Kato, Hisao

Kyoto University Research Information Repository

TitleFunction spaces and isometrical extensions of boundedisometries of separable metric spaces (The present situation ofset-theoretic and geometric topology and its prospects)Author(s)Kato, HisaoCitation数理解析研究所講究録 (2014), 1884: 107-111Issue Date2014-04URL http://hdl.handle.net/2433/195684RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityFunction spaces and isometrical extensions ofbounded isometries of separable metric spacesHisao KatoInstitute of MathematicsUniversity of Tsukuba1 IntroductionIn this note, unless stated otherwise, we assume that all maps are continuous functions.Let $\mathbb{Z},$ $\mathbb{N}$ and $\mathbb{R}$ denote the set of integers, the set of natural numbers and the set of realnumbers, respectively. Also, let $I,$ $\triangle$ and $\mathbb{Q}$ be the unit interval $[0,1]$ , a Cantor set andthe Hilbert cube $I^{\infty}$ , respectively. For any compact metric space $Z,$ $C(Z)$ denotes thefunction space of all (continuous) maps from $Z$ to $\mathbb{R}$ with the supremum metric $\tilde{d}$, i.e.,$\tilde{d}(f, g)=\sup\{|f(z)-g(z)||z\in Z\}$for $f,$ $g\in C(Z)$ .A map $i$ : $(X, d_{X})arrow(Y, d_{Y})$ between separable metric spaces is an isometrical em-bedding from $(X, d_{X})$ into $(Y, d_{Y})$ if $i$ satisfies the condition $d_{Y}(i(x), i(x’))=d_{X}(x, x’)$for each $x,$ $x’\in X.$ $A$ map $g$ : $(X, d_{X})arrow(Y, d_{Y})$ between separable metric spaces isan isometry if $g$ is surjective and $d_{Y}(g(x), g(x’))=d_{X}(x, x’)$ for each $x,$ $x’\in X$ . For aseparable metric space $(X, d)$ , let $Iso(X)$ be the group of all isometries of $X$ equippedwith the pointwise convergent topology, i.e.,$Iso(X)=$ {$g:Xarrow X|g$ is an isometry}.A well-known theorem of Banach and Mazur is the result that $C(I)(I=[0,1])$ is auniversal space of separable metric spaces up to isometry (see [1,3,9]). Als$0$ , Urysohn [11]constructed a complete separable metric space $\mathbb{U}$ that is also universal up to isometry. In[12], Uspenskij proved that for any separable metric space $X$ there is a natural isometricalembedding $i$ : $Xarrow \mathbb{U}$ such that $i$ induces a natural continuous monomorphism $i^{\star}$ :$Iso(X)arrow Iso(\mathbb{U})$ satisfying that $i^{\star}(g)\in$ $I$ so(U) is an extension of $g\in Iso(X)$ (see[2,3,5,7,12,13] for more detailed properties of $\mathbb{U}$).In this note, we study the extension property of “bounded” isometries of separablemetric spaces in function spaces $C(\mathbb{Q})$ and $C(\triangle)$ . Also, we know that $C(I)$ does not havethe extension property. Let $(X, d)$ be a separable metric space and $x_{0}\in X.$ $A$ subgroup$G$ of $Iso(X)$ is bounded if diam $G(x_{0})<\infty$ , where $G(x_{0})=\{g(x_{0})|g\in G\}(\subset X)$ . Thedefinition of “ bounded subgroup” of $Iso(X)$ does not depend on the choice of the point$x_{0}\in X$ . Also, each $g\in Iso(X)$ is bounded if diam$\{g^{n}(x_{0})|n\in \mathbb{Z}\}<\infty$ . Note that if($X$, d) is bounded, i.e., $diam_{d}X<\infty$ , then $Iso(X)$ itself is bounded. In particular, if $X$is a compact metric space, then $Iso(X)$ is bounded. In [6], Mazur and Ulam proved thatif $B$ and $B’$ are Banach spaces, then every isometry $T$ : $Barrow B’$ with $T(O)=0$ is linealy1884 2014 107-111 107isometric and moreover, Banach and Stone proved that if $X$ and $Y$ are compact Hausdorffspaces, then every isometry $T$ : $C(X)arrow C(Y)$ with $T(O)=0$ is linealy isometric andmoreover, $T$ is induced by a homeomorphism $h:Yarrow X$ (see [1,10]).Theorem 1.1. (Banach [1] and Stone [10]) Let $X$ and $Y$ be compact Hausdorff spaces.Then the followings hold.(1) $C(X)$ is isometric to $C(Y)$ if and only if $X$ is homeomorphic to $Y.$(2) If $T:C(X)arrow C(Y)$ is a linear isometry, then there is a homeomorphism $h:Yarrow X$and $a$ (continuous) map $\alpha$ : $Yarrow \mathbb{R}$ with $|\alpha(y)|=1$ for $y\in Y$ such that$(T(f))(y)=\alpha(y)\cdot(f\circ h)(y)$for $f\in C(X)$ and $y\in Y.$ Moreover, if $Y$ is connected, $T(f)=f\circ h$ or $T(f)=-(f\circ h)$ .For any Banach space $B$ , letLinIso(B) $=\{f\in Iso(B)|f$ is linear $\}.$Note that LinIso(B) is bounded, because $LinI_{\mathcal{S}}o(B)(O)=\{0\}.$2 Extensions of bounded isometries in function spacesIn this section, we assume that $(X, d)$ is a separable metric space and $x_{0}$ is a fixedpoint of $X$ . In [9], Sierpi\’{n}ski considered the space$X’=\{f$ : $Xarrow \mathbb{R}|f(x_{0})=0$ and $|f(x)-f(y)|\leq d(x, y)$ for $x,$ $y\in X\}$which is a topological space equipped with the pointwise convergent topology (see also[3] $)$ and by use of the spaces $X’$ , he proved that $C(I)$ is a universal space of separablemetric spaces up to isometry. We modify the Sierpi\’{n}ski’s method of [9]. In this paper,for any bounded subgroup $G$ of $Iso(X)$ , we consider the following more general space$\tilde{X}(=\tilde{X}_{G})=\{f:Xarrow \mathbb{R}|f(z)\in$ [ $-$diam$(G(x_{0}))$ , diam$(G(x_{0}))$ ] for $z\in G(x_{0})$ and$|f(x)-f(y)|\leq d(x, y)$ for $x,$ $y\in X$ }which is a topological space equipped with the pointwise convergent topology. We havethe following lemmas.Lemma 2.1. $\tilde{X}(=\tilde{X}_{G})$ is a compact metric absolute retract $(=AR)$ . Moreover, if $g\in G,$then $\tilde{g}:\tilde{X}arrow\tilde{X}$ is a homeomorphisni, where $\tilde{g}$ is defined by $\tilde{g}(f)=fog$ for $f\in\tilde{X}.$Lemma 2.2. Suppose that $p_{G}$ : $Zarrow\tilde{X}(=\tilde{X}_{G})$ is a map from a compact metric space $Z$onto $\tilde{X}$ such that for each $g\in G$ there is $a$ (lifl) homeomorphism $L_{g}:Zarrow Z$ satisfyingthe following commutative diagram.$Z$ -SL $Z$$Pc\downarrow \downarrow p_{G}$$X arrow^{\overline{g}} \tilde{X}$108Then there is an isometrical embedding $i_{G}$ : $Xarrow C(Z)$ such that for each $g\in G$ , thefollowing commutative diagram holds.$X arrow^{g} X$$i_{G}\downarrow \downarrow i_{G}$$C(Z) -\S\tilde{L} C(Z)$where $\tilde{L}_{g}$ : $C(Z)arrow C(Z)$ is the isometry defined by $\tilde{L}_{g}(f)=f\circ L_{g}$ for $f\in C(Z)$ . Inparticular, $\tilde{L}_{g}\in LinIso(C(Z))$ is an isometrical extension of $g\in G.$Here we have the following theorem of $C(\mathbb{Q})$ which implies that $C(\mathbb{Q})$ is universalconcerning isometrical extensions of bounded isometry groups of separable metric spaces.Theorem 2.3. Let ($X$ , d) be a separable metric space and let $G$ be any bounded subgroupof $Iso(X)$ . Then there is an isometrical embedding $i_{G}$ : $Xarrow C(\mathbb{Q})$ such that $i_{G}$ inducesa continuous monomorphism $i_{G}^{\star}:Garrow Linlso(C(Q))$ such that $i_{G}^{\star}(g)\in LinIso(C(Q))$ isan extension of $g\in G.$Corollary 2.4. Suppose that ($X$ , d) is a bounded separable metric space. Then there is anisometrical embedding $i:Xarrow C(\mathbb{Q})$ such that $i$ induces a continuous monomorphism $i^{\star}$ :$Iso(X)arrow LinIso(C(\mathbb{Q}))$ such that $i^{\star}(g)\in LinIso(C(\mathbb{Q}))$ is an extension of $g\in Iso(X)$ .Remark 1. Note that for any Banach space $B$ , LinIso(B) is a bounded group. Hencein this note, we can not omit the condition that $G$ is bounded.If we observe the proof of Lemma 2.2, we see that some converse assertions of Lemma2.2 are also true. In fact, we have the following.Proposition 2.5. Suppose that $p_{G}$ : $Zarrow\tilde{X}(=\tilde{X}_{G})$ is a map from a compact metricspace $Z$ onto $\tilde{X},$ $i_{G}$ : $Xarrow C(Z)$ is the isometrical embedding as in the proof of Lemma2.2 and $g\in G$ . Let $L_{g}:Zarrow Z$ be a homeomorphism. Then the followings hold.(1) The following diagram is commutative:$z$ $z$$Pc\downarrow \downarrow p_{G}$$\tilde{X} arrow^{\tilde{g}} X$if and only if the following diagram is commutative:$X arrow^{g} X$$i_{G}\downarrow \downarrow i_{G}$$C(Z) -3\tilde{L} C(Z)$(2) The following diagram is commutative:$z$ $z$$p_{G}\downarrow \downarrow p_{G}$$X –\tilde{3} X$109if and only if the following diagram is commutative:$X arrow^{g} X$$i_{G}\downarrow \downarrow i_{G}$$C(Z) -\tilde{L}arrow^{g} C(Z)$Example. Let $X=\{x_{i}|i=0,1,2\}$ be the set of three elements and let $d$ be themetric on $X$ defined by $d(x_{i}, x_{j})=r>0(i\neq j)$ . Define the isometry $g$ : $Xarrow X$ by$g(x_{0})=x_{0},$ $g(x_{1})=x_{2}$ and $g(x_{2})=x_{1}$ . Let $G=\{id_{X}, g\}$ . Note that $G(x_{0})=\{x_{0}\}.$Then there is an isometrical embedding $i_{G}$ : $Xarrow C(\mathbb{Q})$ such that there is no isometricalextension of $g$ on $C(\mathbb{Q})$ . In particular, $C(\mathbb{Q})$ is not equal to the Urysohn universal space$\mathbb{U}$ , because that $\mathbb{U}$ has the following strong property: Any isometry between finite subsetsof $\mathbb{U}$ can be extended to an isometry of $U.$Next we will consider the case of the function space $C(\Delta)$ . Let $H(X)$ be the set of allhomeomorphisms of a space $X.$Proposition 2.6. Let $X$ be a compact metric space and let $G$ be a countable subset of$H(X)$ . Then there is an onto map $Pc:\Deltaarrow X$ such that for any $g\in G$ there is $a$ (lift)homeomorphism $L_{g}:\Deltaarrow\Delta$ of $\triangle$ such that the following diagram is commutative.$\Delta$ -SL $\Delta$$Pc\downarrow \downarrow p_{G}$$X arrow^{9} X$Then we have the following theorem of $C(\triangle)$ .Theorem 2.7. Let ($X$, d) be any separable metric space and let $G$ be a countable boundedsubgroup of $Iso(X)$ . Then there is an isometrical embedding $i_{G}$ : $Xarrow C(\triangle)$ such thatthere exist a countable subgroup $G^{\star}$ of $LinIso(C(\triangle))$ and a continuous epimorphism $r^{\star}$ :$G^{\star}arrow G$ such that each $g^{\star}\in G^{\star}$ is an extension of $r^{\star}(g^{\star})\in G.$ In particular, if $g_{-}\in G,$then there is an extension $g^{\star}\in LinIso(C(\Delta))$ of $g.$Remark 2. Note that the space $H(\Delta)$ of all homeomorphisms of $\Delta$ is homeomorphicto the space $P$ of irrationals, and hence $H(\Delta)$ is zero-dimensional. If $G$ is any boundedsubgroup of $Iso(X)$ with $\dim G\geq 1$ , there is no embedding from $G$ to $H(\Delta)$ .Corollary 2.8. Let $(X, d)$ be any separable metric space. If $g\in Iso(X)$ is periodic i. e.,$g^{n}=id_{X}$ for some $n\in \mathbb{N}$ , then there is an isometrical embedding $i_{g}$ : $Xarrow C(\Delta)$ suchthat there is an extension $g^{\star}\in LinIso(C(\triangle))$ of $g$ with $(g^{\star})^{n}=id_{C(\Delta)}.$Finally, we consider the case of $C(I)$ . We have the following proposition of $C(I)$ .Proposition 2.9. Let $(X, d)$ be any separable metric space and let $g\in Iso(X)$ such that$g$ has a periodic point $x_{0}$ with period $n\in \mathbb{N}$ . If $n\geq 3$ , there is no isometrical embedding$i$ from $X$ to $C(I)$ such that $g$ has an extension in LinIso($C$(I)).Now, we have the following problem.Problem 2.10. Let ($X$ , d) be any separable metric space. Is it true that there is anisometrical embedding $i$ from $X$ to $C(\mathbb{Q})$ such that each $g\in Iso(X)$ has an extensionwhich is an affine isometry of $C(\mathbb{Q})$ ?110References[1] S. Banach, Th\’eories des Op\’etations Lin\’eaires, Hafner, New York 1932, p. 185.[2] P. J. Cameron and A. M. Vershik, Some isometry groups of the Urysohn space, Annalsof Pure and Applied Logic, 143 (2006), 70-78.[3] M. R. Holmes, The universal separable metric spaces of Urysohn and isometric em-beddings thereof in Banach spaces, Fund. Math. 140 (1992), 199-223.[4] Y. Ikegami, H. Kato and A. Ueda, Dynamical systems of finite-dimensional metricspaces and zero-dimensional covers, Topology Appl. 160 (2013), 564-574.[5] $M$, Malicki and S. Solecki, Isometry groups of separable metric spaces, Math. Proc.Camb. Phil.Soc. (2009), 146, 67-81.[6] S. Mazur and S. Ulam, Sur les transformation isom\’etriques d’espace vectoriel normes,C. R. Acad. Sci. Paris, 194 (1932), 946-948.[7] J. Melleray, On the gometry of Urysohn’s universal metric space, Topology Appl. 154(2007), 384-4-3.[8] $J$ . van Mill, Infinite-dimensional topology: prerequisites and introduction,North-Holland publishing Co., Amsterdam, 1989.[9] W. Sierpi\’{n}ski, Sur un espace m\’etrique s\’eparable universel, Fund. Math. 33 (1945),115-122.[10] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans.Amer. Math. Soc. 41 (1937), 375-381.[11] P. Urysohn, Sur un espace m\’etrique universel, Bull. Sci. Math. 51 (1927), 43-64.[12] V. V. Uspenskij, On the group of isometries of the Urysohn universal metric space,Comment. Math. Univ. Carolinae, 31 (1990), 181-182.[13] V. V. Uspenskij, $A$ universal topological group with a countable base, Functionalanalysis and its applications, 20 (1986), 86-87.[14] H. Kato, Isometrical extensions of bounded isometries of separable metric spaces inthe function spaces $C(\mathbb{Q}),$ $C(\triangle)$ and $C(I)$ , preprint.111

Function spaces and isometrical extensions of bounded isometries of separable metric spaces (The present situation of set-theoretic and geometric topology and its prospects)

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/195684/1/1884-14.pdf

Function spaces and isometrical extensions of bounded isometries of separable metric spaces (The present situation of set-theoretic and geometric topology and its prospects)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository