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Isometries of Spaces of Radon Measures.

1. Introduction

We use the notation from the abstract and A denotes the Cantor set endowed with the standard product topology. Throughout this paper, we also assume that every topological space is of cardinality at least 2. For notions and notations undefined here we refer the reader to the monographs [1-5]. Let us recall that a linear injective operator T between two Banach lattices X and Y is said to be an order isomorphism [order-isometry, resp.] if both T and [T.sup.-1] are order-preserving [with T an isometry, resp.]; by [1, Theorem 16.6], in the general case, T is continuous.

In this paper, we deal with surjective isometries of spaces of Radon measures defined on compact Hausdorff spaces. In the monographs and survey papers devoted to isometries on function spaces this topic is either completely overlooked [68] or treated marginally [9, Theorem 7 on p. 177, Exercises 4-7 on p. 229]; cf. [9, pp. 181, 226-227]. Our aim is to show that, in many cases, one can indicate pairs [[OMEGA].sub.1], [[OMEGA].sub.2] of "highly nonhomeomorphic" compact Hausdorff spaces such that the spaces M([[OMEGA].sub.1]) and M([[OMEGA].sub.2]) are order-isometric. It is a little surprising that here a number of results can be obtained immediately by means of a cardinality argument (Propositions A and B below), yet there are cases requiring more advanced knowledge.

In the next section, we list basic notions and results concerning Riesz spaces, that is, linear lattices, which will be applied in proofs of our main results, given in Sections 3 and 4.

2. Preliminaries

Throughout what follows [OMEGA] denotes a compact Hausdorff space with card([OMEGA]) [greater than or equal to] 2, M([OMEGA]) denotes the space of Radon measures on [OMEGA], and C([OMEGA]) stands for the space of continuous functions on [OMEGA], that is, the predual of M([OMEGA]). By N we denote the discrete space of positive integers.

Let F and G be two (real or complex) Banach lattices. The lattice F is said to be an AL-lattice if its norm is additive on [F.sup.+]; that is, [mathematical expression not reproducible] for all [f.sub.1], [f.sub.2] [greater than or equal to] 0. In particular, the classical spaces [L.sub.1]([mu]) and M([OMEGA]) are typical AL-spaces [9, Chapter 6]. Moreover, every AL-space is order-isometric to [L.sub.1]([mu]) for some measure space ([THETA], [summation], [mu]) [9, Theorem 3 on p. 135]. If T is an order-isometry between the real parts of two given AL-spaces F and G, then the extended operator [??] : F [right arrow] G of the form [??]([f.sub.1] + i[f.sub.2]) = T[f.sub.1] + iT[f.sub.2] is an isometry, too [9, p. 139]. This allows us to restrict our considerations to real M([OMEGA])-spaces and apply the theory of real linear lattices [1-3]. A Banach space X is said to be an [L.sub.1]-predual space if its dual space [X.sup.*] is linearly isometric to an AL-space; see [9, Chapter 7].

Let F be a real Banach lattice. A linear projection P in F is said to be an order (or a band) projection if 0 [less than or equal to] Pf [less than or equal to] f for all f [member of] [F.sup.+], and its range, P(F), is called a projection band. We write F [congruent to] G if the Banach lattices F and G are order-isometric. The symbol [beta][GAMMA] denotes the Stone-Cech compactification of a discrete infinite space [GAMMA], and [2.sup.m] denotes m-copies of the two- element discrete space 2, that is, the m-Cantor cube endowed with the product topology; thus [mathematical expression not reproducible].

The following result is an immediate consequence of [10, Theorem 3.4 and Remark 3.5.(ii)].

Lemma 1. Let F and G be two real AL-lattices. If F and G are each order-isometric to a projection band of the other space, then F [congruent to] G.

The symbol [beta]([OMEGA]) denotes the u-algebra of all Borel subsets of a compact space [OMEGA], and [B.sub.0] ([OMEGA]) denotes the Bairesubalgebra of B([OMEGA]) generated by the class of [G.sub.[delta]] subsets of [OMEGA]. In particular, if [OMEGA] is metrizable, then B([OMEGA]) = [B.sub.0]([OMEGA]). Two topological spaces U and V are said to be Borel [Baire, resp.] isomorphic if there is a bijection R:U [right arrow]V such that {R(A) : A [member of] B(U)} = B(V) [resp., {R(A) : A [member of] [B.sub.0](U)} = [B.sub.0](V)]. If D [member of] B([OMEGA]), then D [intersection] B([OMEGA]) denotes the class {D [intersection] A : A [member of] B}.

The next result is proved on page 177 in [9].

Lemma 2. Let G denote the unit circle in the plane endowed with its natural topology. For every infinite discrete space r with m = card([GAMMA]), the spaces [mathematical expression not reproducible] are order-isometric.

3. The Results

It is well known that two Polish spaces are Borel isomorphic if and only if they have the same cardinality ([11, p. 451] or [12, p. 38]). It follows that two compact metrizable spaces [[OMEGA].sub.1], [[OMEGA].sub.2] are Baire isomorphic if and only if

card ([[OMEGA].sub.1]) = card ([[OMEGA].sub.2]), (1)

whence the product of m [greater than or equal to] [N.sub.0] compact metrizable spaces is Baire isomorphic both to the Cantor cube [2.sup.m] and to [I.sup.m]. Hence, by [13, Theorem H, p. 229] and [13, Theorem D, p. 239], we obtain the following.

Proposition A. Let [OMEGA] be a compact metrizable space.

(a) If [OMEGA] is uncountable, then M([OMEGA]) is order-isometric both to M(I) and to M([DELTA]).

(b) If [([[OMEGA].sub.t]).sub.t[member of][theta]] is a family of compact metrizable spaces with card [theta] = m [greater than or equal to] [N.sub.0], then

[mathematical expression not reproducible]. (2)

In particular, each of the spaces M([[OMEGA].sup.m]), M([OMEGA] x [I.sup.m]), M([OMEGA] x [2.sup.m]), and M([I.sup.m]) is order-isometric to M([2.sup.m]).

It is also known that if [OMEGA] is a compact metrizable and uncountable space, then, by Milutin's result [5, Theorem 21.5.10], the spaces of continuous functions C([OMEGA]) and C(I) are linearly homeomorphic yet non-order-isomorphic [5, Theorem 7.8.1], in general. Thus Proposition A may be considered as a dual version of Milutin's theorem, essentially stronger than the classical one.

Let us notice that condition (1) applies also for spaces of Radon measures built on scattered spaces. The classical result [5, Corollary 19.7.7] says that if [OMEGA] is a scattered compact space (i.e., every nonempty closed subset of [OMEGA] has an isolated point), then M([OMEGA]) consists of atomic measures only; that is, M([OMEGA]) is order-isometric to [l.sub.1]([OMEGA]). The examples of scattered spaces are furnished, for example, by order intervals [5, pp. 151-156], Mrowka spaces [14, 15] (cf. [16, Section 3]), and Stone spaces of superatomic Boolean algebras [17, Theorem, p. 1146], [18]. Hence we obtain the following complement to part (a) of Proposition A.

Proposition B. Let [[OMEGA].sub.1], [[OMEGA].sub.2] be two infinite compact scattered spaces. Then condition (1) implies that M([[OMEGA].sub.1]) [congruent to] M([[OMEGA].sub.2]).

In this paper, we shall prove and apply the following theorem which, by the above results, is essential in the class of compact Hausdorff spaces nonhomeomorphic either to products of metrizable spaces or to scattered spaces.

Theorem 3. Let [[OMEGA].sub.1], [[OMEGA].sub.2] be two compact Hausdorff spaces. If

(i) [[OMEGA].sub.1] and [[OMEGA].sub.2] are each homeomorphic to a closed subset of the other space, or

(ii) [[OMEGA].sub.1] and [[OMEGA].sub.2] are each continuously mapped onto the other space and they are extremally disconnected;

then M([[OMEGA].sub.1]) [congruent to] M([[OMEGA].sub.2]).

Proof.

Part (i). Let [[OMEGA].sub.1] be a closed subspace of [[OMEGA].sub.2]. We shall show that M([[OMEGA].sub.1]) is order-isometric to a projection band of M([[OMEGA].sub.2]). The formula

(P[mu]) (B) := [mu](B [intersection] [[OMEGA].sub.1]), B [member of] B ([[OMEGA].sub.2]) (3)

defines an order projection P from M([[OMEGA].sub.2]) onto the projection band [mathematical expression not reproducible] of all Radon measures concentrated on [[OMEGA].sub.1]. Since B([[OMEGA].sub.1]) = [[OMEGA].sub.1] [intersection] B([[OMEGA].sub.2]), the mapping [mathematical expression not reproducible] is an order-isometry from [mathematical expression not reproducible]. Similarly, M([[OMEGA].sub.2]) is order-isometric to a projection band in M([[OMEGA].sub.1]). By Lemma 1, we obtain that M([[OMEGA].sub.1]) [congruent to] M([[OMEGA].sub.2]).

Part (ii). Observe that if [phi] maps [[OMEGA].sub.1] onto [[OMEGA].sub.2], then there is a closed subset V of [[OMEGA].sub.1] such that the restriction [[phi].sub.|V] is a homeomorphism from V onto [[OMEGA].sub.2] (see, e.g., [5, Proposition 7.1.13 and Theorem 24.2.10]). Now we apply part (i).

Corollary 4. Let [OMEGA] be a compact space, and let [GAMMA] be an infinite discrete space with m = card [GAMMA]. Then [mathematical expression not reproducible].

Proof. This is an immediate consequence of Lemma 2 and part (b) of Proposition A.

Corollary 5. Let [OMEGA] be a metrizable compact space, and let K denote an infinite closed subspace of [beta]N. Then [mathematical expression not reproducible].

Proof. Since K contains a homeomorphic copy of [beta]N [19, p. 71], from part (i) of the Theorem we obtain that M(K) [congruent to] M([beta]N). The remaining order isometries follow from Corollary 4.

The corollary below is a generalization of part (b) of Proposition A.

Corollary 6. Let [OMEGA] be a compact space of weight m [greater than or equal to] [N.sub.0]. If [OMEGA] contains a homeomorphic copy of [I.sub.m], or [OMEGA] is 0-dimensional and contains a homeomorphic copy of [2.sup.m], then M([OMEGA]) [congruent to] M([2.sup.m]).

In particular, if K is a compact Hausdorff space with weight [less than or equal to] m, then M(K x [I.sup.m]) [congruent to] M(K x [2.sup.m]) = M([2.sup.m]).

Proof. The first part follows from the universality of the Tychonoff and Cantor cubes, Theorem 3(i) and Proposition A(b).

If [OMEGA] is compact and of weight [less than or equal to] m, the product [OMEGA] x [I.sup.m] is of weight m; thus M([OMEGA] x [I.sup.m]) = M([2.sup.m]) by the first part. Moreover, because [I.sup.m] and [2.sup.m] are Baire isomorphic, the products [OMEGA] x [I.sup.m] and [OMEGA] x [2.sup.m] are Baire isomorphic, too. Hence M([OMEGA] x [I.sup.m]) = M([OMEGA] x [2.sup.m]).

4. Examples

The examples presented in this section are motivated by the result stated in [9, Exercise 5]: If X and Y are [L.sub.1]-predual spaces such that each of them is linearly isomorphically embedded into the other space, then their dual spaces, [X.sup.*] and [Y.sup.*], are linearly isomorphic. We shall show by examples the possibility of [X.sup.*] and [Y.sup.*] being order-isometric with no linear isomorphic embedding of either of these spaces into the other one. To this end, we shall apply the following strengthening of Pelczynski's observation [5, pp. 366-367] that if [GAMMA] is an uncountable set then there is no isometric embedding of C([beta][GAMMA]) into C([I.sup.m]) for every infinite cardinal m.

Lemma 7. Let [OMEGA] denote either one of the spaces: [N.sup.*] = [beta]N \ N or [beta][GAMMA], where [GAMMA] is an uncountable set. Then there is no linear isomorphic embedding of C([OMEGA]) into C([I.sup.m]), for every infinite cardinal number m.

Proof. We follow an idea of the proof of the above-mentioned Pelczynski's result. By the remark of Pelczynski [5, p. 367], the space C([I.sup.m]) has an equivalent strictly convex norm. By Partington's results [20, 21], in each of the either cases, C([OMEGA]) contains an isometric copy of [l.sub.[infinity]]. Hence, C([OMEGA]) cannot be embedded into C([I.sup.m]).

Example 8. Let [GAMMA] be a discrete space with card [GAMMA] = m > [N.sub.0]. From Corollary 4 we know that the spaces [mathematical expression not reproducible] are order-isometric.

On the other hand, by Lemma 7, there is no linear isomorphic embedding from C([beta][GAMMA]) into [mathematical expression not reproducible].

In the next example we show that a similar property holds for the pair [mathematical expression not reproducible].

Example 9. By Corollary 5, the spaces [mathematical expression not reproducible] are order-isometric, yet, by Lemma 7, C([N.sup.*]) does not embed linearly isomorphically into [mathematical expression not reproducible].

https://doi.org/10.1155/2017/3850817

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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Marek Wojtowicz

Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, Pl. Weyssenhoffa 11, 85- 072 Bydgoszcz, Poland

Correspondence should be addressed to Marek Wojtowicz; mwojt@ukw.edu.pl

Received 2 May 2017; Accepted 5 June 2017; Published 4 July 2017

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Title Annotation:Research Article
Author:Wojtowicz, Marek
Publication:Journal of Function Spaces
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Date:Jan 1, 2017
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