Abstract
Let and denote a compact metrizable space with and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces of Radon measures on compact Hausdorff spaces . In particular, we obtain the following results: () for every infinite closed subset of the spaces , , and are order-isometric; () for every discrete space with the spaces and are order-isometric, whereas there is no linear homeomorphic injection from into .
1. Introduction
We use the notation from the abstract and 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 between two Banach lattices and is said to be an order isomorphism [order-isometry, resp.] if both and are order-preserving [with an isometry, resp.]; by [1, Theorem 16.6], in the general case, 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 , of “highly nonhomeomorphic” compact Hausdorff spaces such that the spaces and 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 denotes a compact Hausdorff space with , denotes the space of Radon measures on , and stands for the space of continuous functions on , that is, the predual of . By we denote the discrete space of positive integers.
Let and be two (real or complex) Banach lattices. The lattice is said to be an -lattice if its norm is additive on ; that is, for all . In particular, the classical spaces and are typical -spaces [9, Chapter 6]. Moreover, every -space is order-isometric to for some measure space [9, Theorem on p. 135]. If is an order-isometry between the real parts of two given -spaces and , then the extended operator of the form is an isometry, too [9, p. 139]. This allows us to restrict our considerations to real -spaces and apply the theory of real linear lattices [1–3]. A Banach space is said to be an -predual space if its dual space is linearly isometric to an -space; see [9, Chapter 7].
Let be a real Banach lattice. A linear projection in is said to be an order (or a band) projection if for all , and its range, , is called a projection band. We write if the Banach lattices and are order-isometric. The symbol denotes the Stone-Čech compactification of a discrete infinite space , and denotes -copies of the two-element discrete space , that is, the -Cantor cube endowed with the product topology; thus .
The following result is an immediate consequence of [10, Theorem 3.4 and Remark 3.5.(ii)].
Lemma 1. Let and be two real -lattices. If and are each order-isometric to a projection band of the other space, then .
The symbol denotes the -algebra of all Borel subsets of a compact space , and denotes the Baire-subalgebra of generated by the class of subsets of . In particular, if is metrizable, then . Two topological spaces and are said to be Borel [Baire, resp.] isomorphic if there is a bijection such that [resp., ]. If , then denotes the class .
The next result is proved on page 177 in [9].
Lemma 2. Let denote the unit circle in the plane endowed with its natural topology. For every infinite discrete space with , the spaces and 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 , are Baire isomorphic if and only ifwhence the product of compact metrizable spaces is Baire isomorphic both to the Cantor cube and to . Hence, by [13, Theorem H, p. 229] and [13, Theorem D, p. 239], we obtain the following.
Proposition A. Let be a compact metrizable space.(a)If is uncountable, then is order-isometric both to and to .(b)If is a family of compact metrizable spaces with , then In particular, each of the spaces , , , and is order-isometric to .
It is also known that if is a compact metrizable and uncountable space, then, by Milutin’s result [5, Theorem 21.5.10], the spaces of continuous functions and 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 ] says that if is a scattered compact space (i.e., every nonempty closed subset of has an isolated point), then consists of atomic measures only; that is, is order-isometric to . 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. Let , be two infinite compact scattered spaces. Then condition (1) implies that .
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 , be two compact Hausdorff spaces. If(i) and are each homeomorphic to a closed subset of the other space, or(ii) and are each continuously mapped onto the other space and they are extremally disconnected; then .
Proof.
Part (i). Let be a closed subspace of . We shall show that is order-isometric to a projection band of . The formula defines an order projection from onto the projection band of all Radon measures concentrated on . Since , the mapping is an order-isometry from onto . Similarly, is order-isometric to a projection band in . By Lemma 1, we obtain that .
Part (ii). Observe that if maps onto , then there is a closed subset of such that the restriction is a homeomorphism from onto (see, e.g., [5, Proposition 7.1.13 and Theorem 24.2.10]). Now we apply part (i).
Corollary 4. Let be a compact space, and let be an infinite discrete space with . Then .
Proof. This is an immediate consequence of Lemma 2 and part (b) of Proposition A.
Corollary 5. Let be a metrizable compact space, and let denote an infinite closed subspace of . Then .
Proof. Since contains a homeomorphic copy of [19, p. 71], from part (i) of the Theorem we obtain that . The remaining order isometries follow from Corollary 4.
The corollary below is a generalization of part (b) of Proposition A.
Corollary 6. Let be a compact space of weight . If contains a homeomorphic copy of , or is 0-dimensional and contains a homeomorphic copy of , then .
In particular, if is a compact Hausdorff space with weight , then .
Proof. The first part follows from the universality of the Tychonoff and Cantor cubes, Theorem 3(i) and Proposition A(b).
If is compact and of weight ≤, the product is of weight ; thus by the first part. Moreover, because and are Baire isomorphic, the products and are Baire isomorphic, too. Hence .
4. Examples
The examples presented in this section are motivated by the result stated in [9, Exercise 5]: If and are -predual spaces such that each of them is linearly isomorphically embedded into the other space, then their dual spaces, and , are linearly isomorphic. We shall show by examples the possibility of and 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 is an uncountable set then there is no isometric embedding of into for every infinite cardinal .
Lemma 7. Let denote either one of the spaces: or , where is an uncountable set. Then there is no linear isomorphic embedding of into , for every infinite cardinal number .
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 has an equivalent strictly convex norm. By Partington’s results [20, 21], in each of the either cases, contains an isometric copy of . Hence, cannot be embedded into .
Example 8. Let be a discrete space with . From Corollary 4 we know that the spaces and are order-isometric.
On the other hand, by Lemma 7, there is no linear isomorphic embedding from into .
In the next example we show that a similar property holds for the pair and .
Example 9. By Corollary 5, the spaces and are order-isometric, yet, by Lemma 7, does not embed linearly isomorphically into .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.