing the map φ to be complex. • Avoiding freedom in the choice of the map a ∈ C(X\{1}), in particular, setting a ≡ 1. 6

Concerning the first way, we give the following definition. Definition 4.1 Let X be compact, and suppose that T = T [a, φ, ∆] : C(X) → C(X) is a non-primitive isometric shift of type I. For n ∈ N, we say that T is n-generated if n is the least number with the following property: There exist n points x1 , . . . , xn ∈ X \ clX (N ) such that the set {φk (xi ) : k ∈ Z, i ∈ {1, . . . , n}} is dense in X \ clX (N ). We say that T is ∞-generated if it is not n-generated for any n ∈ N. Notice that the above definition does not make sense when T is an isometric shift which is not of type I. On the other hand, it is proved in [10, Theorem 2.5] that for such isometric shifts, there exists a point x ∈ X such that {φk (x) : n ∈ Z} is dense in X. In particular, if T is a non-primitive isometric shift of both types I and II, then it is 1-generated. Obviously, ∞-generated examples are necessarily very complex. The first example of such a shift was given in [11, Theorem 3.5]. ∞}. There exists an isometric shift on Theorem 4.1 Let X := βN + N ∪ {∞ C(X) which is ∞-generated. Our goal here is to give spaces with many infinite connected components admitting isometries which are n-generated for different n. This cannot be done in general, and we must put some restrictions on the number of components. For this, we need some definitions. We say that P ⊂ N is an initial subset if either P = N or P = {1, . . . , N } for some N ∈ N. Definition 4.2 Given an initial subset P, a sequence (pn )n∈P of natural numbers is said to be P-compatible if • if P = {1}, then p1 > 1, and • if P 6= {1}, then pn+1 is an even multiple of pn for every n (allowing the possibility p1 = 1).

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The following result gives us P a picture of how complicated X and T can be. In it the symbols + and denote the topological sum of spaces. Of course, when there are many connected components we have to take a compactification so as to ensure that X is compact. Even if the space X given in Theorem 4.1 is a compactification of integers, some of the ideas in it were taken to prove part of Theorem 4.2. Theorem 4.2 and Corollaries 4.3, 4.4 and 4.5 were proved in [2]. Below T0 denotes an isolated point. Theorem 4.2 Let P be an initial subset of N and let (pn )n∈P be a P-compatible sequence. Suppose that (κn )n∈P and (Zn )n∈P are a sequence of cardinals and a sequence of (nonempty) compact spaces satisfying at least one of the following two conditions: • Condition 1. For every n ∈ P, 0 ≤ κn ≤ c and Zn := Tκn ; • Condition 2. For every n ∈ P, ℵ0 ≤ κn ≤ c and Zn := Kn κn , where Kn is separable. Then, for N = card P, there exists a compactification ωX0 of X Zn + · · · + Zn X0 := {z } | n∈P

pn

such that ωX0 \ X0 is either countable or empty, and such that C(X) admits ∞}). an N -generated isometric shift of type I (where X := ωX0 + N ∪ {∞ The next corollary shows that things are very different with respect to the case where we do not allow X to contain isolated points, and should be compared with Theorem 2.6. Corollary 4.3 Every infinite-dimensional normed space contains a compact subset X with infinitely many pairwise nonhomeomorphic components such that C(X) admits an ∞-generated isometric shift of type I. Corollary 4.4 With the same notation as in Theorem 4.2, if s := sup{κn : n ∈ P} = ℵ0 (and further each Kn is metrizable if we are under Condition 2), then X may be taken to be contained in `2 , endowed with the norm topology. Moreover, if we are under Condition 1 and s < ℵ0 , then X may be taken to be contained in Cs . 8

We easily deduce the following corollary, using the case when s = 1. Corollary 4.5 In R2 , we can find a compact set X having a countably infinite number of components (each of them being infinite), and such that C(X) admits an isometric shift which is ∞-generated. The above corollary should be compared with the following very closed result given previously in [11]: Theorem 4.6 There exists a compact subset X of R2 and a type I isometric shift on C(X) such that X \ N is infinite. Remark 4.1 Notice that each copy of a power of T given in Theorem 4.2 is a connected component of X \ clX (N ), and they are indeed all the infinite connected components of X (if we assume κn 6= 0 for every n). The same comment applies to copies of powers of Kn when every Kn is connected. This implies that we can construct examples where we may decide at will on the number of (different) infinite connected components of X \ clX (N ). Remark 4.2 In Theorem 4.2, we are not assuming that spaces Kn or cardinals κn are necessarily pairwise different.

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Question 2d for a = 1

Theorem 4.2 and its corollaries in Section 4 involve spaces X = ωX0 + N ∪ ∞} for a suitable compact space ωX0 , that is, N appears with its one-point {∞ compactification as a topological summand. We next see that in such a way a is always different from the constant map 1. Results in this section can be found in [2]. Proposition 5.1 Let X be compact, and suppose that X = X1 + X2 , with X1 6= ∅ and N being a (countable) dense subset of X2 . Suppose also that φ : X \ {1} → X is a homeomorphism (with φ(n + 1) = n for n ∈ N ). Let T : C(X) → C(X) be a codimension 1 linear isometry such that (T f )(x) = f (φ(x)) for every f ∈ C(X) and x ∈ X \ {1}. Then T is not an isometric shift. Consequently, when we try to obtain a = 1, we need to embed N in X in a different way. This can be done and leads to the following result. 9

Theorem 5.2 C(X) admits an isometric shift of type I for which a ≡ 1 in the following cases, being X = W ∪ N : • if W is a separable infinite power of a compact space with at least two points, • if W is a compact n-manifold (with or without boundary), for n ≥ 2, • if W is the Sierpi´ nski curve. Also in the case when the weight a is equal to 1, it is possible to find examples where the number of infinite connected components is n, for every n ∈ N. Corollary 5.3 Let n ∈ N. Let Y0 be a connected and compact space with more than one point, and suppose that φ : Y0 → Y0 is a homeomorphism having a periodic point, and such that φn is transitive. Then there exist a compact space X and an isometric shift of type I on C(X) with a ≡ 1, such that X \ N consists exactly of n connected components with more than one point, each homeomorphic to Y0 .

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Questions 3 and 1b

The question on whether there is or not an isometric shift on C(K) was raised in [10], where some hints were also given about the existence of such a map. According to the representation we have given, we first see that, if there is one, then it cannot be of type I. The answer to the question (for the real case) appears in [12]. Theorem 6.1 There is an isometric shift on CR (K). Once again, when a sequence is adjoined to the Cantor set, we can find many different isometric shifts, all of them of type I (see [2]). Theorem 6.2 Suppose that (xn ) is a nonconstant sequence in R \ K which converges to a point L ∈ R. Let X := K ∪ {xn : n ∈ N} ∪ {L}. We have • If L ∈ / K, then for each n ∈ N ∪ {∞}, there exists an isometric shift of type I on C(X) which is n-generated. 10

• If L ∈ K, then there exists an isometric shift of type I on C(X) which also satisfies the additional condition that a ≡ 1. Remark 6.1 Recall that, as mentioned by the authors, by [11, Theorem 1.9] we can conclude that if X consists of a convergent sequence adjoined to a non-separable Cantor cube, then C(X) does not admit an isometric shift. This is not true in the separable case, as shown in [21, Example 20] for an isometric shift of both types I and II. Theorem 5.2 and Theorem 6.2 say also the contrary in the separable case for isometric shifts which are not of type II. Theorem 6.2 provides indeed completely different families of isometric shifts of type I. Some other special examples were also studied in [12], where the two following results appear (in particular, an answer to Question 1b). Theorem 6.3 There is a Peano continuum X such that CC (X) admits an isometric shift. Theorem 6.4 There is a one-dimensional, connected, compact, metric space X such that CC (X) admits an isometric shift.

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Question 2

Question 2 was first raised in [10], and not much was known about its final answer until very recently. As we mentioned above, the question was known to be positive for type II isometric shifts, so that if X has no isolated points and C(X) admits a shift, then X must be separable (see Corollary 3.2). Something similar can be said if we know that C(X) admits an isometric shift T that is also disjointness preserving, that is, (T f )(T g) ≡ 0 whenever f g ≡ 0, as it was shown in [10, Theorem 5.1]; in this case X must be separable. Another positive answer was given in [4], and can be stated as follows. Theorem 7.1 Let M be a complete metric space, and suppose that C(βM ) admits an isometric shift. Then M is separable. Almost at the same time, another result giving a clue on a general property spaces must satisfy so as to admit isometric shifts, was given in [11, Theorem 1.4]. 11

Theorem 7.2 If C(X) admits an isometric shift of type I, then X has the countable chain condition. Related to this, we also have that if C(X) admits an isometric shift of type I, then C0 (X \ clX (N )) (the space of K-valued continuous functions vanishing at infinity) must have cardinality at most equal to c (see [11, Theorem 1.9]). Question 2 was recently solved in the negative in [1]. The following theorems provide examples of X not separable for which C(X) admits an isometric shift (necessarily of type I). All the examples given in [1] are based on M, the maximal ideal space of the algebra L∞ (T) of all Lebesgue-measurable essentially bounded complex-valued functions on T. It is not hard to see that M is not separable. Consequently, since it has no isolated points, C(M) admits no isometric shift (Corollary 3.2). A different conclusion is obtained when we adjoin to M a convergent sequence and its limit. A first example is the following (see [1, Theorem 3.1]). ∞}) admits an isometric shift. Theorem 7.3 C (M + N ∪ {∞ M is homeomorphic to an infinite closed subset of βN \ N, and consequently its cardinal must be 2c (see [17] and [9, Corollary 9.2]). Now we can obtain more examples, in particular with many infinite connected components, as is the case of the following result ([1, Theorem 3.2]), which has 2c infinite connected components (see again Question 1). ∞}) Theorem 7.4 Let κ be any cardinal such that 1 ≤ κ ≤ c. Then C (M × Tκ + N ∪ {∞ admits an isometric shift. We can also give examples with just one infinite component (see [1, Theorem 3.3]). ∞}) Theorem 7.5 Let κ be any cardinal such that 1 ≤ κ ≤ c. Then C (M + Tκ + N ∪ {∞ admits an isometric shift. All the above results are valid both for K = R and K = C. We next see some depending on the scalar field (see [1, Theorems 5.1, 5.2, and Example 5.3]). Theorem 7.6 Let K = C. Suppose that n ∈ N, and that (κj )nj=1 is a finite sequence of cardinals satisfying 0 ≤ κj ≤ c for every j. Let Xn := Tκ1 + . . . + ∞}. Then Tκn + N ∪ {∞ 12

• CC (Xn ) admits an n-generated isometric shift. • CC (M + Xn ) admits an isometric shift. The above is in general not true in the real case. ∞}) nor Example 7.7 If K = R, then neither CR (T + T2 + T3 + N ∪ {∞ 2 3 ∞}) admit an isometric shift. CR (M + T + T + T + N ∪ {∞

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A final question

We end with the following question, which was posed in [12] and does not seem to have been answered yet: Question Is there a finite-dimensional Peano continuum X such that CC (X) admits an isometric shift?

References [1] J. Araujo, On the separability problem for isometric shifts on C(X). J. Funct. Anal. 256 (2009) 1106-1117. [2] J. Araujo, Examples of isometric shifts on C(X). Preprint. [3] J. Araujo and J.J. Font, Codimension 1 linear isometries on function algebras. Proc. Amer. Math. Soc. 127 (1999), 2273-2281. [4] J. Araujo and J.J. Font, Isometric shifts and metric spaces. Monatshefte Math. 134 (2001), 1-8. [5] L.-S. Chen, J.-S. Jeang, and N.-C. Wong, Disjointness preserving shifts on C0 (X). J. Math. Anal. Appl. 325 (2007), 400-421. [6] R. M. Crownover, Commutants of shifts on Banach spaces. Michigan Math. J. 19 (1972), 233-247. [7] F.O. Farid and K. Varadajaran, Isometric shift operators on C(X). Can. J. Math. 46 (1994), 532-542. 13

[8] J.J. Font, Isometries on function algebras with finite codimensional range. Manuscripta Math. 100 (1999), 13-21. [9] L. Gillman and M. Jerison, Rings of continuous functions. Springer Verlag, New York, 1976. [10] A. Gutek, D. Hart, J. Jamison and M. Rajagopalan, Shift operators on Banach spaces. J. Funct. Anal. 101 (1991), 97-119. [11] A. Gutek and J. Norden, Type 1 shifts on C(X). Topology Appl. 114 (2001), 73-89. [12] R. Haydon, Isometric shifts on C(K). J. Funct. Anal. 135 (1996), 157-162. [13] H. Holszty´ nski, Continuous mappings induced by isometries of spaces of continuous functions. Studia Math. 26 (1966), 133-136. [14] J.R. Holub, On shift operators. Canad. Math. Bull. 31 (1988), 85-94. [15] K. Izuchi, Douglas algebras which admit codimension 1 linear isometries. Proc. Amer. Math. Soc. 129 (2001), 2069-2074. [16] J.-S. Jeang and N.-C. Wong, Isometric shifts on C0 (X). J. Math. Anal. Appl. 274 (2002), 772-787. [17] S. Negrepontis, On a theorem by Hoffman and Ramsay. Pacific J. Math. 20 (1967), 281-282. [18] M. Rajagopalan, T. M. Rassias, and K. Sundaresan, Generalized backward shifts on Banach spaces C(X, E). Bull. Sci. Math. 124 (2000), 685-693. [19] M. Rajagopalan and K. Sundaresan, Backward shifts on Banach spaces C(X). J. Math. Anal. Appl. 202 (1996), 485-491. [20] M. Rajagopalan and K. Sundaresan, An account of shift operators. J. Anal. 8 (2000), 1-18. [21] M. Rajagopalan and K. Sundaresan, Generalized shifts on Banach spaces of continuous functions. J. Anal. 10 (2002), 5-15.

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[22] T. M. Rassias and K. Sundaresan, Generalized backward shifts on Banach spaces. J. Math. Anal. Appl. 260 (2001), 36-45. Departamento de Matem´aticas, Estadistica y Computaci´on Facultad de Ciencias Universidad de Cantabria Avenida de los Castros s.n. E-39071, Santander, Spain. E-mail address: [email protected]

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