A Rigid Space Homeomorphic to Hilbert Space

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on `2, and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to `2. Rigid topological vector spaces first appeared in the literature in 1977 with an example by Waelbroeck, in the paper [11]. This first rigid space, however, was not complete, and the existence of a complete rigid space was first confirmed by Kalton and Roberts in [5]. In that paper, the constructed space is not only complete and rigid, but is also quotient-rigid and a subspace of L0[0, 1] (quotient-rigid meaning that every quotient of the space inherits the rigid character). A rigid space which serves as the domain space of a non-trivial compact operator was constructed in [9], illustrating that rigid spaces can have relatively rich topologies. In this paper we demonstrate that rigid spaces can in fact be topologically homeomorphic to Hilbert space, thus illustrating the degree to which the two concepts of topological homeomorphism and topological vector space isomorphism can differ. To do this, we will first modify slightly the rigid space constructed in [5], and then employ a characterization of ANR-spaces (absolute neighborhood retract spaces) due to the first author, which appeared in its original form in [6] and in a refined form in [7]. It is this second version that we will apply in this paper. The first section of the paper contains the characterization we will apply, as well as an explanation of the modifications we must make to the first rigid space. We will then show in the second and third sections that the rigid space under consideration is an AR-space. 1. Theorems and a construction We begin with an explanation of the characterization of ANR spaces to be found in [7]. Received by the editors January 23, 1996. 1991 Mathematics Subject Classification. Primary 46A16, 54F65; Secondary 46C05, 54G15. c ©1998 American Mathematical Society 85 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 86 NGUYEN TO NHU AND PAUL SISSON Let {Un} be a sequence of open covers of a metric space X . For a given open cover Un, let mesh(Un) = sup{diam(U) : U ∈ Un}. We say that Un is a zero sequence if mesh(Un) → 0 as n → ∞. For a given open cover U , we let N (U) denote the nerve of U . The nerve of an open cover is the simplicial complex {σ : σ = 〈U1, . . . , Un〉, Ui ∈ U , n ∈ N} made up of all σ = 〈U1, . . . , Un〉 for which ⋂n i=1 Ui 6= ∅. N (U) is endowed with the Whitehead (or weak) topology (see [1] or [3] for a discussion). Finally, define U = ⋃∞i=1 Un and let K(U) = ⋃∞n=1N (Un ∪ Un+1), and for any σ ∈ K(U) let n(σ) = max{n ∈ N : σ ∈ N (Un ∪ Un+1)}. We use the following version of the ANR-characterization theorem; see [6], [7], [8]. Theorem 1.1. A metric space X with no isolated points is an ANR if and only if there is a zero sequence {Un} of open covers of X and a map g : K(U) → X such that g|U → X is a selection (i.e. g(U) ∈ U for every U ∈ U) and such that for any sequence of simplices {σk} in K(U) with n(σk) → ∞ and g(σ k) → x0 ∈ X we have g(σk) → x0 ∈ X. Here, σ k represents the collection of vertices {U1, . . . , Un} making up the simplex σk, and g(σ 0 k) and g(σk) represent the sets of images of, respectively, the vertices of σk and the convex combinations of the vertices of σk. The goal in this paper is to show that the rigid space to be constructed has this property. This then shows that the rigid space is an AR space, and so by the result of Dobrowolski and Torunczyk [2] the space is homeomorphic to `2 (Dobrowolski and Torunczyk showed that for a complete, separable infinite-dimensional linear metric space, X ∼= `2 ⇐⇒ X is an AR). The topology of the rigid space will be generated by an F -norm on the space. Recall that an F -norm is defined as follows. Definition 1.1. Let X be a vector space. A map ||·|| : X → [0,∞) is an F -norm if (1) ||x|| = 0 ⇐⇒ x = 0, (2) ||x+ y|| ≤ ||x||+ ||y||, (3) ||αx|| ≤ ||x|| whenever |α| ≤ 1, and (4) ||αx|| → 0 whenever |α| → 0. In addition, the construction in [5] often makes use of quasi-norms, similar to F -norms but with the following characteristics: (1) ||x|| = 0 ⇐⇒ x = 0, (2) ||x+ y|| ≤ C(||x||+ ||y||), C independent of x and y, and (3) ||αx|| = |α| ||x||, α a scalar. The type of “norm” in use at a given point in the construction will be clear from the context. We now proceed with an overview of the construction of the first rigid space, along with our modifications. For simplicity of terminology we first introduce the following definition. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A RIGID SPACE HOMEOMORPHIC TO HILBERT SPACE 87 Definition 1.2. We say that a sequence {pn} is a 1-approaching sequence if the following conditions hold: 1/2 = p0 < p1 < ... < pn < ... < 1, lim n→∞ pn = 1. Let {pn} be a 1-approaching sequence. We define the space `({pn}) by `({pn}) = {x ≡ ∞ ∑ n=0 xnen : ‖x‖ ≡ ∞ ∑ n=0 |xn|n <∞}, (1) where en denotes the characteristic function of [n, n+ 1]. The space `({pn}) is equipped with the F -norm ‖ · ‖ defined in (1). In [5], the existence of a sequence of finite-dimensional spaces {Vn}∞n=0 is demonstrated, each a subspace of Lpn and each with basis {vn,k} k=1, with the basis elements possessing certain properties and with 1/2 = p0 < p1 < · · · < 1. For the purposes of that paper an explicit construction of the basis elements was not necessary, but for our purposes we will find it convenient to be more precise. To that end, let vn,k be the characteristic function of the k th sub-interval of [0, 1] of length l(n)−1, where [0, 1] has been sub-divided into l(n) essentially disjoint sub-intervals of equal length. Note that with this specification of the basis elements, the spaces Vn have the properties of Lemma 3.1 of [5]. In [5], each Vn is translated by the map τn, which takes functions of [0, 1] to functions of [n, n + 1] by τnf(x) = f(x − n). Keeping the notation of [5], we will let Un = τnVn. Let M be the closed linear span of {en}, and let Y be the closure of ⋃ Un, where closure in both cases is relative to the F -norm defined on the space Z of real-valued functions on [0,∞) by ||f || = ∞ ∑ n=0 ∫ n+1 n |f(x)|pn dx. As in [5], let Z(a, b) be the subspace of Z of functions with support on [a, b]. Lemma 3.2 of [5] proved the following: Lemma. Suppose f ∈ Z(0, n) with ||f || = 1. Then there exists a linear operator A : Z(n, n+ 1) → Z(0, n) with Aen = f and ||A|| = 1. (In this context, Z(0, n) and Z(n, n+ 1) are equipped with quasi-norms and, as in Banach spaces, ||A|| = sup||x||≤1 ||Ax||.) We will have to modify this lemma slightly, so that in addition to the above, A : Un → (U0 + · · · + Un−1). To do this, we have to assume of course that f ∈ (U0 + · · ·+Un−1), but this will be the case in the application of the lemma. As in the proof of Lemma 3.2 of [5], decompose f so that f = h0 + · · ·+ hn−1, where hi ∈ Ui. For each i, decompose hi so that hi = h1 + · · ·+ hl(i), with the support of hj being the j th sub-interval of [i, i + 1] (recall that [i, i + 1] has been partitioned into l(i) sub-intervals of equal length). Let ej denote the characteristic function of the j sub-interval of [n, n+1], for each j = 1, . . . , l(n). For the moment, fix i. By Lemma 2.2 of [5] there exist linear operators F i j , each mapping the pn –integrable functions with support on the j sub-interval of [n, n + 1] to the pi –integrable License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 88 NGUYEN TO NHU AND PAUL SISSON functions with support on the j sub-interval of [i, i + 1], for each j = 1, . . . , l(i). Furthermore, F i j e n j = h i j and ∣∣∣∣F i j ∣∣∣∣ = ∣∣∣∣hij∣∣∣∣ . For j = l(i) + 1, . . . , l(n), let F i j be the zero operator, and let Fi = F i 1 + · · ·+F i l(n). Then Fien = hi and Fi : Un → Ui. Also, due to the partitioning of the intervals, ||Fi|| = sup ||x||≤1 ∫ |F i 1x+· · ·+F i l(n)x| = sup ||x||≤1 (∫ |F i 1x|i + · · ·+ ∫ |F i l(n)x| ) ≤ ∣∣∣∣F i 1∣∣∣∣pi + · · ·+ ∣∣∣∣∣∣F i l(n)∣∣∣∣∣∣pi = ∣∣∣∣hi1∣∣∣∣pi + · · ·+ ∣∣∣∣∣∣hil(i)∣∣∣∣∣∣pi = ||hi|| . At this point the remainder of the proof is as in Lemma 3.2 of [5], with the operator A being defined by A = F0 + · · ·+ Fn−1. The next step in the construction of the rigid space involves defining an operator S mapping Z to Z. This begins with the selection of a sequence of elements {γk}k=1, with γk ∈ U0+· · ·+Uk−1. By Lemma 3.2 of [5], operators Ak : Z(k, k+1) → Z(0, k) with ||Ak|| = 1 and Akek = γk can be chosen. Note that by our modification to Lemma 3.2 above, we can assume also that Ak : Uk → (U0 + · · · + Uk−1). A map T : Z → Z is then defined by T = ∑∞k=1 ckAkEk (each Ek is the projection map from Z onto Z(k, k+ 1), and {ck} is a sequence of reals). Our modification to Lemma 3.2 implies that, in addition, T maps Y into Y . Let T̃ denote the restriction of T to the subspace Y . Finally, the map S : Z → Z is defined by S = I − T . We will want to work with the restriction map S̃ : Y → Y as well, defined by S̃ = I− T̃ . In [5] it is shown that ||T || ≤ 1/4, and that therefore S is invertible. Combining this argument with the fact that we have made T : Y → Y , we obtain ∣∣∣∣∣∣T̃ ∣∣∣∣∣∣ ≤ 1/4, and therefore S̃ : Y → Y is also invertible. Kalton and Roberts then showed in [5] that the sequences {l(n)} and {pn} can be chosen so that the quotient space X = Y/S(M), where M = `({pn}), is a rigid space. Remark 1.1. In [5], the sequences {pn} and {l(n)} are constructed inductively. In the inductive step, pn is chosen sufficiently close to 1 so as to obtain the desired behavior, and l(n) then depends on pn. In our modification of the rigid space, we will want to choose pn possibly closer to 1 in the inductive step, for reasons given in the proof of Theorem 3.5. Since our choice of pn is, if anything, larger than the choice of pn in [5], this has no impact on the construction of the rigid space. 2. The AR-property for Y In this section we prove the following theorem: Theorem 2.1. Y is an AR. Proof. We aim to verify the conditions of Theorem 1.1. Let {Un} be a zero sequence of open covers of Y . Let U = ⋃∞n=1 Un, K(U) = ⋃∞n=1N (Un ∪ Un+1), and let f0 : U → Y be a selection. We extend f0 to a map f : K(U) → Y as follows. For any simplex σ = 〈U1, . . . , Um〉 ∈ K(U), Uj ∈ U for j = 1, ...,m. Since f0(Uj) ∈ Y , we have f0(Uj) = ∞ ∑ n=0 l(n) ∑ i=1 xj,ie n i , j = 1, ...,m, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A RIGID SPACE HOMEOMORPHIC TO HILBERT SPACE 89 where ei represents the characteristic function of i th sub-interval of [n, n+ 1]. For every x ∈ σ, with x = m ∑ j=1 λjUj, λj ≥ 0, j = 1, . . . ,m and m ∑ j=1 λj = 1, we define f(x) = ∞ ∑ n=0 l(n) ∑ i=1 ∣∣∣∣∣ m ∑ j=1 λj |xj,i|τ(xj,i) ∣∣∣∣∣ 1/pn τ i e n i , (2) where τ : R → R is the sign function: τ(t) =  1 if t > 0, 0 if t = 0, −1 if t < 0, (3)