# Isometries of Spaces of Radon Measures

Authors: Marek Wójtowicz
Journal: Journal of Function Spaces
 (2017)

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## Abstract

LetΩ and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m).


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## Body

## 1. Introduction

We use the notation from the abstract andΔ 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-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 [6–8] 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 Ω1, Ω2 of “highly nonhomeomorphic” compact Hausdorff spaces such that the spaces M(Ω1) and M(Ω2) 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 Sections3 and 4.


## 2. Preliminaries

Throughout what followsΩ denotes a compact Hausdorff space with card(Ω)≥2, M(Ω) denotes the space of Radon measures on Ω, and C(Ω) stands for the space of continuous functions on Ω, that is, the predual of M(Ω). By N we denote the discrete space of positive integers.LetF 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+; that is, f1+f2=f1+f2 for all f1,f2≥0. In particular, the classical spaces L1(μ) and M(Ω) are typical AL-spaces [9, Chapter 6]. Moreover, every AL-space is order-isometric to L1(μ) for some measure space (Θ,Σ,μ) [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 T~:F→G of the form T~(f1+if2)=Tf1+iTf2 is an isometry, too [9, p. 139]. This allows us to restrict our considerations to real M(Ω)-spaces and apply the theory of real linear lattices [1–3]. A Banach space X is said to be an L1-predual space if its dual space X∗ is linearly isometric to an AL-space; see [9, Chapter 7].LetF be a real Banach lattice. A linear projection P in F is said to bean order (or a band)projection if 0≤Pf≤f for all f∈F+, and its range, P(F), is calleda projection band. We write F≅G if the Banach lattices F and G are order-isometric. The symbol βΓ denotes the Stone-Čech compactification of a discrete infinite space Γ, and 2m denotes m-copies of the two-element discrete space 2, that is, the m-Cantor cube endowed with the product topology; thus Δ=2ℵ0.The following result is an immediate consequence of [10, Theorem 3.4 and Remark 3.5.(ii)].Lemma 1.
LetF 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≅G.The symbolB(Ω) denotes the σ-algebra of all Borel subsets of a compact space Ω, and B0(Ω) denotes the Baire-subalgebra of B(Ω) generated by the class of Gδ subsets of Ω. In particular, if Ω is metrizable, then B(Ω)=B0(Ω). Two topological spaces U and V are said to beBorel [Baire, resp.] isomorphic if there is a bijection R:U→V such that {R(A):A∈B(U)}=B(V) [resp., {R(A):A∈B0(U)}=B0(V)]. If D∈B(Ω), then D∩B(Ω) denotes the class {D∩A:A∈B}.The next result is proved on page 177 in [9].Lemma 2.
LetG denote the unit circle in the plane endowed with its natural topology. For every infinite discrete space Γ with m=card(Γ), the spaces M(βΓ) and M(G2m) 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 thattwo compact metrizable spacesΩ1,Ω2are Baire isomorphic if and only if(1)card⁡Ω1=card⁡Ω2,whence the product of m≥ℵ0 compact metrizable spaces is Baire isomorphic both to the Cantor cube 2m and to Im. Hence, by [13, Theorem H, p. 229] and [13, Theorem D, p. 239], we obtain the following.Proposition A.
LetΩ be a compact metrizable space.(a)
IfΩ is uncountable, then M(Ω) is order-isometric both to M(I) and to M(Δ).(b)
If(Ωt)t∈θ is a family of compact metrizable spaces with card⁡θ=m≥ℵ0, then(2)M∏t∈θΩt≅MIm≅M2m.In particular, each of the spacesM(Ωm), M(Ω×Im), M(Ω×2m), and M(Im) is order-isometric to M(2m).It is also known that ifΩ is a compact metrizable anduncountable space, then, by Milutin’s result [5, Theorem 21.5.10], the spaces of continuous functions C(Ω) 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 thatifΩis a scattered compact space (i.e., every nonempty closed subset of Ω has an isolated point), then M(Ω) consists of atomic measures only; that is, M(Ω)is order-isometric tol1(Ω). The examples of scattered spaces are furnished, for example, by order intervals [5, pp. 151–156], Mrówka 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.
LetΩ1, Ω2 be two infinite compact scattered spaces. Then condition (1) implies that M(Ω1)≅M(Ω2).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.
LetΩ1, Ω2 be two compact Hausdorff spaces. If(i)
Ω1 and Ω2 are each homeomorphic to a closed subset of the other space, or(ii)
Ω1 and Ω2 are each continuously mapped onto the other space and they are extremally disconnected;
thenM(Ω1)≅M(Ω2).Proof.
Part (i). Let Ω1 be a closed subspace of Ω2. We shall show that M(Ω1) is order-isometric to a projection band of M(Ω2). The formula (3)PμB≔μB∩Ω1,B∈BΩ2defines an order projection P from M(Ω2) onto the projection band MΩ1(Ω2) of all Radon measures concentrated on Ω1. Since B(Ω1)=Ω1∩B(Ω2), the mapping MΩ1(Ω2)∋ν→ν|Ω1 is an order-isometry from MΩ1(Ω2) onto M(Ω2). Similarly, M(Ω2) is order-isometric to a projection band in M(Ω1). By Lemma 1, we obtain that M(Ω1)≅M(Ω2).
Part (ii). Observe that if ϕ maps Ω1 onto Ω2, then there is a closed subset V of Ω1 such that the restriction ϕ|V is a homeomorphism from V onto Ω2 (see, e.g., [5, Proposition 7.1.13 and Theorem 24.2.10]). Now we apply part (i).Corollary 4.
LetΩ be a compact space, and let Γ be an infinite discrete space with m=card⁡Γ. Then M(βΓ)≅M(Ωm)≅M(I2m).Proof.
This is an immediate consequence of Lemma2 and part (b) of Proposition A.Corollary 5.
LetΩ be a metrizable compact space, and let K denote an infinite closed subspace of βN. Then M(K)≅M(βN)≅M(Ω2ℵ0)≅M(I2ℵ0).Proof.
SinceK contains a homeomorphic copy of βN [19, p. 71], from part (i) of the Theorem we obtain that M(K)≅M(βN). The remaining order isometries follow from Corollary 4.The corollary below is a generalization of part (b) of Proposition A.Corollary 6.
LetΩ be a compact space of weight m≥ℵ0. If Ω contains a homeomorphic copy of Im, or Ω is 0-dimensional and contains a homeomorphic copy of 2m, then M(Ω)≅M(2m).
In particular, ifK is a compact Hausdorff space with weight ≤m, then M(K×Im)≅M(K×2m)≅M(2m).Proof.
The first part follows from the universality of the Tychonoff and Cantor cubes, Theorem3(i) and Proposition A(b).IfΩ is compact and of weight ≤m, the product Ω×Im is of weight m; thus M(Ω×Im)≅M(2m) by the first part. Moreover, because Im and 2m are Baire isomorphic, the products Ω×Im and Ω×2m are Baire isomorphic, too. Hence MΩ×Im≅M(Ω×2m).


## 4. Examples

The examples presented in this section are motivated by the result stated in [9, Exercise 5]:IfXandYareL1-predual spaces such that each of them is linearly isomorphically embedded into the other space, then their dual spaces,X∗andY∗, are linearly isomorphic. We shall show by examples the possibility of X∗ and Y∗ 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 Pełczyński’s observation [5, pp. 366-367] that if Γ is an uncountable set then there is no isometric embedding of C(βΓ) into C(Im) for every infinite cardinal m.Lemma 7.
LetΩ denote either one of the spaces: N∗≔βN∖N or βΓ, where Γ is an uncountable set. Then there is no linear isomorphic embedding of C(Ω) into C(Im), for every infinite cardinal number m.Proof.
We follow an idea of the proof of the above-mentioned Pełczyński’s result. By the remark of Pełczyński [5, p. 367], the space C(Im) has an equivalent strictly convex norm. By Partington’s results [20, 21], in each of the either cases, C(Ω) contains an isometric copy of l∞. Hence, C(Ω) cannot be embedded into C(Im).Example 8.
LetΓ be a discrete space with card⁡Γ=m>ℵ0. From Corollary 4 we know that the spaces M(βΓ) and M(I2m) are order-isometric.
On the other hand, by Lemma7, there is no linear isomorphic embedding from C(βΓ) into C(I2m).In the next example we show that a similar property holds for the pairM(N∗) and M(I2ℵ0).Example 9.
By Corollary5, the spaces M(N∗) and M(I2ℵ0) are order-isometric, yet, by Lemma 7, C(N∗) does not embed linearly isomorphically into C(I2ℵ0).



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