Isometries of Spaces of Radon Measures

Hindawi Journal of Function Spaces Volume 2017, Article ID 3850817, 4 pages https://doi.org/10.1155/2017/3850817

Research Article Isometries of Spaces of Radon Measures

Marek Wójtowicz

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

Correspondence should be addressed to Marek Wojtowicz;´ [email protected]

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

Academic Editor: Adrian Petrusel

Copyright © 2017 Marek Wojtowicz.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let Ω 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 2ℵ0 following results: (1) for every infinite closed subset 𝐾 of 𝛽N the spaces M(𝐾), M(𝛽N),andM(Ω ) are order-isometric; (2) 2m for every discrete space Γ with m ﬂ card(Γ) > ℵ0 the spaces M(𝛽Γ) and M(𝐼 ) are order-isometric, whereas there is no linear 2m homeomorphic injection from 𝐶(𝛽𝑇) into 𝐶(𝐼 ).

1. Introduction appliedinproofsofourmainresults,giveninSections3and 4. We use the notation from the abstract and Δ denotes the Cantor set endowed with the standard product topology. 2. Preliminaries Throughout this paper, we also assume that every topological space is of cardinality at least 2. For notions and notations Throughout what follows Ω denotes a compact Hausdorff undefined here we refer the reader to the monographs [1–5]. space with card(Ω) ≥ 2, M(Ω) denotes the space of Radon Let us recall that a linear injective operator 𝑇 between two measures on Ω,and𝐶(Ω) stands for the space of continuous Banach lattices 𝑋 and 𝑌 is said to be an order isomorphism Ω M(Ω) N −1 functions on ,thatis,thepredualof .By we denote [order-isometry, resp.] if both 𝑇 and 𝑇 are order-preserving the discrete space of positive integers. [with 𝑇 an isometry, resp.]; by [1, Theorem 16.6], in the Let 𝐹 and 𝐺 be two (real or complex) Banach lattices. The + general case, 𝑇 is continuous. lattice 𝐹 is said to be an 𝐴𝐿-latticeifitsnormisadditiveon𝐹 ; In this paper, we deal with surjective isometries of spaces that is, ‖𝑓1 +𝑓2‖=‖𝑓1‖+‖𝑓2‖ for all 𝑓1,𝑓2 ≥0.Inparticular, of Radon measures defined on compact Hausdorff spaces. In the classical spaces 𝐿1(𝜇) and M(Ω) are typical 𝐴𝐿-spaces the monographs and survey papers devoted to isometries on [9, Chapter 6]. Moreover, every 𝐴𝐿-space is order-isometric function spaces this topic is either completely overlooked [6– to 𝐿1(𝜇) for some measure space (Θ,Σ,𝜇)[9, Theorem 3 on 8] or treated marginally [9, Theorem 7 on p. 177, Exercises p. 135]. If 𝑇 is an order-isometry between the real parts of 4–7onp.229];cf.[9,pp.181,226-227].Ouraimisto two given 𝐴𝐿-spaces 𝐹 and 𝐺, then the extended operator Ω Ω ̃ ̃ show that, in many cases, one can indicate pairs 1, 2 of 𝑇:𝐹→𝐺of the form 𝑇(𝑓1 +𝑖𝑓2)=𝑇𝑓1 +𝑖𝑇𝑓2 is an isometry, “highly nonhomeomorphic” compact Hausdorff spaces such too [9, p. 139]. This allows us to restrict our considerations to that the spaces M(Ω1) and M(Ω2) are order-isometric. It real M(Ω)-spacesandapplythetheoryofreallinearlattices is a little surprising that here a number of results can be [1–3]. A Banach space 𝑋 is said to be an 𝐿1-predual space if ∗ obtained immediately by means of a cardinality argument its dual space 𝑋 is linearly isometric to an 𝐴𝐿-space; see [9, (Propositions A and B below), yet there are cases requiring Chapter 7]. more advanced knowledge. Let 𝐹 bearealBanachlattice.Alinearprojection𝑃 in 𝐹 In the next section, we list basic notions and results is said to be an order (or a band) projection if 0≤𝑃𝑓≤𝑓for + concerning Riesz spaces, that is, linear lattices, which will be all 𝑓∈𝐹 ,anditsrange,𝑃(𝐹), is called aprojectionband.We 2 Journal of Function Spaces write 𝐹≅𝐺if the Banach lattices 𝐹 and 𝐺 are order-isometric. considered as a dual version of Milutin’s theorem, essentially The symbol 𝛽Γ denotes the Stone-Cechˇ compactification of a stronger than the classical one. m discrete infinite space Γ,and2 denotes m-copies of the two- Let us notice that condition (1) applies also for spaces element discrete space 2,thatis,the𝑚-Cantor cube endowed ℵ of Radon measures built on scattered spaces. The classical with the product topology; thus Δ=2 0 . result [5, Corollary 19.7.7]saysthatif Ω is a scattered compact The following result is an immediate consequence of [10, space (i.e., every nonempty closed subset of Ω has an isolated Theorem 3.4 and Remark 3.5.(ii)]. point), then M(Ω) consists of atomic measures only; that is, M(Ω) is order-isometric to ℓ1(Ω). The examples of scattered Lemma 1. Let 𝐹 and 𝐺 be two real 𝐴𝐿-lattices. If 𝐹 and 𝐺 are spaces are furnished, for example, by order intervals [5, pp. each order-isometric to a projection band of the other space, 151–156], Mrowka´ spaces [14, 15] (cf. [16, Section 3]), and then 𝐹≅𝐺. Stone spaces of superatomic Boolean algebras [17, Theorem, p. 1146], [18]. Hence we obtain the following complement to B(Ω) 𝜎 The symbol denotes the -algebra of all Borel part (a) of Proposition A. subsets of a compact space Ω,andB0(Ω) denotes the Baire- B(Ω) 𝐺 Ω subalgebra of generated by the class of 𝛿 subsets of . Proposition B. Let Ω1, Ω2 be two infinite compact scattered Ω B(Ω) = B (Ω) In particular, if is metrizable, then 0 .Two spaces. Then condition (1) implies that M(Ω1)≅M(Ω2). topological spaces 𝑈 and 𝑉 are said to be Borel [Baire,resp.] isomorphic if there is a bijection 𝑅:𝑈→𝑉such that {𝑅(𝐴) : In this paper, we shall prove and apply the following 𝐴∈B(𝑈)} = B(𝑉) [resp., {𝑅(𝐴) : 𝐴∈ B0(𝑈)} = B0(𝑉)]. If theorem which, by the above results, is essential in the class 𝐷∈B(Ω),then𝐷∩B(Ω) denotes the class {𝐷∩𝐴 : 𝐴∈ B}. of compact Hausdorff spaces nonhomeomorphic either to The next result is proved on page 177 in [9]. products of metrizable spaces or to scattered spaces.

Lemma 2. Let 𝐺 denotetheunitcircleintheplaneendowed Theorem 3. Let Ω1, Ω2 be two compact Hausdorff spaces. If with its natural topology. For every infinite discrete space Γ 2m (i) Ω1 and Ω2 are each homeomorphic to a closed subset with m = card(Γ),thespacesM(𝛽Γ) and M(𝐺 ) are order- of the other space, or isometric. (ii) Ω1 and Ω2 are each continuously mapped onto the other space and they are extremally disconnected; 3. The Results then M(Ω1)≅M(Ω2). 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, Proof. p. 38]). It follows that two compact metrizable spaces Ω1, Ω2 Part (i). Let Ω1 beaclosedsubspaceofΩ2.Weshallshowthat are Baire isomorphic if and only if M(Ω1) is order-isometric to a projection band of M(Ω2).The card (Ω1)=card (Ω2), (1) formula

(𝑃𝜇) (𝐵) ﬂ 𝜇(𝐵∩Ω1), 𝐵∈B (Ω2) (3) whence the product of m ≥ℵ0 compact metrizable spaces m m is Baire isomorphic both to the Cantor cube 2 and to 𝐼 . defines an order projection 𝑃 from M(Ω2) onto the projec- Hence,by[13,TheoremH,p.229]and[13,TheoremD,p.239], tion band MΩ (Ω2) of all Radon measures concentrated on we obtain the following. Ω B(Ω1 )=Ω ∩ B(Ω ) M (Ω )∋ 1.Since 1 1 2 ,themapping Ω1 2 ] → ]|Ω is an order-isometry from MΩ (Ω2) onto M(Ω2). Proposition A. Let Ω be a compact metrizable space. 1 1 Similarly, M(Ω2) is order-isometric to a projection band in (a) If Ω is uncountable, then M(Ω) is order-isometric both M(Ω1). By Lemma 1, we obtain that M(Ω1)≅M(Ω2). to M(𝐼) and to M(Δ). Part (ii).Observethatif𝜙 maps Ω1 onto Ω2, then there is (b) If (Ω𝑡)𝑡∈𝜃 is a family of compact metrizable spaces with a closed subset 𝑉 of Ω1 such that the restriction 𝜙|𝑉 is a card 𝜃=m ≥ℵ0,then homeomorphism from 𝑉 onto Ω2 (see, e.g., [5, Proposition 7.1.13 and Theorem 24.2.10]). Now we apply part (i). m m M (∏Ω𝑡)≅M (𝐼 )≅M (2 ). (2) 𝑡∈𝜃 Corollary 4. Let Ω be a compact space, and let Γ be an infinite m discrete space with m = card Γ.ThenM(𝛽Γ) ≅ M(Ω )≅ m m m 2 In particular, each of the spaces M(Ω ), M(Ω × 𝐼 ), M(𝐼 ). m m m M(Ω × 2 ),andM(𝐼 ) is order-isometric to M(2 ). Proof. ThisisanimmediateconsequenceofLemma2and It is also known that if Ω is a compact metrizable and part (b) of Proposition A. uncountable space, then, by Milutin’s result [5, Theorem Corollary 5. Ω 21.5.10], the spaces of continuous functions 𝐶(Ω) and 𝐶(𝐼) Let be a metrizable compact space, and let 𝐾 𝛽N M(𝐾) ≅ are linearly homeomorphic yet non-order-isomorphic [5, denote an infinite closed subspace of .Then 2ℵ0 2ℵ0 Theorem 7.8.1], in general. Thus Proposition A may be M(𝛽N)≅M(Ω )≅M(𝐼 ). Journal of Function Spaces 3

∗ 2ℵ0 Proof. Since 𝐾 contains a homeomorphic copy of 𝛽N [19, p. Example 9. By Corollary 5, the spaces M(N ) and M(𝐼 ) ∗ 71], from part (i) of the Theorem we obtain that M(𝐾) ≅ are order-isometric, yet, by Lemma 7, 𝐶(N ) does not embed M(𝛽N) 2ℵ0 . The remaining order isometries follow from Corol- linearly isomorphically into 𝐶(𝐼 ). lary 4.

The corollary below is a generalization of part (b)of Conflicts of Interest Proposition A. The author declares that there are no conflicts of interest

Corollary 6. Let Ω be a compact space of weight m ≥ℵ0.If regarding the publication of this paper. 𝑚 Ω contains a homeomorphic copy of 𝐼 ,orΩ is 0-dimensional m and contains a homeomorphic copy of 2 ,thenM(Ω) ≅ m References M(2 ). In particular, if 𝐾 is a compact Hausdorff space with weight [1] C. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces, m m m ≤ m,thenM(𝐾 × 𝐼 )≅M(𝐾 × 2 )≅M(2 ). Academic Press, New York, NY, USA, 1978. [2]W.A.LuxemburgandA.C.Zaanen,Riesz Spaces. Vol. I,North- Proof. The first part follows from the universality of the Holland Publishing Co., Amsterdam, Netherlands, 1971. Tychonoff and Cantor cubes, Theorem 3(i) and Proposition [3] P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, New A(b). York, NY, USA, 1991. [4] H. H. Schaefer, Banach Lattices and Positive Operators,Springer, m If Ω is compact and of weight ≤ m,theproductΩ×𝐼 Berlin, Germany, 1974. m m is of weight m;thusM(Ω × 𝐼 )≅M(2 ) by the first [5] Z. Semadeni, Banach Spaces of Continuous Functions. Vol. I, m m part. Moreover, because 𝐼 and 2 are Baire isomorphic, the Polish Scientific Publishers, Warsaw, Poland, 1971. m m products Ω×𝐼 and Ω×2 are Baire isomorphic, too. Hence [6] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: m m M(Ω × 𝐼 )≅M(Ω×2 ). Function Spaces,vol.129ofMonographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. 4. Examples [7] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces. The examples presented in this section are motivated by Vol. 2,vol.138ofMonographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, the result stated in [9, Exercise 5]: If 𝑋 and 𝑌 are 𝐿1- 2008. predual spaces such that each of them is linearly isomorphically ∗ [8] A. Koldobsky and H. Konig,¨ “Aspects of the isometric theory of embedded into the other space, then their dual spaces, 𝑋 and ∗ BANach spaces,”in Handbook of the geometry of BANach spaces, 𝑌 , are linearly isomorphic. We shall show by examples the ∗ ∗ Vol. I,pp.899–939,Elsevier,Amsterdam,Netherlands,2001. possibility of 𝑋 and 𝑌 being order-isometric with no linear [9]H.E.Lacey,The Isometric Theory of Classical Banach Spaces, isomorphic embedding of either of these spaces into the other Springer, Berlin, Germany, 1974. one. To this end, we shall apply the following strengthening Γ [10] M. Wojtowicz,´ “On Cantor-Bernstein type theorems in Riesz of Pełczynski’s´ observation [5, pp. 366-367] that if is an spaces,” Indagationes Mathematicae,vol.50,no.1,pp.93–100, uncountable set then there is no isometric embedding of 𝐶(𝛽Γ) 𝐶(𝐼m) m 1988. into for every infinite cardinal . [11] K. Kuratowski, Topology. Vol. II, Polish Scientific Publishers, ∗ Warszawa, Poland, Academic Press, New York, NY, USA, 1976. Lemma 7. Let Ω denote either one of the spaces: 𝑁 ﬂ 𝛽N\N or 𝛽Γ,whereΓ is an uncountable set. Then there is no linear [12] J. P. Christensen, Topology and Borel Structure,North-Holland m Publishing Company, Amsterdam, Netherlands, 1974. isomorphic embedding of 𝐶(Ω) into 𝐶(𝐼 ), for every infinite cardinal number m. [13] P.R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, NY, USA, 1950. Proof. We follow an idea of the proof of the above-mentioned [14] A. Dow and J. E. Vaughan, “Mrowka´ maximal almost disjoint Pełczynski’s´ result. By the remark of Pełczynski´ [5, p. 367], families for uncountable cardinals,” Topology and Its Applica- m tions,vol.157,no.8,pp.1379–1394,2010. the space 𝐶(𝐼 ) has an equivalent strictly convex norm. By Partington’sresults[20,21],ineachoftheeithercases,𝐶(Ω) [15] S. Mrowka,´ “Some set-theoretic constructions in topology,” Polska Akademia Nauk. Fundamenta Mathematicae,vol.94,no. contains an isometric copy of ℓ∞.Hence,𝐶(Ω) cannot be m 2, pp. 83–92, 1977. embedded into 𝐶(𝐼 ). [16] J. Ferrer and M. Wojtowicz,´ “The controlled separable projec- Example 8. Let Γ be a discrete space with card Γ=m > tion property for Banach spaces,” Central European Journal of Mathematics,vol.9,no.6,pp.1252–1266,2011. ℵ0.FromCorollary4weknowthatthespacesM(𝛽Γ) and m M(𝐼2 ) [17] G. W. Day, “Free complete extensions of Boolean algebras,” are order-isometric. Pacific Journal of Mathematics, vol. 15, pp. 1145–1151, 1965. On the other hand, by Lemma 7, there is no linear m [18] L. Soukup, “Scattered Spaces,”in Encyclopedia of General Topol- 𝐶(𝛽Γ) 𝐶(𝐼2 ) isomorphic embedding from into . ogy, K. P. Hart, J. Nagata, and J. E. Vaughan, Eds., pp. 350–353, Elsevier,Amsterdam,Netherlands,2004. In the next example we show that a similar property holds [19] R. C. Walker, The Stone-Cechˇ Compactification, Springer, Berlin, ∗ 2ℵ0 for the pair M(N ) and M(𝐼 ). Germany, 1974. 4 Journal of Function Spaces

[20] J. R. Partington, “Equivalent norms on spaces of bounded functions,” Israel Journal of Mathematics,vol.35,no.3,pp.205– 209, 1980. [21] J. R. Partington, “Subspaces of certain Banach sequence spaces,” The Bulletin of the London Mathematical Society,vol.13,no.2, pp. 163–166, 1981. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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