aa r X i v : . [ m a t h . L O ] J a n AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS
MARCIN SABOK
Abstract.
We present a general framework for automatic continuityresults for groups of isometries of metric spaces. In particular, weprove automatic continuity property for the groups of isometries of theUrysohn space and the Urysohn sphere, i.e. that any homomorphismfrom either of these groups into a separable group is continuous. Thisanswers a question of Melleray. As a consequence, we get that the groupof isometries of the Urysohn space has unique Polish group topology andthe group of isometries of the Urysohn sphere has unique separable grouptopology. Moreover, as an application of our framework we obtain newproofs of the automatic continuity property for the group Aut([0 , , λ ),due to Ben Yaacov, Berenstein and Melleray and for the unitary groupof the infinite-dimensional separable Hilbert space, due to Tsankov. Theresults and proofs are stated in the language of model theory for metricstructures. Introduction
It is well known that every group is isomorphic to the group of isometriesof a metric space (or even of a graph). Moreover, if G is the group ofisometries of a metric space X , then G carries the topology of pointwiseconvergence on X . If the space X is separable and its metric is complete,then G is separable and completely metrizable (i.e. Polish ). In fact, thecoverse is also true and any Polish group is isomorphic to the group ofisometries of a separable complete metric space [14, Theorem 3.1]. Notethat if a group is isomorphic to the group of isometries of a space X , thenthe structure of this group is completely determined by the metric propertiesof X . In this paper, we exploit this observation to study the structure ofvarious groups of isometries.Automatic continuity is a phenomenon that connects the algebraic andtopological structures and typically says that any map which preserves analgebraic structure must automatically be continuous. One of the first in-stances of this phenomenon appears in C*-algebras and Banach algebras,where it is known that any homomorphism from an abelian Banach algebrainto C is continuous. More nontrivial results concern continuity of deriva-tions on C*-algebras. Sakai [46] (proving a conjecture of Kaplansky [26]) Mathematics Subject Classification. showed that any derivation on a C*-algebra is norm-continuous. This wasgeneralized first by Kadison [23] who improved it to the continuity in theultraweak topology and then by Ringrose [41] for derivations of C*-algebrasinto Banach modules. Johnson and Sinclair [22] on the other hand, showedautomatic continuity for derivations on semi-simple Banach algebras. Adetailed account on this subject can be found in [7, 8].In the context of groups and their homomorphisms, one of the first au-tomatic continuity results has been proved by Dudley [9], who showed thatany homomorphism from a complete metric or a locally compact group intoa normed (e.g. free) group is continuous (see also [47] for a recent general-ization to homomorphisms into free products). A general form of automaticcontinuity phenomenon for groups has appeared in the work of Kechris andRosendal [30], with connection to the results of Hodges, Hodkinson, Lascarand Shelah [19].A topological group G has the automatic continuity property if for everyseparable topological group H , any group homomorphism from G to H iscontinuous. Recall [27, Theorem 9.10] that any measurable homomorphismfrom a Polish group to a separable group must be continuous and the exis-tence of non-measurable homomorphisms on groups such as ( R , +) can bederived from the axiom of choice. So, similarly as amenability, automaticcontinuity property for a given group can be interpreted in terms of nonex-istence (on this group) of pathological phenomena that can follow from theaxiom of choice.Kechris and Rosendal [30] showed that automatic continuity is a conse-quence of the existence of comeager orbits in the diagonal conjugacy actionsof G on G n for each n ∈ N (i.e. ample generics , cf [30, Section 1.6]) and dis-covered a connection between ample generics for the automorphism groupsof homogeneous structures and the Fra¨ıss´e theory. In consequence, manyautomorphism groups of homogeneous structures turned out to have the au-tomatic continuity property. However, automatic continuity can hold alsofor groups which do not have ample generics (and even have meager conju-gacy classes). Rosendal and Solecki [45] proved that automatic continuityproperty holds for the groups of homeomorphisms of the Cantor space andof the real line, and for the automorphism group of ( Q , < ). Rosendal [42]showed automatic continuity for the groups of homeomorphisms of compact2-manifolds. A survey on recent results in this area can be found in [43].The Urysohn space U is the separable complete metric space which is homogeneous (i.e. any finite partial isometry of U extends to an isometry of U ) and such that any finite metric space embeds into U isometrically. It isknown that these properties of U determine it uniquely up to isometry andthat any separable metric space embeds into U [38, Theorem 5.1.29]. Theanalogue of the Urysohn space of diameter 1 also exists and is called the Urysohn sphere (or the bounded Urysohn space of diameter 1 ) and denotedby U (see [38, Remark 5.1.31]). The group of isometries of U is univer-sal among Polish groups, i.e. any Polish group is its closed subgroup [13, UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 3
Theorem 2.5.2]. The Urysohn space, as well as its group of isometries, havereceived a considerable amount of attention recently. Tent and Ziegler [52]showed that the quotient of the group of isometries of U modulo the normalsubgroup of bounded isometries is a simple group and recently [51] provedthat the group of isometries of the bounded Urysohn space is simple. Formore on the structure of the Urysohn space and its group of isometries, seethe recent monographs [38, 14, 13] on this subject.Kechris and Rosendal showed that the group of automorphisms of the rational Urysohn space (which is the Fra¨ıss´e limit of the class of finite metricspaces with rational distances) has ample generics and deduced from it thatthe group Iso( U ) has a dense conjugacy class [31, Theorem 2.2]. The questionwhether the group of isometries of the Urysohn space has the automaticcontinuity property has been asked by Melleray (cf. [5, Section 6.1]). Oneof the main applications of the results of this paper is the following. Theorem 1.1.
The groups of isometries of the Urysohn space and theUrysohn sphere have the automatic continuity property.
Theorem 1.1 has some immediate consequences on the topological struc-ture of the above groups, in spirit of the results of Kallman [24, 25] andAtim and Kallman [2]. The first one is an abstract consequence of auto-matic continuity
Corollary 1.2.
The group
Iso( U ) has unique Polish group topology. Recall that a group is minimal if it does not admit any strictly corser(Hausdorff) group topology (in this paper we consider only Hausdorff topolo-gies on groups). The second corollary follows from minimality of the groupof isometries of the Urysohn sphere, proved by Uspenskij [54]
Corollary 1.3.
The group
Iso( U ) has unique separable group topology. Theorem 1.1 will follow from the following abstract result, which isolatesmetric (or model-theoretic) properties of a metric structure that imply thatthe group of automorphisms (with the pointwise convergence topology) ofthe structure has the automatic continuity property. The definitions of ametric structure and the notions appearing in the statement of the theoremare given in Sections 2 and 3.
Theorem 1.4.
Suppose M is a homogeneous complete metric structure thathas locally finite automorphisms, the extension property and admits weaklyisolated sequences. Then the group Aut( M ) has the automatic continuityproperty. Theorem 1.4 can be also applied to give a unified treatment of previouslyknown automatic continuity results for automorphism groups of some metricstructures. It is worth mentioning that up to now, these results have beenproved with different methods, varying from case to case. In this paper, weapply Theorem 1.4 to show the automatic continuity property for the group
MARCIN SABOK
Aut([0 , , λ ) (the group of measure-preserving automorphism of the unitinterval) and the group U ( ℓ ) (unitary operators of the infinite-dimensionalseparable Hilbert space).Automatic continuity for the group Aut([0 , , λ ) has been proved in aseries of two papers [32, 5]. Kittrell and Tsankov [32] showed first that anyhomomorphism from Aut([0 , , λ ) to a separable group must be continu-ous in the strong topology of Aut([0 , , λ ) (see [28, Section 1]) and later,Ben Yaacov, Berenstein and Melleray [5] proved a general result which im-plies that any homomorphism which is continuous in the strong topologyon Aut([0 , , λ ) must be continuous in the weak topology on Aut([0 , , λ )(this approach has been recently simplified by Malicki [34]). The groupAut([0 , , λ ) (with the weak topology) is isomorphic to the group of auto-morphism of the measure algebra and applying Theorem 1.4 to the measurealgebra we get a new proof of automatic continuity. Corollary 1.5 (Ben Yaacov, Berenstein, Melleray) . The group
Aut([0 , , λ ) has the automatic continuity property. After the work of Ben Yaacov, Berenstein and Melleray [5], Tsankov [53]further showed the automatic continuity property for the infinite-dimensionalunitary group. Given that the group U ( ℓ ) is the automorphism group ofthe Hilbert space ℓ (or the isometry group of the sphere in ℓ ), we canapply Theorem 1.4 to the Hilbert space and get a new proof of this result. Corollary 1.6 (Tsankov) . The group U ( ℓ ) has the automatic continuityproperty. Our proof of Theorem 1.4 builds on the work of Kechris and Rosendal[31] and Rosendal and Solecki [45]. In particular, we use the notion of amplegenerics introduced in [31] and exploit some ideas of [45, Section 3]. Theverification of conditions of Theorem 1.4 in the case of the Urysohn spaceuses a result of Solecki [48] that is based on earlier results of Herwig andLascar [17]. These result in turn, are connected to a theorem of Ribesand Zalesski˘ı [40], who showed separability of products of finitely-generatedsubgroups of the free groups (this was later generalized by Minasyan [36] tohyperbolic groups). In Section 8 we present a new proof of Solecki’s theorem[48] in the style of Mackey’s construction of induced actions and based onthe separability result of Ribes and Zalesski˘ı.This paper is organized as follows. In Sections 2 and 3 we give an overviewof model-theoretic notions that appear in the statement of Theorem 1.4. InSections 4 and 5 we prove a weak version of ample generics for the automor-phism groups of metric structures and Section 6 includes a proof of Theorem1.4. Section 7 contains a further result on triviality of homomorphisms togroups admitting complete left-invariant metrics. In Section 8 we verify thatthe assumptions of Theorem 1.4 are satisfied by the Urysohn space, whichproves Theorem 1.1. Sections 9 and 10 contain discussion of the cases of themeasure algebra and the Hilbert space and proofs of Corollaries 1.5 and 1.6.
UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 5 First-order continuous model theory
The techniques used in this paper are motivated by the framework andlanguage of model theory for metric structures, developed recently by BenYaacov, Berenstein, Henson, Usvyatsov [4] and others. There are, however,some details, that will vary from the original setting. In this paper, a met-ric structure is a tuple (
X, d X , f , f , . . . ) where X equipped with d X is aseparable metric space and f , f , . . . are either closed subsets of X n × R m ( relations ), for some n, m ∈ N , or continuous functions, from X n × R m to X k × R l for some n, m, k, l ∈ N (here we consider the discrete topology on R , i.e. demand continuity only on the arguments from X ). Thus, a metricstructure is a two-sorted structure with the second sort being a subset of thereal line R . Let us note here that in the examples considered in this paper,the structures will contain no relational symbols (only functions) and weallow them in the definition only for sake of generality.We do not require our metric structures to be complete (as metric spaces)and we say that a metric structure X is complete if it is complete with respectto d X .Given a metric structure M we write Aut( M ) for the group of automor-phisms of M (i.e. bijections of the first sort of M which preserve boththe metric and each f n ). Aut( M ) is always endowed with the topology ofpointwise convergence and if M is complete, then Aut( M ) is a Polish group.Say that a metric structure X is homogeneous if every partial isomorphismbetween finitely generated substructures of X extends to an automorphismof X . Note that in case X is a metric space, this coicides with the usualnotion of homogeneity (sometimes also referred to as ultrahomogeneity ).The main difference between continuous logic and our approach lies thesyntax. We will consider only a first-order variant of the language. Terms are either variables, elements of a structure (of the first sort or the secondsort), or expressions of the form f ( τ , . . . , τ n ), where τ , . . . , τ n are termsand f is a function symbol (e.g. the symbol for the distance function) ofappropriate kind (where the numbers and sorts of the variables are correct).The first-order formulas in our language will be of the form • τ = σ , or R ( τ , . . . , τ n ), where τ, σ, τ , . . . , τ n are terms and R is arelational symbol, • if ϕ and ψ are first-order formulas and x is a variable of the first sort,then ∃ xϕ , ∀ xϕ , ¬ ϕ , ϕ ∨ ψ are first-order formulas as well (quantifi-cation is only allowed over the first sort).As usual, a first-order sentence is a first-order formula without free vari-ables. The truth value of a first-order sentence in a metric structure isdefined as in the classical setting (it is either 0 or 1). We will use thesymbol ≺ for an elementary substructure, in the following (classical) sense:given a metric structure X and its substructure Y ⊆ X we write Y ≺ X iffor every first order sentence σ with parameters in Y , if σ is true in Y , then σ is true in X . MARCIN SABOK
Given a metric structure X and a tuple ¯ a = ( a , . . . , a m ) of elements of X , a quantifier-free type over ¯ a is a set of quantifier-free formulas ϕ (¯ x, ¯ a ) fora fixed sequence of variables ¯ x = ( x , . . . , x n ) of the first sort. A quantifier-free type is a quantifier-free type over the empty tuple. Quantifier-free typeswill be denoted by p (¯ x ) (to indicate the variables), or simply p . If ¯ x =( x , . . . , x n ), then we also say that p (¯ x ) is a quantifier-free n -type over ¯ a .We say that an n -tuple ¯ b = ( b , . . . , b n ) of elements of a metric structure X realizes a given quantifier-free n -type p over ¯ a (write ¯ b | = p ) if X | = ϕ (¯ b, ¯ a )for every ϕ (¯ x, ¯ a ) ∈ p (the definition of satisfaction in a model is the naturalone). Given Y ⊆ X with ¯ a ∈ Y n and a quantifier-free type n -type p over¯ a we say that p is realized in Y if there is ¯ b ∈ Y n such that ¯ b | = p . Also,abusing notation, if Y ⊆ X k , then we say that p is realized in Y if if there is¯ b ∈ Y such that ¯ b | = p . The quantifier-free type of a tuple ¯ b over ¯ a , denotedby qftp(¯ b/ ¯ a ) is then the set of all quantifier-free formulas ϕ (¯ x, ¯ a ) such that X | = ϕ (¯ b, ¯ a ). If ¯ a is the empty tuple, then we write qftp(¯ b ) for qftp(¯ b/ ¯ a ).A quantifier-free type is consistent if there is a model that realizes it, anda consistent quantifier-free type p is complete if whenever p ⊆ q and q is aconsistent quantifer-free type, then p = q . Definition 2.1.
Given n ∈ N , a quantifier-free n -type p over ¯ a = ( a , . . . , a n )and ε >
0, say that p is an ε -quantifier-free n -type over ¯ a if qftp(¯ a ) ⊆ p and d ( x i , a i ) = ε i belongs to p for each i ≤ n and for some 0 ≤ ε i < ε .Given a metric structure X , k ∈ N and a complete quantifier-free k -type p write p ( X ) = { ¯ a ∈ X k : ¯ a | = p } and note that p ( X ) ⊆ X k is G δ (closedif there are no relation symbols) in the topology of X k , so if X is complete,then p ( X ) becomes a Polish space.Given three tuples ¯ a, ¯ b and ¯ c in a metric structure X , write ¯ b ≡ ¯ a ¯ c ifqftp(¯ b/ ¯ a ) = qftp(¯ c/ ¯ a ). Also, write ¯ b ≡ ¯ c to denote ¯ b ≡ ∅ ¯ c . If X is ahomogeneous metric space, then the former is equivalent to the fact thatthere is g ∈ Iso( X ) with g ↾ ¯ a = id and g (¯ c ) = ¯ b . Definition 2.2.
Given a metric structure X , k ∈ N , a tuple ¯ a ∈ X k with p = qftp(¯ a ) and ε >
0, say that a subset Y ⊆ p ( X ) is relatively ε -saturatedover ¯ a if every ε -quantifier-free k -type over ¯ a which is realized in X , is alsorealized in Y .Note that if Y is relatively ε -saturated over ¯ a , then in particular, Y con-tains ¯ a . Given two tuples ¯ a, ¯ b ∈ X m and ε > d X (¯ a, ¯ b ) < ε if d X ( a k , b k ) < ε for every k ≤ m . Definition 2.3.
Suppose X is a homogeneous metric structure and ¯ a ∈ X k for some k ∈ N . Write p for qftp(¯ a ). Say that a sequence (¯ a n : n ∈ N ) ofelements of X k is an isolated sequence in p if every ¯ a n realizes p and thereexists a sequence of ε n > Y n ⊆ p ( X ) such that for every n ∈ N the set Y n is relatively ε n -saturated over ¯ a n and for every sequence ¯ b n ∈ Y n such that qftp(¯ b n ) = qftp(¯ a n ) and d X (¯ a n , ¯ b n ) < ε n there is an automorphism ϕ of X with ϕ (¯ a n ) = ¯ b n for every n ∈ N . UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 7
The following definition will be generalized in Definition 2.8 below.
Definition 2.4.
Say that a metric structure X admits isolated sequences iffor every k ∈ N , every complete quantifier-free k -type p realized in X , forevery nonmeager set Z ⊆ p ( X ) there is a sequence (¯ a n : n ∈ N ) which isisolated in p and ¯ a n ∈ Z for every n .The above definitions are enough for the study of the Urysohn space andthe measure algebra but in order to deal with the Hilbert space we need tointroduce slightly more general notions. Definition 2.5.
Given a metric structure X , k ∈ N , ε > a ∈ X k write p = qftp(¯ a ). Suppose T ⊆ p ( X ). Say that Y ⊆ X k is T -relatively ε -saturated over ¯ a if for every ¯ b ∈ T with d X (¯ b, ¯ a ) < ε there is ¯ b ′ ∈ Y suchthat qftp(¯ b/ ¯ a ) = qftp(¯ b ′ / ¯ a ). Definition 2.6.
Suppose X is a metric structure, ε > k, m ∈ N , ¯ a ∈ X k and p = qftp(¯ a ). Say that a subset T ⊆ p ( X ) ( m, ε ) -generates an open setover ¯ a if there is a nonempty open set U ⊆ p ( X ) such that for every ¯ b ∈ U there is a sequence g , . . . , g m ∈ Aut( X ) such that • g m . . . g (¯ a ) = ¯ b , • g i (¯ a ) ∈ T and d X ( g i (¯ a ) , ¯ a ) < ε for each i ≤ m .Note that, in particular, if T contains an open ball of radius ε around ¯ a (i.e. { ¯ b ∈ p ( X ) : d X (¯ b, ¯ a ) < ε } ), then it (1 , ε )-generates an open set over ¯ a .Therefore, the following definition is a generalization of Definition 2.2. Definition 2.7.
Suppose X is a metric structure, ε > k, m ∈ N , ¯ a ∈ X k and p = qftp(¯ a ). Say that Y ⊆ p ( X ) is m -relatively ε -saturated over ¯ a ifthere is T ⊆ p ( X ) such that • Y is T -relatively ε -saturated over ¯ a , • T ( m, ε )-generates an open set over ¯ a .Thus, if Y is relatively ε -saturated over ¯ a , then it is 1-relatively ε -saturatedover ¯ a . Definition 2.8.
Suppose X is a homogeneous metric structure and ¯ a ∈ X k for some k ∈ N . Write p for qftp(¯ a ). Say that a sequence (¯ a n : n ∈ N ) ofelements of X k is a weakly isolated sequence in p if every ¯ a n realizes p andthere exists m ∈ N and a sequence of ε n > Y n ⊆ p ( X ) suchthat for every n ∈ N the set Y n is m -relatively ε n -saturated over ¯ a n and forevery sequence ¯ b n ∈ Y n such that qftp(¯ b n ) = qftp(¯ a n ) and d X (¯ a n , ¯ b n ) < ε n there is an automorphism ϕ of X with ϕ (¯ a n ) = ¯ b n for every n ∈ N . Given m as above we will also say that the sequence is m -weakly isolated . Definition 2.9.
Say that a metric structure X admits weakly isolated se-quences if there is m ∈ N such that for every k ∈ N , every completequantifier-free k -type p realized in X , for every nonmeager set Z ⊆ p ( X )there is a sequence (¯ a n : n ∈ N ) which is m -weakly isolated in p and ¯ a n ∈ Z MARCIN SABOK for every n . Given m as above we will also say that the structure admits m -weakly isolated sequences.Now, Definition 2.4 is a special case of Definition 2.9 since an isolatedsequence is obviously weakly isolated.3. Metric structures
Automatic continuity for automorphism groups of metric structures willdepend on the model-theoretic properties of the structure. The key defini-tions, which stem from the analysis of the work of Kechris and Rosendal[31] are given below.Below, and throughout of this paper, we use the convention that a finitelygenerated substructure of a metric structure is always enumerated (a finitelygenerated substructure is a tuple if there are no function symbols).
Definition 3.1.
Let M be a metric structure, B, C ⊆ M be finitely gener-ated substructures. Given a finitely generated substructure A ⊆ B ∩ C saythat B and C are independent over A and write B | ⌣ A C if for every pair of automorphisms ϕ : B → B , ψ : C → C such that A isclosed under ϕ and ψ and ϕ ↾ A = ψ ↾ A , the function ϕ ∪ ψ extends to anautomorphism of the substructure generated by B and C .An abstract notion of stationary independence has been considered byTent and Ziegler in [52]. In general, the above notion is not a stationaryindependence relation in the sense of [52, Definition 2.1] and satisfies onlythe Invariance and Symmetry conditions (see also the remarks [52, Example2.2]). However, in all concrete cases, the examples of independence relationconsidered in this paper will be the same as in [52]. The following is moti-vated by a standard property of the independence relation in stable theories(see [4, Theorem 14.12]). Definition 3.2.
Say that a metric structure M has the extension property if for every pair B, C ⊆ M of finitely generated substructures and a finitelygenerated substructure A ⊆ B ∩ C there is a finitely generated substructure C ′ ⊆ M with C ′ ≡ A C such that B | ⌣ A C ′ .Another property of the metric structures that we will need for automaticcontinuity is connected with the extension theorems proved by Hrushovski[20], Herwig and Lascar [17] and Solecki [48]. Definition 3.3.
Say that a metric structure M has locally finite automor-phisms if for every n ∈ N , for any finitely generated substructure N of M ,for any partial automorphisms ϕ , . . . , ϕ n of N , there is a finitely generatedstructure N ′ of M containing N such that every ϕ i extends to an automor-phism of N ′ , for each i ≤ n . UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 9
Note that if finitely generated substructures of M are finite (e.g. whenthere are no function symbols for functions from M m to M ), then M haslocally finite automorphisms if and only if for any finite substructure N of M there is is a finite structure N ′ of M containing N such that everyisomorphism between finite substructures of N extends to an automorphismof N ′ . 4. First order logic for metric structures
Similarly as in the classical case, the L ω ω -formulas in first-order logicfor metric structures are formed by allowing countable infinite conjunctionsand disjunctions of first order formulas as well as finite quantification. Notethat the formula qftp(¯ x ) = qftp(¯ y ) belongs to L ω ω .A property Φ of a metric structures is called a first-order property if thereis a set ˜Φ of L ω ω -sentences such that a metric structure M satisfies Φ ifand only if M | = φ for every φ ∈ ˜Φ. Note that if φ is an L ω ω -sentence and N ≺ M , then we have that N | = φ if and only if M | = φ . Lemma 4.1 (L¨owenheim–Skolem) . Suppose M is a metric structure andfor each ¯ a, ¯ b ∈ M <ω let ϕ ¯ a ¯ b be an automorphism of M . For every countable M ⊆ M there is a countable metric structure N ⊆ M with M ⊆ N , suchthat N ≺ M and N is closed under ϕ ¯ a ¯ b for each ¯ a, ¯ b ∈ N <ω .Proof. The standard L¨owenheim–Skolem argument shows that there is acountable M with M ⊆ M and M ≺ M . Construct a chain of countablefirst-order elementary substructures M n ≺ M with M n ⊆ M n +1 such that M n +1 is closed under ϕ ¯ a ¯ b for every ¯ a, ¯ b ∈ M <ωn . Write N = S n M n . Then N is as needed. (cid:3) Lemma 4.2.
The property saying that a metric structure has locally finiteautomorphisms is a first-order property for homogeneous metric structures.Proof.
For each n ∈ N write x ∈ h x , . . . , x n i for the L ω ω -formula (in variables x, x , . . . , x n ) saying that x belongs to thesubstructure generated by x , . . . , x n . The formula is of the form W k ∈ N x = g k ( x , . . . , x n ) where g k enumerate all compositions of function symbolsin the language. We also write y , . . . , y m ∈ h x , . . . , x n i for V i ≤ m y i ∈h x , . . . , x n i .Note that if a homogeneous metric structure M has locally finite auto-morphisms and N is its finitely generated substructure, say by x , . . . , x n and k ∈ N , then there is a number n ( p, k ) ∈ N , depending only on thequantifier-free type p of x , . . . , x n and k such that any for any substructure N isomorphic to N in M , and any set of partial automorphisms ϕ , . . . , ϕ k of N there is a substructure N ′ of M containing N and generated by m ≤ n ( p ) elements such that every ϕ i extends to an automorphism of N ′ . Thus, a homogeneous metric structure M has locally finite automor-phisms if and only if it satisfies the following L ω ω -sentences, for every n, k ∈ N , quantifier-free n -type p and every n , . . . , n k ≤ n . ∀ x , . . . , x n [qftp( x , . . . , x n ) = p ] ⇒∀ y , . . . , y n , z , . . . , z n y , . . . , y n , z , . . . , z n , . . . , y k , . . . , y kn k , z k , . . . , z kn k (cid:20) k ^ i =1 (cid:0) n i ^ l =1 y il , z il ∈ h x , . . . , x n i (cid:1) ∧ qftp( y i , . . . , y in i ) = qftp( z i , . . . , z in i ) (cid:21) ⇒ (cid:20) _ n ≤ m ≤ n ( p,k ) ∃ x ′ , . . . , x ′ m x ′ = x ∧ . . . ∧ x ′ n = x n ^ j ≤ k ∃ x k , . . . , x km ∈ h x ′ , . . . , x ′ m i (cid:18) x ′ , . . . , x ′ m ∈ h x k , . . . , x km i∧ qftp( x ′ , . . . , x ′ m , y k , . . . , y kn k ) = qftp( x k , . . . , x km , z k , . . . , z kn k ) (cid:19)(cid:21) (cid:3) Lemma 4.3.
The extension property is a first-order property for metricstructures.Proof.
The extension property is the conjunction of the following sentences,for all n, m ∈ N and k ≤ min( n, m ). Below, for a tuple ¯ x = ( x , . . . , x n )and σ ∈ S n (the group of permutations of n ) we write ¯ x σ for the tuple( x σ (1) , . . . , x σ ( n ) ). ∀ x , . . . , x n ∀ y , . . . , y m ( x = y ∧ . . . ∧ x k = y k ) ⇒ (cid:20) ∃ y ′ . . . , y ′ m ( y ′ = y ∧ . . . ∧ y ′ k = y k ) ∧ qftp( y . . . y m ) = qftp( y ′ . . . y ′ m ) ∧ ^ σ ∈ S n ^ τ ∈ S m (cid:16) σ ↾ k : k → k ∧ σ ↾ k = τ ↾ k (cid:0) qftp(¯ x ) = qftp(¯ x σ ) ∧ qftp(¯ y ′ ) = qftp(¯ y ′ τ ) (cid:1) ⇒ qftp(¯ x, ¯ y ) = qftp(¯ x σ , ¯ y τ ) (cid:17)(cid:21) (cid:3) Corollary 4.4.
Suppose M is a homogeneous metric structure which haslocally finite automorphisms and the extension property. Let ϕ ¯ a be an auto-morphism of M for each ¯ a ∈ M <ω . Then there is a countable homogeneousmetric structure N ⊆ M which is dense in M , has locally finite automor-phisms, the extension property and is closed under all automorphisms ϕ ¯ a for ¯ a ∈ N <ω .Proof. Let M be countable dense in M and for each ¯ a, ¯ b ∈ M <ω whichgenerate isomorphic substructures, let ϕ ¯ a ¯ b be an automorphism of M which UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 11 maps ¯ a to ¯ b . If ¯ a, ¯ b ∈ M <ω are of different cardinality or do not generateisomorphic substructures, then put ϕ ¯ a ¯ b = ϕ ¯ a . By Proposition 4.1, there isa countable substructure N ≺ M which contains N and is closed under all ϕ ¯ a ¯ b for ¯ a, ¯ b ∈ N <ω . The latter clearly implies that N is homogeneous andclosed under ϕ ¯ a for ¯ a ∈ N <ω . Lemmas 4.3 and 4.2 imply that N has theextension property and locally finite automorphisms. (cid:3) Ample generics
If a metric structure is countable and has a discrete metric, then itsautomorphism group is a subgroup of S ∞ and for such groups Kechris andRosendal [31] developed machinery for proving automatic continuity. Recallthat a topological group G has ample generics if for every n ∈ N there is acomeager class in the diagonal conjugacy action of G on G n (i.e. the action g · ( g , . . . , g n ) = ( gg g − , . . . , gg n g − )).Recall [18] that given a continuous action of a Polish group G on a Polishspace X , a point x ∈ X is turbulent if for every open neighborhood U ⊆ G of the identity and every open V ∋ x , the local orbit O ( x, U, V ) = { x ′ ∈ X : ∃ g , . . . , g n ∈ U ∀ i ≤ n g i . . . g x ∈ U ∧ x ′ = g n . . . g x } is somewhere dense.If G is a subgroup of S ∞ , then a point x ∈ X is turbulent if and only if forevery open subgroup U ≤ G the set U x = { gx : g ∈ U } is somewhere dense[31, Proposition 3.2]. Also, in the case of a continuous action of a Polishgroup, if a point is turbulent and has a dense orbit, then its orbit is actuallycomeager. This is because in such case the orbit of a turbulent point cannotbe meager and hence has to be meager in its closure [31, Proposition 3.2].The groups of automorphisms of metric structures can be endowed withmany topologies and this is the starting point of the analysis of Ben Yaacov,Berenstein and Melleray [5], who consider two types of topologies: the Polishtopologies of pointwise convergence and variants of strong (non separable)topologies. We are, however, primarily interested in separable topologieson these groups. By default, the topology on Aut( M ) is that of pointwiseconvergence with respect to the metric on M . However, if M is countable(but perhaps not complete with respect to its metric), we will also considerthe topology inherited from the group S ∞ , which coincides with the point-wise convergence topology with respect to a discrete metric on M . If M iscountable, then we refer to this topology by saying that the group Aut( M )is treated as a subgroup of S ∞ . It can be also viewed as the topology ofpointwise convergence on the automorphism group of the structure M en-dowed with a discrete metric and the original distance function d M treatedas a part of the structure.Below, and throughout this paper, we use the following notation. If G actson M and ¯ a = ( a , . . . , a n ) ∈ M <ω , then G ¯ a = { g ∈ G : ∀ i ≤ n g ( a i ) = a i } . Lemma 5.1.
Suppose M is a countable homogeneous metric structure. If M has locally finite automorphisms and the extension property, then Aut( M ) has ample generics as a subgroup of S ∞ . Proof.
Write G for Aut( M ). Fix n ∈ N to find a generic tuple in G n .Enumerate as ( a n : n ∈ N ) with infinite repetitions all tuples a = ( A, ~ϕ, B, ~ψ )with ~ϕ = ( ϕ , . . . , ϕ n ), ~ψ = ( ψ , . . . , ψ n ) such that A ⊆ B are finitelygenerated substructures of M (possibly generated by the empty set) andfor each i ≤ n we have ϕ i ⊆ ψ i , and ϕ i : A → A , ψ i : B → B areautomorphisms.By induction on k construct a sequence of increasing finitely generatedsubstructures D k ⊆ M together with tuples of increasing automorphisms ~γ k = ( γ k , . . . , γ nk ) with γ ik : D k → D k for each i ≤ n . Using back-and-forthand homogeneity make sure that S k D k = M and for each i ≤ n the function S k γ ik is an automorphism of M . Additionally, for each k make sure thatif a k = ( A k , ~ϕ k , B k , ~ψ k ) and A k ⊆ D k are such that ~ϕ k = ( ϕ k , . . . , ϕ nk ) , ~ψ k = ( ψ k , . . . , ψ nk ) ,ϕ ik ↾ A k : A k → A k , ψ ik ↾ A k = ϕ ik ↾ A k and ϕ ik ⊆ γ ik for each i ≤ n, then there is g ∈ G A k with gγ ik +1 g − ⊇ ψ ik for each i ≤ n. ( ∗ )At the induction step k first use locally finite automorphisms to find D ′ k ⊇ D k such that the first k -many elements of M are in D ′ k and each ϕ ik ex-tends to an automorphism γ ik ′ of D ′ k . Next, use the extension property tofind a finitely generated substructure B ′ k of M with B ′ k ≡ A k B k such that B ′ k | ⌣ A k D ′ k . Let g ∈ G A k witness that B ′ k ≡ A k B k , i.e. g ( B ′ k ) = B k . Define D k +1 to be the substructure generated by D ′ k and B ′ k and for each i ≤ n define γ ik +1 so that γ ik +1 ↾ D ′ k = γ ik ′ and γ ik +1 ↾ B ′ k = g − ψ i g and use theassumption B ′ k | ⌣ A k D ′ k to extend it to D k +1 . Note that g witnesses that( ∗ ) is satisfied at the step k .After this construction is done, write g i = S k γ ik and note that g i is anautomorphism of M , for each i ≤ n . To see that ~g = ( g , . . . , g n ) is generic,it is enough to see that ~g is turbulent under the diagonal conjugacy actionof Aut( M ) on Aut( M ) n and has a dense orbit.To see the turbulence, fix an open neighborhood O of the identity inAut( M ), and, say, O = G ¯ a for a finite tuple ¯ a ⊆ M . We need to see that O · ~g is somewhere dense. Find m such that ¯ a ⊆ D m and write V i for { f ∈ Aut( M ) : f ↾ D m = g i ↾ D m } . Note that g i ↾ D m : D m → D m is anautomorphism for each i ≤ n . We claim that O · ~g is dense in V × . . . × V n .To see that, fix a nonempty open subset W ⊆ V × . . . × V n and withoutloss of generality assume W = W × . . . × W n . Since M has locally finiteautomorphisms, we can assume that W i = { f ∈ Aut( M ) : f ⊇ ψ i } for each i ≤ n , where each ψ i : B → B is an automorphism of a finitely generatedsubstructure B of M . Note that ψ i ⊇ g i ↾ D m for each i ≤ n .We need to see that ( O · ~g ) ∩ ( W × . . . × W n ) = ∅ . Let k ∈ N be such that a k = ( A k , ~ϕ k , B k , ~ψ k ) with A k = D m , B k = B , ~ϕ k = ( g ↾ D m , . . . , g n ↾ D m ) UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 13 and ~ψ = ( ψ , . . . , ψ n ). By ( ∗ ) at the step k , there is g ∈ G A k ⊆ G ¯ a suchthat gg i g − ⊇ ψ i for each i ≤ n and so we are done.The fact that the orbit of ~g is dense follows by analogous arguments. (cid:3) Recall that a subset S of a group G is called countably syndetic if G canbe written as S n a n S for some sequence a n ∈ G . A subset S of G is called symmetric if S − = S . Corollary 5.2.
Suppose M is a countable homogeneous metric structureand G = Aut( M ) . If M has locally finite automorphisms and the extensionproperty, and W ⊆ Aut( M ) is symmetric and countably syndetic, then thereis a finite tuple ¯ a ⊆ M such that G ¯ a ⊆ W .Proof. This is an abstract consequence of ample generics [31, Lemma 6.15]and thus follows from Lemma 5.1. (cid:3) Automatic continuity
In [45] Rosendal and Solecki isolated an abstract property of a group thatimplies automatic continuity. We say that G is Steinhaus (cf. [49]) if thereis a natural number k ≥ S ⊆ G the set S k = { s · . . . · s k : s , . . . , s k ∈ S } contains a nonemptyopen set in G . In such a case G is also called k -Steinhaus . If a group G isSteinhaus, then G has the automatic continuity property [45, Proposition2].We will need the following lemma. Lemma 6.1.
Suppose X is a complete homogeneous metric structure thathas locally finite automorphisms and the extension property and let G =Aut( X ) . If W ⊆ G is symmetric and countably syndetic, then there is ¯ a ∈ X <ω such that G ¯ a ⊆ W .Proof. Suppose otherwise. This means that for each k ∈ N and a ∈ X k there is f ¯ a ∈ G with f ¯ a (¯ a ) = ¯ a and f ¯ a / ∈ W . Let g n ∈ G be such that G = S n g n W . Use Corollary 4.4 to find a countable, dense X ⊆ X suchthat X ≺ X and X is closed under each g n as well as under f ¯ a for each¯ a ∈ ( X ) <ω . Since X is elementary in X , it has locally finite automorphismsand the extension property by Lemmas 4.2 and 4.3.Write W for the set of those automorphisms of X whose unique extensionto X belongs to W . Claim 6.2. W is symmetric and countably syndetic in Aut( X ) .Proof. It is clear that W is symmetric. To see that it is countably syndetic,pick f ∈ Aut( X ) and let f ∈ Aut( X ) be the unique extension of f to X .Now, there is n ∈ N and s ∈ W such that f = g n s . Since s = g − n f leaves X invariant, we have that s = s ↾ X ∈ Aut( X ) and s ∈ W . (cid:3) Now, this gives a contradiction since by Corollary 5.2, ( W ) contains anopen neighborhood of the identity in Aut( X ), i.e. there is ¯ a ∈ ( X ) <ω such that every automorphism of X which fixes ¯ a belongs to ( W ) . Write f for f ¯ a ↾ X and note that f / ∈ ( W ) , by the density of X , contradiction.This proves the lemma. (cid:3) Theorem 6.3.
Suppose X is a complete homogeneous metric structure thatadmits weakly isolated sequences, has locally finite automorphisms and theextension property. Then the group Aut( X ) is Steinhaus.Proof. Write G for Aut( X ). Suppose m ∈ N is such that X admits m -weakly isolated sequences. We will show that G is (24 m + 10)-Steinhaus.The same argument also shows that if X admits isolated sequences then G is 24-Steinhaus. Let W ⊆ G be symmetric and countably syndentic. Let g n ∈ G be such that G = S n g n W .Let ¯ a be such as in Lemma 6.1 and k ∈ N such that ¯ a ∈ X k . Write¯ a = ( a , . . . , a k ), p for qftp(¯ a ) and Z = { w (¯ a ) : w ∈ W } ⊆ p ( X ). Notethat since p ( X ) = S n g n Z , the set Z is nonmeager in p ( X ). Choose an m -weakly isolated sequence ¯ a n such that each ¯ a n belongs to Z and for each n ∈ N let v n ∈ W be such that ¯ a n = v n (¯ a ). Let ε n >
0, and T n , X n ⊆ p ( X )witness that the sequence of ¯ a n is m -weakly isolated (i.e. X n is T n -relatively ε n -saturated in X and T n ( m, ε n )-generates an open set over ¯ a n ).Given two subspaces X ′ , X ′′ of X and a set C of partial automorphismsfrom X ′ to X ′′ , say that a subset G of G is full for C if every element of C can be extended to an element of G (cf. [45, Claim 1]). Claim 6.4.
There is n ∈ N such that W is full for C n = { ϕ : ¯ a n → ball X (¯ a n , ε n ) ∩ X n : ϕ is an isomorphic embedding } . Proof.
First note that there is n ∈ N such that g n W is full for C n . If not,then for each n ∈ N there is ϕ n ∈ C n such that ϕ n cannot be extended toan element of g n W . Since ¯ a n are isolated, there is an automorphism ϕ of X which extends all the ϕ n . Then ϕ / ∈ S n g n W , which is a contradiction.Now, if g n W is full for C n , then so is W = ( g n W ) − ( g n W ) as C n containsthe identity. (cid:3) Fix n as in Claim 6.4 and write T = v − n ( T n ). Claim 6.5.
We have { g ∈ G : d X (¯ a, g (¯ a )) < ε n and g (¯ a ) ∈ T } ⊆ W . Proof.
Let g ∈ G be such that g (¯ a ) ∈ T and d X ( a i , g ( a i )) < ε n for each i ≤ k .Let Y = v − n ( X n ). Note that since v n is an automorphism and X n is T n -relatively ε n -saturated over ¯ a n , we get that Y is T -relatively ε n -saturatedover ¯ a . Thus, there is ¯ b ∈ Y withqftp(¯ b/ ¯ a ) = qftp( g (¯ a ) / ¯ a ) . By homogeneity of X , there is w ∈ G ¯ a such that w ( g (¯ a )) = ¯ b . Note that w ∈ W as G ¯ a ⊆ W . UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 15
Look at v n w − gv − n ∈ G and note that it maps ¯ a n to ball X (¯ a n , ε n ) ∩ X n because v n maps Y to X n . By Claim 6.4 and the choice of n , there is w ∈ W which is equal to v n w − gv − n on ¯ a n . This means that w − v n w − gv − n ∈ G ¯ a n and thus v − n w − v n w − g ∈ G ¯ a . Therefore, v − n w − v n w − g ∈ W andthus g ∈ W . (cid:3) Now note that T ( m, ε n )-generates an open set over ¯ a , so let U ⊆ p ( X )be nonempty and such that for every ¯ b ∈ U there is a sequence g , . . . , g m ∈ Aut( X ) such that • g m . . . g (¯ a ) = ¯ b , • g i (¯ a ) ∈ T and d X ( g i (¯ a ) , ¯ a ) < ε n for each i ≤ m .We claim that { g ∈ G : g (¯ a ) ∈ U } ⊆ W m +10 . To see this, let g ∈ G be such that g (¯ a ) ∈ U and let ¯ b = g (¯ a ). Find g , . . . , g m as above and note that by Claim 6.5 we have that g i ∈ W for each i ≤ m .Now, g − g m . . . g ∈ G ¯ a , so g ∈ W m +10 , as needed.The set { g ∈ G : g (¯ a ) ∈ U } is open in G , so we have that G is (24 m + 10)-Steinhaus. This ends the proof. (cid:3) Triviality of homomorphisms
In this section we study the circumstances under which one can excludenontrivial homomorphisms from the groups of the form Aut( M ) to certaintopological groups H . Given two groups G and H say that G is H -trivial ifany homomorphism from G to H is trivial. Tsankov [53] concluded (fromthe minimality of the unitary group) that whenever H is a separable groupwhich admits a complete left-invariant metric, then the unitary group is H -trivial. In this section, we isolate an abstract property of a metric structure M which implies that Aut( M ) is H -trivial for H as above and in Section 10we will see that this property is satisfied by the Hilbert space. The same istrue for the group Aut( X, µ ) but here it follows immediately from the factthat Aut(
X, µ ) is simple [10]. The unitary group is not simple but (similarlyas the group of isometries of the Urysohn space [52]) has a maximal propernormal subgroup [11]. We do not know, however, if the methods below applyto U .Given a subset N of a metric structure M and ¯ a ∈ M <ω , say that N is relatively saturated over ¯ a if it is α -relatively saturated over ¯ a for every α ∈ [0 , ∞ ). Definition 7.1.
Suppose M is a homogeneous metric structure and ¯ a ∈ M k for some k ∈ N . Write p for qftp(¯ a ). Say that a sequence (¯ a n : n ∈ N ) ofelements of M k is an independent sequence in p if every ¯ a n realizes p andthere exists a sequence of subsets N n such that N n is relatively saturatedover ¯ a n such that for every sequence ¯ b n ∈ N n with qftp(¯ b n ) = qftp(¯ a n ) thereis an automorphism ϕ of M with ϕ (¯ a n ) = ¯ b n for every n ∈ N . Definition 7.2.
Say that a metric structure M admits independent se-quences if for every k ∈ N and ¯ a ∈ M k , for every sequence (¯ s n : n ∈ N ) offinite tuples of elements of M , there is a sequence ¯ a n which is independentin qftp(¯ a ) and is such that ¯ a n +1 ≡ ¯ s n ¯ a n holds for each n ∈ N . Theorem 7.3.
Suppose X is a complete metric structure that admits inde-pendent sequences, has locally finite automorphisms and the extension prop-erty. Then the group Aut( X ) is H -trivial for every Polish group H whichhas a complete left-invariant metric.Proof. Write G for Aut( X ) and suppose H has a complete left-invariantmetric. Fix ϕ : G → H and we will show that ϕ is trivial. To do this, it isenough to see that whenever U ⊆ H is an open neighborhood of the identity,then ϕ ( G ) ⊆ U .Let then U ⊆ H be an open neighborhood of the identity. Find V ⊆ H open neighborhood of the identity such that V ⊆ U and write T = ϕ − ( V ).Note that T ⊆ G is countably syndetic, so by Lemma 6.1, there is k ∈ N and ¯ a ∈ X k such that G ¯ a ⊆ T . Write p for the quantifier-free type of ¯ a .Let d H be a complete left-invariant metric on H and pick a decreasingsequence of open neighborhoods V n of the identity in H with V = V suchthat diam d H ( V n ) ≤ − n . Let also W n ⊆ H be open symmetric neighbor-hoods of the identity in H with ( W n ) ⊆ V n . Note that each ϕ − ( W n )is countably syndetic in G . Using Lemma 6.1, by induction on n > s n of finite tuples of elements of X such that G ¯ s n ⊆ ϕ − ( W n ) ⊆ ϕ − ( V n ). Let ¯ s be the empty tuple.Write ¯ a = ¯ a . Using the assumption that X admits independent se-quences, pick a sequence g n ∈ G for n > g n ∈ G ¯ s n and thesequence (¯ a n : n >
0) defined as a n +1 = g n (¯ a n ), forms an independent se-quence in p . Let X n ⊆ X witness the that the sequence is independent, sothat each X n is relatively saturated over ¯ a n .Let f n = g n . . . g and write h n = ϕ ( f n ). Note that h − n is d H -Cauchy in H and hence convergent. Thus, h n is convergent in H too. Let h = lim n h n . Claim 7.4.
There is n such that f n T f − n is full for C n = { ϕ : ¯ a n → X n : ϕ is an isomorphic embedding } . Proof.
We will first prove that there is a sequence b n ∈ G such that G = S n b n T f − n . This will follow from the fact that there is a sequence a n ∈ H such that H = S n a n V h − n by taking b n such that ϕ ( b n ) = a n .To see the latter fact, pick a n so that they are dense in H and we claimthat H = S n a n V h − n . Indeed, if x ∈ H , then note that the sequence z n = xh n = xϕ ( g n ) . . . ϕ ( g )is convergent in H to z = xh . Pick a subsequence a k n converging to z . Since z k n converges to z , we get that d H ( z k n , a k n ) →
0. Thus, d H ( a − k n z k n , H ) → n such that a − k n xh k n ∈ V and then x ∈ a k n V h − k n . UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 17
Now, note that there is n ∈ N such that b n T f − n is full for C n . If not,then for each n ∈ N there is ϕ n ∈ C n such that ϕ n cannot be extended toan element of b n T f − n . As the sequence ¯ a n is independent, there is an auto-morphism ϕ of X which extends all the ϕ n . But then ϕ / ∈ S n a n T f − n ,which is a contradiction. Finally, if a n T f − n is full for C n , then so is f n T f − n = ( a n T f − n ) − ( a n T f − n ) since C n contains the identity. This provesthe claim. (cid:3) We need to prove that G ⊆ ϕ − ( U ). Let g ∈ G be arbitrary. Fix n as inClaim 7.4 and note that f n (¯ a ) = ¯ a n . Let Y = f − n ( X n ). Note that Y and X realize the same quantifier-free n -types over ¯ a . This follows from the factthat X n is relatively saturated over ¯ a n and f n is an automorphism.Thus, there is ¯ b ∈ Y such thatqftp(¯ b/ ¯ a ) = qftp( g (¯ a ) / ¯ a ) . Let w ∈ G ¯ a be such that w ( g (¯ a )) = ¯ b . Note that w ∈ T as G ¯ a ⊆ W .Look at f n w − gf − n ∈ G and note that it maps ¯ a n to X n as f n maps Y to X n . By Claim 7.4, there is y ∈ f n T f − n which is equal to f n w − gf − n on¯ a n . Write y = f n w f − n for some w ∈ W and note that y − ( f n w − gf − n ) = f n w w − gf − n ∈ G ¯ a n . Therefore, w w − g = f − n ( y − ( f n w − gf − n )) f n fixes ¯ a , which means that w w − g ∈ T and thus g ∈ T ⊆ ϕ − ( U ).This shows that G ⊆ ϕ − ( U ) and since U ⊆ H was an arbitrary openneighborhood of the identity, we have that ϕ is trivial. This ends the proof. (cid:3) The Urysohn space
There is no essential difference in verifying that the Urysohn space andthe Urysohn sphere satisfy the assumptions of Theorem 1.4. We will focusonly on the Urysohn space.Locally finite automorphisms for the Urysohn space have been alreadyshown by Solecki [48], who proved that for any finite metric space X andfinitely many partial isometries of X , there is a metric space Y containing X such that all these partial isometries extend to isometries of Y . Soleckiderived his result from an extension theorem of Herwig and Lascar [17].The theorem of Herwig and Lascar is connected to the Rhodes’ Type IIConjecture proved independently by Ash [1] and by Ribes and Zalesskiˇı[40]. The latter results concern the profinite topology on the free groups (cf.also [39]).Recall that the profinite topology on a free group F n is the one withthe basis at the identity consisting of finite-index subgroups of F n . In theliterature, the fact that a set A ⊆ F n is closed in the profinite topology isusually referred to as by saying that A is separable . A classical result of M. Hall, Jr. [16] says that any finitely generated subgroup of F n is separable.Note that it also implies that any coset of a finitely generated subgroup of F n is separable (since the multiplication is continuous in the profinte topology).The main result of Ribes and Zalesskiˇı [40] states that products of finitelymany finitely generated subgroups of F n are also separable. Again, notethat it immediately implies that products of finitely many cosets of finitelygenerated subgroups of F n are separable as well.An abstract connection between the theorem of Ribes and Zalesskiˇı andextensions of partial isometries was discovered by Rosendal [44], who ex-pressed it in the language of finitely approximable actions and, in particu-lar, gave a new proof of the result of Solecki [48]. On the other hand, thepaper of Solecki [48] contains a very elegant argument on the extensions ofone isometry. That argument is done in the style of Mackey’s constructionsof induced actions [33, Page 190] (cf. [3, 2.3.5]) and a similar argumenthas been used by Hrushovski [20] in the context of extensions of partialisomorphisms of graphs.Below, we present a new proof of Solecki’s theorem [48], which exploitsthe ideas used in the case of one isometry in [48, Section 3] and is also donein the style of Mackey’s construction of induced actions. Theorem 8.1 (Solecki) . The Urysohn space has locally finite automor-phisms.Proof (`a la Mackey).
By the finite extension property of the Urysohn space,it is enough to show that for every finite metric space X , for every tuple ϕ , . . . , ϕ n of partial isometries of X there is a finite metric space Y ⊇ X such all ϕ , . . . , ϕ n extend to isometries of Y .Let X be a finite metric space. Write δ = min { d X ( x, y ) : x, y ∈ X, x = y } and let ∆ = diam( X ). Suppose ϕ , . . . , ϕ n are partial isometries of X .Write a , . . . , a n for the generators of the free group F n . Write also A forthe set { a , . . . , a n , a − , . . . , a − n } and W n for A ∗ (the set of all words over A ). For a word w ∈ W n with w = v . . . v k , v i ∈ A and x ∈ X say that w ( x ) is defined if there is a sequence of points x j ∈ X ( j ≤ k ) with x = x and x j +1 = ϕ i ( x j ) if v k − j = a i and x j +1 = ϕ − i ( x j ) if v k − j = a − i . If w ( x ) isdefined, then write w ( x ) = y for y = x k as above. We also use the notation w ( x ) if w belongs to F n (using the reduced word for w ).For each x, y ∈ X write T yx for the set of w ∈ F n such that w ( x ) = y . Claim 8.2.
For every x, y ∈ X the set T yx is either empty or a coset of afinitely generated subgroup of F n .Proof. If T yx is nonempty, then let w ∈ F n be such that w ( x ) = y . Notethat whenever w ′ ∈ T yx , then w − w ′ ∈ T xx . Now, T xx is a finitely generatedsubgroup of F n : it is the fundamental group of the graph whose verticesare the points in X and (labelled) edges connect x, y if ϕ i ( x ) = y for some i ≤ n . Therefore, T yx = wT xx is a coset of a finitely generated subgroup. (cid:3) UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 19
Claim 8.3.
For every m ∈ N and x , y , . . . , x m , y m ∈ X the set T y x · . . . · T y m x m is closed in F n in the profinite topology.Proof. This follows from the Ribes–Zalesskiˇı theorem [40] and Claim 8.2. (cid:3)
We need to define an extension of X . It will be obtained by dividing X × F n by certain equivalence relation ≃ so that x ( x, e ) / ≃ is an embedding.We will have to make sure that the extension is finite and define a metric onit so that the embedding is isometric. Before we make this definition precise,let us comment on how the metric on ( X × F n ) / ≃ will be defined. Note thatthere is a partial distance function d on X × F n , namely for ( x, w ) , ( y, w )with x, y ∈ X and w ∈ F n we put d (( x, w ) , ( y, w )) = d X ( x, y ). Now, if ≃ is an equivalence relation on X × F n , then there is a natural distancefunction on ( X × F n ) / ≃ defined as follows. If C, D ∈ ( X × F n ) / ≃ , thenput d ( X × F n ) / ≃ ( C, D ) to be the minimum of ∆ and the sums of the form( ∗∗ ) m − X i =0 d ( z i , z ′ i +1 )such that z , z , z ′ , . . . , z m − , z ′ m − , z ′ m ∈ X × F n , the value d ( z i , z ′ i +1 ) isdefined for each 0 ≤ i < m and there is a sequence C , . . . , C m of elementsof X × F n / ≃ with C = C, C m = D and z ∈ C , z ′ m ∈ C m and z j , z ′ j ∈ C j for 0 < j < m .Now we will define the equivalence relation ≃ and check the details ofthe construction described above. For that, we need a couple of definitions.Given x, y ∈ X , a chain from x to y is a sequence z , z , z ′ , . . . , z m − , z ′ m − , z m , z ′ m ∈ X such that z = x, z m = y and for each 1 ≤ i ≤ m there exists w i ∈ W n suchthat w i ( z i ) = z ′ i . The distance of a chain z , z , z ′ , . . . , z m − , z ′ m − , z m , z ′ m is defined as P m − i =0 d A ( z i , z ′ i +1 ). A word realization of a chain as aboveis a sequence of words w , . . . , w m ∈ W n such that w i ( z i ) = z ′ i for each1 ≤ i ≤ m . Call a chain trivial if it has a word realization w , . . . , w m ∈ W n such that w · . . . · w m = e holds in F n .Let now M ∈ N be such that M δ > ∆. Note that since X is finite,there are only finitely many nontrivial chains z , z , z ′ , . . . , z m , z ′ m ∈ X with m ≤ M . For each nontrivial chain z , z , z ′ , . . . , z m , z ′ m ∈ X with m ≤ M ,the set T z ′ z · . . . · T z ′ m z m is a closed subset of F n which does not contain e . UsingClaim 8.3, find a finite index normal subgroup H ⊳ F n which is disjoint fromevery T z ′ z · . . . · T z ′ m z m as above.Write Z for X × F n and define an equivalence relation ≃ on Z as follows.Given w , w ∈ F n write ( x , w ) ≃ ( x , w )if there is v ∈ F n with w − w H = vH and v ( x ) = x . Given ( x, w ) ∈ Z write [ x, w ] for its ≃ -class. Write Y for Z/ ≃ and note that Y is finite. The latter follows from thefact that if F n /H = { d H, . . . , d t H } , then Y = { [ x, d i ] : x ∈ X, i ≤ t } . Now,define a metric d Y on Y as follows. Let d Y ([ x, w ] , [ y, v ]) be the minimum of∆ and the set of sums of the form m − X i =0 d X ( z i , z ′ i +1 )for sequences z , z , z ′ , . . . , z m , z ′ m of elements of X such that • z = x, z m = y , • and there are w i ∈ F n (for 0 ≤ i ≤ m ) with w = w, w m = v and( z ′ i , w i − ) ≃ ( z i , w i ) for each 1 ≤ i ≤ m .Note that a sum as above is equal to 0 exactly when z i = z ′ i +1 for every i < m and hence the definition of d Y does not depend on the representativesof ≃ -classes and defines a metric on Y . Note that this definition coincideswith the formula given by ( ∗∗ ).Define an embedding of X into Y via x [ x, e ]. We claim that thisis an isometric embedding and that each ϕ i extends to an isometry of Y .The second part is clear given the first one since for each i ≤ n the map[ x, w ] [ x, a i w ] is well-defined and is easily seen to be an isometry of Y which extends ϕ i . Thus, we only need to show that x [ x, e ] is an isometricembedding. Claim 8.4.
For any x, y ∈ X and w ∈ F n the distance d Y ([ x, e ] , [ y, w ]) isequal to the minimum of ∆ and the minimal distance of a chain from x to y which has a word realization v , . . . , v k such that v · . . . · v k H = wH .Proof. If z , z , z ′ , . . . , z m , z ′ m in X are such that d Y ([ x, e ] , [ y, w ]) = m − X i =0 d X ( z i , z ′ i +1 )and w , . . . , w n in F n are such that w = e, w m = w and ( z ′ i , w i − ) ≃ ( z i , w i )for each 1 ≤ i ≤ m , then find v i ∈ F n (for 1 ≤ i ≤ m ) such that v i H = w − i − w i H and v i ( z ′ i ) = z i . Then z , z , z ′ , . . . , z m , z ′ m is a chain and v , . . . , v m is its word realization with v . . . v m H = w − w . . . w − m − w m H = w m H = wH .On the other hand, if z , z , z ′ , . . . , z m , z ′ m in X forms a chain from x to y with a word realization v , . . . , v k such that v · . . . · v k H = wH , then write w = e and w i = w i − v i for 1 ≤ i < m and w m = w . Then the sequence z , z , z ′ . . . , z m , z ′ m together with w , . . . , w m satisfies the conditions in thedefinition of d Y . (cid:3) Consequently, as d X ( x, y ) ≤ ∆, by Claim 8.4, we have that d Y ([ x, e ] , [ y, e ])is the minimal distance of the chains from x to y which have a word real-ization w , . . . , w k with w · . . . · w k ∈ H . Say that a chain c from x to y realizes the distance if the distance of c is equal to d Y ([ x, e ] , [ y, e ]). We need UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 21 to show that if a chain c realizes the distance from x to y , then its distanceis equal to d X ( x, y ). Claim 8.5.
Suppose x, y ∈ X and c is a chain from x to y with a wordrealization w , . . . , w m ∈ W n . If w i = v i a and w i +1 = a − v i +1 for some ≤ i < m with v i , v i +1 ∈ W n and a ∈ A , then there is a chain c ′ from x to y which has the same distance as c and a word realization w ′ , . . . , w ′ m ∈ W n such that w ′ j = w j for j = i, i + 1 , w j = v i for j = i, i + 1 .Proof. Write c = ( z , z , z ′ , . . . , z m , z ′ m ). Let ϕ = ϕ k if a = a k and ϕ = ϕ − k if a = a − k . Consider the chain c ′ = ( y , y , y ′ , . . . , y m , y ′ m ) with y j = z j for j = i and y ′ j = z ′ j for j = i + 1, and y i = ϕ ( z i ), y ′ i +1 = v i +1 ( z i +1 ). Thedistance of c ′ is the same as that of c since d ( y i , y ′ i +1 ) = d ( ϕ ( z i ) , v i +1 ( z i +1 )) = d ( z i , ϕ − ( v i +1 ( z i +1 )))as ϕ is an isometry. And we have ϕ − ( v i +1 ( z i +1 )) = w i +1 ( z i +1 ) = z ′ i +1 . (cid:3) Given two chains c = ( z , z , z ′ . . . , z m , z ′ m ) and c ′ = ( z , z , z ′ . . . , z k , z ′ k ),both from x to y , say that c is shorter than c ′ if m < k and the distance of c is not greater than that of c ′ . Say that a word realization w , . . . , w m of achain has a trivial element if there is 0 < i < m with w i = e . Claim 8.6.
If a chain c = ( z , z , z ′ . . . , z m , z ′ m ) from x to y has a wordrealization w , . . . , w m with a trivial element, then there is a chain from x to y which is shorter than c and has a word realization w ′ , . . . , w ′ k with w . . . w m = w ′ . . . w ′ k .Proof. Suppose w i = e , i.e. z i = z ′ i . Consider the chain z , . . . , z i − , z ′ i − , z i +1 , z ′ i +1 , . . . , z m , z ′ m and note that w , . . . , w i − , w i +1 , . . . , w m ∈ W n is its word realization. Thefact that this chain is shorter than c follows from the triange inequality. (cid:3) Claim 8.7.
If a chain c = ( z , z , z ′ . . . , z m , z ′ m ) from x to y has a wordrealization w , . . . , w m and z i = z ′ i +1 for some < i < m , then there isa chain from x to y which is shorter than c and has a word realization w ′ , . . . , w ′ k with w . . . w m = w ′ . . . w ′ k .Proof. If z i = z ′ i +1 , then consider the chain c ′ of y , y , y ′ , . . . , y m − , y ′ m − with y j = z j for j < i , y j = z j +1 for j ≥ i , y ′ j = z ′ j for j ≤ i and y ′ j = z ′ j +1 for j > i . Note that it is still a chain from x to y with a word realization w ′ , . . . , w ′ m − with w ′ j = w j for j < i , w ′ i = w i w i +1 and w ′ j = w j +1 for j > i . (cid:3) Claim 8.8.
If chain c = ( z , z , z ′ . . . , z m , z ′ m ) from x to y realizes thedistance from x to y and cannot be made shorter, then m = 1 and z m = z ′ m .Proof. Note that by Claim 8.7 and the assumption that
M δ > ∆ we havethat m ≤ M . First note that the chain must be trivial. Indeed, sinceotherwise, for any word realization w , . . . , w m of c we have w . . . w m ∈ T z ′ z · . . . · T z ′ m z m and the latter set is disjoint from H if the chain is nontrivial.Now, since the chain is trivial, it has a word realization w , . . . , w m suchthat w . . . w m = e . Now, if m ≥
2, then Claims 8.5 and 8.6 imply that thechain can be made shorter. Therefore, m = 1 and w m = e . (cid:3) Note finally that since ( x, y, y ) is a chain from x to y , Claim 8.8 impliesthat d Y ([ x, e ] , [ y, e ]) = d X ( x, y ) and we have that x [ x, e ] is an isometricembedding, as needed. This ends the proof. (cid:3) Lemma 8.9.
The Urysohn space has the extension property.Proof.
This is a standard amalgamation argument. Note that since thelanguage of metric spaces does not have any function symbols, instead offinitely generated structures, we talk about finite tuples. Suppose then that¯ b = ( b , . . . , b n ) , ¯ c = ( c , . . . , c m ) , ¯ a = ( a , . . . , a k ) are finite tuples in U .Write B = { b , . . . , b n } , C = { c , . . . , c m } , A = { a , . . . , a k } and suppose A ⊆ B ∩ C . Let C ′ be copy of C with B ∩ C ′ = A and let D = B ∪ C ′ be a metric space with the metric d D such that d D ↾ B = d U ↾ B , d D ↾ C ′ = d U ↾ C (under the natural identification) and if b ∈ B, c ∈ C ′ ,then d D ( b, c ) = min { d U ( b, a ) + d U ( a, c ) : a ∈ A } . Assume without loss ofgenerality that D is embedded into U over B and note that C ′ ≡ A C and C ′ | ⌣ A B . This ends the proof. (cid:3) To check that the Urysohn space admits isolated sequences, we need tointroduce a couple of definitions. Given a metric structure M and a tuple¯ a ∈ M k and ε > M (¯ a, ε ) for { x ∈ M : d M ( x, a i ) < ε for some i ≤ k } . Suppose M is a homogeneous metric structure, ¯ a ∈ M k for some k ∈ N and p = qftp(¯ a ). We say that a sequence (¯ a n : n ∈ N ) of elements of p ( M ) is isometrically isolated if there exists a sequence of ε n ∈ (0 , ∞ ) and isometricembeddings η n : ball M (¯ a, ε n ) → M such that η n (¯ a ) = ¯ a n and for every sequence ¯ b n ∈ rng( η n ) such that qftp(¯ b n ) =qftp(¯ a n ) and d M (¯ a n , ¯ b n ) < ε n there is an automorphism ϕ of M with ϕ (¯ a n ) = ¯ b n for every n ∈ N .Note that any isometrically isolated sequence is isolated since if p =qftp(¯ a ) and η : ball M (¯ a, ε ) → M is an isometric embedding for some ε > { ¯ b = ( b , . . . , b k ) ∈ p ( M ) : ∀ i ≤ k b i ∈ rng( η n ) } is relatively ε -saturated over η (¯ a ).Given k ∈ N , say that a sequence (¯ a n : n ∈ N ) of k -tuples of elements of ametric structure M is nontrivial convergent if it is convergent as a sequencein M k and if ¯ a ∞ = ( a ∞ , . . . , a ∞ k ) is its limit and ¯ a n = ( a n , . . . , a nk ), then a ni = a ∞ j and a ni = a mj for any ( n, i ) = ( m, j ) ∈ N . In particular, a ni = a nj for every n ∈ N and i = j . Note that, in case k = 1, a nontrivial convergentsequence is a convergent sequence such that all its elements are distinct anddifferent from its limit. UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 23
A basic property of the Urysohn space that we will use in the argumentsbelow, due to Huhunaiˇsvili [21] (cf. [38, Proposition 5.1.20]), says that anypartial isometry between compact subspaces of U can be extended to anisometry of U . Lemma 8.10.
For every k ∈ N and a quantifier-free k -type p , any nontrivialconvergent sequence in p is isometrically isolated in p .Proof. Let p be the quantifier-free type of ¯ a ∈ U k . Write ¯ a = ( a , . . . , a k )and let δ ij = d U ( a i , a j ) for i, j ≤ k .Let ¯ a n be a nontrivial convergent sequence in p . Assume that ¯ a n convergesto ¯ a ∞ = ( a ∞ , . . . , a ∞ k ). Write ¯ a n = ( a n , . . . , a nk ) for each n ∈ N . For each n, m ∈ N and i, j ≤ k let δ nmij = d U ( a ni , a mj ) and δ n ∞ ij = d U ( a ni , a ∞ j ) and notethat lim m,n →∞ δ nmij = δ ij and lim n →∞ δ n ∞ ij = δ ij .For each n ∈ N choose ε n > ε n < δ nmij for each m = n and i, j ≤ k as well as ε n < δ nnij = δ ij for all i = j , i, j ≤ k . Such an ε n > a n : n ∈ N ) is nontrivial and lim m →∞ δ nmij = d U ( a ni , a ∞ j ) > n ∈ N write A n for ball( { a n , . . . , a nk } , ε n ). Note that A n is adisjoint union of balls around the points a ni . Consider the metric space B ′ which is the disjoint union S n ∈ N B ′ n with each B ′ n a copy of A n (say the copyof a ni in B k is a ni ′ ) and let the metric on B ′ be defined so that it is equalto the original metric d U on each B ′ n and if x, y ∈ S n ∈ N B ′ n are such that x ∈ B ′ n and y ∈ B ′ m with n = m and x ∈ ball( a ni ′ , ε n ), y ∈ ball( a mj ′ , ε m ),then d B ′ ( x, y ) = δ nmij . Note that d B ′ is a metric by the choice of the numbers ε n . Now let B ∞ = { a ∞ ′ , . . . , a ∞ k ′ } be a copy of { a ∞ , . . . , a ∞ k } and let B = B ′ ∪ B ∞ with themetric d B = d B ′ on B ′ , d B = d B ∞ on B ∞ and if x ∈ B ′ is such that x ∈ B n with x ∈ ball( a nj ′ , ε n ) and i ≤ k , then d B ( x, a ∞ i ′ ) = δ n ∞ ji . Since the subspace of B consisting of the points a ni ′ and a ∞ i ′ for n ∈ N and i ≤ k is compact and isometric to the subspace of U consisting of the points a ni and a ∞ i for n ∈ N and i ≤ k , the Huhunaiˇsvili theorem [21] implies thatthe map a ni ′ a ni and a ∞ i ′ a ∞ i extends to an isometric embedding η of B into U .For each n ∈ N write B n for the image of B ′ n under η . Note that each B n is an isometric copy of ball(¯ a, ε n ). We claim that the sets B n (treatedas the embeddings of ball(¯ a, ε n )), together with the numbers ε n witnessthat ¯ a n is isometrically isolated. For that, pick a sequence of isometricembeddings ϕ n : { a n , . . . , a nk } → B n with d U ( ϕ n ( a ni ) , a ni ) < ε n for each i ≤ k .Consider a partial isometry ϕ ′ of U with dom( ϕ ′ ) = S n ∈ N { a n . . . , a nk } ∪{ a ∞ , . . . , a ∞ k } such that ϕ ′ ( a ∞ i ) = a ∞ i and ϕ ′ ( a ni ) = ϕ n ( a ni ). Note that ϕ ′ isa partial isometry of the Urysohn space with compact domain, so again by the Huhunaiˇsvili theorem [21], there is an isometry ϕ ∈ Iso( U ) that extends ϕ ′ . Clearly, ϕ extends each ϕ n , which shows that B n are as needed and thesequence is isometrically isolated. This ends the proof. (cid:3) Proposition 8.11.
The Urysohn space admits isolated sequences.Proof.
Suppose p is the quantifier-free k -type of a tuple ¯ a = ( a , . . . , a k ).First note that we can assume that ¯ a consists of distinct elements. Indeed,otherwise one can remove repetitions from ¯ a and work with a quantifier-free m -type q for some m < k . Then, for every m -tuple ¯ b ∈ q ( M ) there is aunique tuple ¯ b ′ ∈ p ( M ), which contains ¯ b such that • if (¯ b n : n ∈ N ) is isolated in q , then (¯ b ′ n : n ∈ N ) is isolated in p , • the map ¯ b ¯ b ′ is a homeomorphism of q ( M ) and p ( M ).Now, suppose Z ⊆ p ( U ) is nonmeager. Without loss of generality (re-stricting to an open subset of p ( U ) if neccessary), assume that Z is non-meager in every nonempty open set. Pick any ¯ a ∞ ∈ p ( U ) with ¯ a ∞ =( a ∞ , . . . , a ∞ k ) and note that a ∞ i = a ∞ j for i = j . Using the assumptionthat Z ∩ V is nonmeager for every open neighborhood V of ¯ a ∞ , construct asequence ¯ a n of elements of Z convergent to ¯ a ∞ such that if ¯ a n = ( a n , . . . , a nk ),then a ni = a ∞ j and a ni = a mj for any n, m ∈ N and i, j ≤ k with ( n, i ) = ( m, j ).This sequence is then nontrivial convergent and hence isolated by Proposi-tion 8.10. (cid:3) The measure algebra
Recall that given a standard probability space ( X, B , µ ) we define theequivalence ≈ on B by A ≈ B if µ ( A ∆ B ) = 0 and the measure algebra isthe family of equivalence classes of sets in B . Given A ∈ B write [ A ] for its ≈ -equivalence class (although we will often abuse notation and write only A in-stead of [ A ]). The measure algebra is then the family of ≈ -classes of the setsin B . It becomes a metric space with the metric d MALG ([ A ] , [ B ]) = µ ( A ∆ B )and we treat it as a metric structure together with this metric, the operationof symmetric difference ∆ and the empty set as a constant. We write MALGfor the structure ( B / ≈ , d MALG , ∆ , ∅ ). The Sikorski duality [27, Theorem15.9] connects automorphisms of MALG and measure-preserving bijectionson the space X . In particular, it implies that the group of automorphismsof MALG with the topology of pointwise convergence is isomorphic to thegroup of measure-preserving bijections Aut( X, µ ) with the weak topology(see [28, Section 1]). For more details about the measure algebra and thestandard measure space we refer the reader to [37, Chapter 22] and to [12,Chapter 32].Throughout the proofs below we often use the fact (cf [29, Lemma 7.10])that whenever
A, B ⊆ X have the same measure, then there is a measure-preserving bijection f : X → X such that f ( A ) = B . Below, given a finitesubalgebra A of MALG, we write atom( A ) for the set of atoms of A . Lemma 9.1.
The measure algebra
MALG has locally finite automorphisms.
UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 25
Proof.
Note that finitely generated substructures of MALG are finite sub-algebras. Thus, to show locally finite automorphism we need to prove thefollowing. For every finite subalgebra
A ⊆
MALG there exists a finite al-gebra
B ⊆
MALG with
A ⊆ B such that every partial automorphism of A extends to an automorphism of B . To see this, we need a couple of notions.Given finite A ⊆ B ⊆
MALG and A , A ∈ A say that A and A are iden-tically partitioned by B if the sets { µ ( A ∩ B ) : B ∈ atom( B ) , B ⊆ A } and { µ ( A ∩ B ) : B ∈ atom( B ) , B ⊆ A } (both counted with repetitions) areequal (up to a permutation). Note that if A ⊆ B ⊆
MALG are finite andsuch that every A , A ∈ A of the same measure are identically partitionedby B , then any partial automorphism of A extends to an automorphism of B . Moreover, it is enough to guarantee this for A and A disjoint.Say that a finite extension A ⊆ B is good if every two atoms of A of thesame measure are identically partitioned by B . Note that this is a transitiverelation and if A ⊆ B is good and A , A ∈ A are identically partitioned by A , then A and A are identically partitioned by B .Enumerate as (( A i , B i ) : 1 ≤ i < N ) the set of all pairs A, B of disjointsets in A of the same measure. By induction on i ≤ N construct a sequenceof finite subalgebras A i ⊆ MALG with A = A such that for every 1 ≤ i ≤ N we have • A i − ⊆ A i is good • A i and B i are identically partitioned by A i +1 .After this is done, the algebra B = A N will be as needed.It is enough to describe the induction step construction of A i +1 from A i .Note first that (by shrinking A i and B i if neccessary) we can assume thatfor every A, B ∈ atom( A i ) with A ⊆ A i and B ⊆ B i we have µ ( A ) = µ ( B ). Write R = { µ ( A ) : A ∈ atom( A i ) , A ⊆ A i } and S = { µ ( B ) : B ∈ atom( A i ) , B ⊆ B i } , so that R ∩ S = ∅ . Write a = µ ( A i ) = µ ( B i ) and let U ⊆ X be a measurable set of measure a . Let C be the algebra of subsets of A i equal to A i ↾ A i and C be the algebra of subsets of B i equal to A i ↾ B i .Find two algebras C ′ and C ′ of subsets of U such that C ′ is isomorphic to C , C ′ is isomorphic to C and C ′ , C ′ are (stochastically) independent. Fixmeasure-preserving bijections ϕ : A i → U , ψ : B i → U such that ϕ maps C to C ′ and ψ maps C to C ′ . Write C ′ for the algebra of subsets of U generatedby C ′ and C ′ and let C be the algebra of subsets of A i ∪ B i generated by thepreimages ϕ − ( C ) and ψ − ( C ) for C ∈ C ′ . Note that the sets A i and B i areidentically partitioned by the algebra generated by A i and C . Note also thatif C ⊆ A i is an atom of A i of measure r , then C is partitioned by C intosets of measures r · µ ( D ) for D ∈ atom( C ) and analogously, if E ⊆ B i is anatom of A i of measure s , then D is partitioned by C into sets of measures s · µ ( F ) for F ∈ atom( C ). In order to construct a good extension of A i ,partition every atom A ∈ atom( A i ) that is disjoint from A i ∪ B i as follows:(i) if µ ( A ) ∈ R , then partition A into sets of measures µ ( A ) · µ ( D ), for D ∈ atom( C ), (ii) if µ ( A ) ∈ S , then partition A into sets of measures µ ( A ) · µ ( F ), for F ∈ atom( C ).Let A i +1 be an extension of A i generated by all the partitions as in (i) and(ii) above and by C . Now A i +1 is a good extension of A i and the sets A i and B i are identically partitioned by A i +1 . This ends the construction andthe proof. (cid:3) Lemma 9.2.
The measure algebra
MALG has the extension property.Proof.
Suppose
A, B, C are finitely generated subalgebras of MALG with A ⊆ B ∩ C . Write A , . . . , A n for the set of atoms of A . Find an auto-morphism ϕ of the measure space which fixes A , . . . , A n and within each A i sends the atoms of C contained in A i to sets which are (stochastically)independent from the atoms of B contained in A i . It is easy to see that ϕ ( C ) | ⌣ A B . (cid:3) To see that MALG admits isolated sequences, we need to understandwhich quantifier-free ε -types are realized over finite tuples in MALG. Definition 9.3.
Suppose k ∈ N and P = ( A , . . . , A k ) is a partition of X into positive measure sets. Let E = ( e ij : 1 ≤ i, j ≤ k ) be a matrix of reals.Say that E is P -additive if the following conditions hold: • e ii ≥ ≤ e ij ≤ µ ( A i ) + µ ( A j ) for every i, j ≤ k , • the following equations are satisfied: e ii = X j = i µ ( A i ) + µ ( A j ) − e ij e ii = X j = i µ ( A i ) + µ ( A j ) − e ji Claim 9.4.
Suppose P = ( A , . . . , A k ) is a partition of X into positivemeasure sets and ϕ ∈ Aut(MALG) . Let e ij = d MALG ( A i , ϕ ( A j )) . Then thematrix E = ( e ij : 1 ≤ i, j ≤ k ) is P -additive.Proof. Let f : X → X be a measure-preserving bijection that induces ϕ .For i = j write ε ij = µ ( f ( A i ) ∩ A j ). Note that e ij = µ ( f ( A i )∆ A j ) = µ ( A j ) + µ ( A j ) − ε ij . This implies that e ii = µ ( f ( A i )∆ A i ) = 2 X j = i ε ij = X j = i µ ( A i ) + µ ( A j ) − e ij On the other hand, e ii = µ ( f ( X \ A i )∆( X \ A i )) = 2 X j = i ε ji = X j = i µ ( A i ) + µ ( A j ) − e ji . (cid:3) UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 27
Lemma 9.5.
Let ¯ a = ( A , . . . , A k ) be a partition of X into positive measuresets and let p = qftp(¯ a ) . Suppose C , . . . , C k are such that C i ⊆ A i for each i ≤ k and µ ( C ) = . . . = µ ( C k ) > . Let M = { ( B , . . . , B k ) ∈ p (MALG) : ∀ i = j ≤ k B i ∩ A j ⊆ C j ∧ A i \ B i ⊆ C i } . Then M is relatively µ ( C ) -saturated over ¯ a .Proof. Write ε = 2 µ ( C ). Let E , . . . , E k ∈ MALG be such that µ ( E i ∆ A i ) <ε and qftp( E , . . . , E k ) = p . Write e ij = µ ( E i ∆ A j ) and note that E = ( e ij : i, j ≤ k ) is ¯ a -additive by Claim 9.4. We need to find ( B , . . . , B k ) ∈ M suchthat µ ( B i ∆ A j ) = e ij for each i, j ≤ k . For each i = j write ε ij = 12 ( µ ( A i ) + µ ( A j ) − e ij )and note that by ¯ a -additivity we have X j = i ε ji = 12 e ii < ε = µ ( C i )Thus, we can find disjoint measurable sets D ji ⊆ C i such that µ ( D ji ) = ε ji .Write D i = S j = i D ji and note that µ ( D i ) = e ii . Put B i = A i \ D i ∪ S j = i D ij and note that since (by ¯ a -additivity) X j = i ε ji = X j = i ε ij we have that µ ( B i ) = µ ( A i ). The sets B i are pairwise disjoint since the sets D ij are disjoint and so qftp( B , . . . , B k ) = qftp( A , . . . , A k ). Also, we have µ ( B i ∆ A j ) = µ ( B i ) + µ ( A j ) − ε ij = e ij if j = i and µ ( B i ∆ A i ) = 2 X j = i ε ij = e ii . This shows that ( B , . . . , B k ) is as needed and M is relatively ε -saturatedover ¯ a . (cid:3) Definition 9.6.
Let ¯ a = ( A , . . . , A k ) be a tuple in MALG such that ¯ a isa partition of X into positive measure sets. Write p for qftp(¯ a ). Given asequence ¯ a n = ( A n , . . . , A nk ) in p say that it is weakly independent if there isa sequence (( C n , . . . , C nk ) : n ∈ N ) such that • C ni ⊆ A ni for each i ≤ k and n ∈ N , • µ ( C n ) = . . . = µ ( C nk ) > n ∈ N , • all sets { C ni : i ≤ k, n ∈ N } are pairwise disjoint, • if m = n , then C ni ⊆ A m for every i ≤ k . Lemma 9.7.
If a sequence ¯ a n in MALG is weakly independent, then it isisolated.
Proof.
Let k ∈ N be such that each ¯ a n = ( A n , . . . , A nk ) is an k -elementpartition of X . Suppose (( C n , . . . , C nk ) : n ∈ N ) witnesses that the sequenceis weakly independent and let ε n = 2 µ ( C n ). Write p for the quantifier-freetype of ¯ a n and let M n = { ( B , . . . , B k ) ∈ p (MALG) : ∀ i = j ≤ k B i ∩ A nj ⊆ C nj ∧ A ni \ B i ⊆ C ni } . We claim that the sequence of sets M n together with ε n witness that thesequence ¯ a n is isolated. The fact that M n is relatively ε n -saturated over ¯ a n follows directly from Lemma 9.5.Suppose now that ¯ b n = ( B n , . . . , B nk ) are such that each ¯ b n belongs to M n and qftp(¯ b n ) = qftp(¯ a n ). We need to find ϕ ∈ Aut(MALG) such that ϕ (¯ a n ) = ¯ b n for each n ∈ N . For every n ∈ N and i = j let D nij = B ni ∩ A nj ⊆ C nj and let ε nij = µ ( D nij ). For each i ≤ k write D ni = S j = i D nji . Let E ni = A ni \ B ni ⊆ C ni and note that µ ( E ni ) = X j = i ε nij = X j = i ε nji since µ ( B ni ) = µ ( A ni ). For every j = i find measurable sets E nji ⊆ E ni such that E ni = S j = i E nji and µ ( E nji ) = ε nji . Now, for each n ∈ N find ameasure-preserving bijection f n : [ i ≤ k D ni ∪ E ni → [ i ≤ k D ni ∪ E ni such that • f n ( E ni ) = D ni for each i ≤ k , • f n ( D nij ) = E nij for every i = j ≤ k Let f : X → X be a measure-preserving bijection such that f ⊇ f n foreach n ∈ N and f is equal to the identity on the complement of the set S n ∈ N S i ≤ k D ni ∪ E ni . Note that for m = n and i ≤ k , the set D mi ∪ E mi iscontained in C mi , so the function f m maps A n into itself and the domain of f m is disoint from A mi for i >
1. This implies that f ( A ni ) = B ni and hencethe autmomorphism of MALG induced by f is as needed. This ends theproof. (cid:3) Proposition 9.8.
The measure algebra
MALG admits isolated sequences.Proof.
Suppose p is a quantifier-free k -type of a tuple ¯ a = ( A , . . . , A k )in MALG. First note that we can assume that the elements of ¯ a form apartition of X into positive measure sets. Otherwise, one can consider theatoms of the algebra generated by ¯ a and work with a quantifier-free m -type q for m equal to the number of these atoms. Then, for every m -tuple ¯ b ∈ q ( M )there is a unique tuple ¯ b ′ ∈ p ( M ), such that the algebras generated by ¯ b and¯ b ′ are the same and such that • if (¯ b n : n ∈ N ) is isolated in q , then (¯ b ′ n : n ∈ N ) is isolated in p , • the map ¯ b ¯ b ′ is a homeomorphism of q ( M ) and p ( M ). UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 29
Now, suppose Z ⊆ p (MALG) is nonmeager and assume (restricting to anopen subset if neccessary) that Z is nonmeager in every nonempty open set.Construct a sequence of ( A n , . . . , A nk ) ∈ Z and positive measure pairwisedisjoint sets D ni ⊆ A ni (for i ≤ k ) together with positive reals δ ni such thatfor every n ∈ N we have(i) D n , . . . , D nk ⊆ A m if m < n ,(ii) µ ( D ni ∩ A n +11 ) = δ ni for every i ≤ k ,(iii) d MALG ( A n + l +1 k , A n + lk ) < δ nk / l +1 for every l ≥ µ ( T m ≤ n A m \ D m ) > i ≤ k and n ∈ N put C ni = D ni ∩ T m>n A m and note that by (ii) and (iii) above we have that µ ( C ni ) > i ≤ k and n ∈ N . By shrinking the sets C ni if neccessary, we can assume that µ ( C n ) = . . . = µ ( C nk ). Then the definition of C ni and condition (i) aboveimply that if m = n , then C ni ⊆ A m . Given that D ni are pariwise disjoint,so are the sets C ni and so the sequence ( A n , . . . , A nk ) is weakly independent,as witnessed by C ni and hence isolated by Lemma 9.7.To perform the induction step, suppose we have constructed ( A j , . . . , A jk ) ∈ Z for j ≤ n , the sets D ji for i ≤ n and i ≤ k as well as δ ji for j ≤ n − i ≤ k . Consider the open set U = { ( A , . . . , A k ) ∈ p (MALG) : ∀ i ≤ k d MALG ( A i , A ni ) ≤ min m
0. Using the fact that for any positive measure set E and i ≤ k the set { ( A , . . . , A n ) ∈ p (MALG) : µ ( A i ∩ E ) = 0 } is closednowhere dense, find ( A n +11 , . . . , A n +1 k ) ∈ U ∩ Z such that(a) µ ( F ∩ A n +1 i ) > i ≤ k ,(b) µ ( A n +11 ∩ D ni ) > i ≤ k .Now, using (a) above, find D n +1 i ⊆ A n +1 i ∩ F such that0 < µ ( D n +1 i ) < µ ( A n +1 i ∩ F ) . This implies that the inductive condition (iv) will be satisfied at the nextstep. Put δ ni = µ ( A n +11 ∩ D ni ) and note that δ ni > A n +11 , . . . , A n +1 k ) ∈ U and thus, this concludes theinduction step. This ends the proof. (cid:3) The Hilbert space
The orthogonal group O ( ℓ ) is the group of automorphism of the (real)Hilbert space. The Hilbert space here is treated as the metric structure withthe first sort being ( ℓ , , +) and the second sort being the real line with thefield structure (including the inverse function defined on non-zero elementsby x x − and mapping 0 to 0, as well as the function x
7→ − x ) andconstants for the rationals. We also add to the language the multiplication by scalars function · : R × ℓ → ℓ (i.e. ( a, v ) a · v ) as well as the innerproduct function h· , ·i : ℓ × ℓ → R .Recall that by the Mazur–Ulam theorem [35] any isometry of a normedvector space which preserves zero, is a linear isomorphism (in case of theHilbert space this is even simpler than the general case of the Mazur–Ulamtheorem), so we could also consider the structure only with the constant 0and the inner product function. Still another way would be to look at theunit sphere in the Hilbert space equipped only with the metric (as a metricspace with no additional structure) and then the orthogonal group wouldbe the group of isometries of the sphere. We will however, use the abovelanguage, as it seems the most natural, and we will make use of it in orderto talk about substructures of the Hilbert space.The unitary group U ( ℓ ) is the automorphism group of the complexHilbert space and the arguments below apply in the same way to the complexHilbert space, so we will focus only of the real Hilbert space. Claim 10.1. If A is a finitely generated substructure of the Hilbert space,then there exists a countable field K ⊆ R such that Q ⊆ K and A is afinite-dimensional K -vector spaceProof. Let K consist of the elements of A which are of the second sort. Sincethe language contains constants for the rationals, we have Q ⊆ K and sincethe language contains the language of fields, K is a field. Clearly then A is a K -vector space and the dimension is bounded by the number of generatorsof A . (cid:3) Lemma 10.2.
The Hilbert space ℓ has locally finite automorphisms.Proof. In fact, ℓ has the following stronger property. For any finitely gener-ated substructure A ⊆ ℓ , any isomoprhism between finitely generated sub-structures of A extends to an automorphism of A . To see this, let A , A ⊆ A be finitely generated substructures and ϕ : A → A be an isomorphism.Let K and K , K ⊆ K be such that A is a K -vector space, A is K -vectorspace and A is a K -vector space. Write A ′ for the K -vector space gener-ated by A and A ′ for the K -vector space generated by A . Note that since ϕ preserves the inner product, it is an isometry and since both K and K contain Q , the map ϕ can be extended to an isomorphism ϕ ′ : A ′ → A ′ .Now, since K is a field, the usual Gram–Schmidt orthogonalization processgives orthogonal bases { b , . . . , b k } and { b , . . . , b k } for the orthogonal com-plements of A ′ in A and A in A (respectively). The map which extends ϕ and maps b i to b i extends to an automorphism of A . (cid:3) Note that the above proof also shows that given a finitely generatedsubstructure A of the Hilbert space and its finitely generated substructure C ⊆ A , we can form the orthogonal complement A ⊖ C inside A using thestandard Gram–Schmidt process. The extension property for the Hilbertspace is then straightforward and based on the following claim. UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 31
Claim 10.3.
Given finitely generated substructures
A, B, C ⊆ ℓ with C ⊆ A ∩ B , if A ⊖ C ⊥ B ⊖ C , then A | ⌣ C B .Proof. This is elementary linear algebra and the proof is analogous to thatof Lemma 10.2. (cid:3)
Corollary 10.4.
The Hilbert space ℓ has the extension property.Proof. Given finite-dimensional subspaces
A, B, C ⊆ ℓ with C ⊆ A ∩ B finda copy D ⊆ ℓ of B ⊖ C which is orthogonal to A . Then C ⊕ D witnessesthe extension property by Claim 10.3. (cid:3) Before we show that the Hilbert space admits isolated sequences, we needa couple of lemmas. Below, given a closed subspace V ⊆ ℓ and a vector v ∈ ℓ write π V ( v ) for the projection of v onto V . Also, ball ℓ ( v, ε ) standsfor the open ball { w ∈ ℓ : || w − v || < ε } and S ℓ ( v, ε ) stands for the sphere { w ∈ ℓ : || w − v || = ε } . Recall also that ¯ v = ( v , . . . , v k ) ∈ ℓ is an orthonormal tuple if || v i || = 1 and v i ⊥ v j for i = j Lemma 10.5.
Suppose ¯ v = ( v , . . . , v k ) is an orthonormal tuple in ℓ and let H ⊆ ℓ be an infinite-dimensional closed subspace. Suppose V , . . . , V k ⊆ ℓ are closed infinite-dimensional subspaces such that v i ∈ V i and V i ⊥ V j for i = j . Write H i = V i ∩ H and suppose H i is infinite-dimensional and that π H i ( v i ) = 0 for each i ≤ k . Then there exists ε > such that for every ¯ v ′ = ( v ′ , . . . , v ′ k ) such that ¯ v ′ ≡ ¯ v, v ′ i ∈ V i and || v ′ i − v i || < ε for each i ≤ k , there exists ¯ v ′′ = ( v ′′ , . . . , v ′′ k ) such that ¯ v ′′ ≡ ¯ v ¯ v ′ and v ′′ i − v i ∈ H i for each i ≤ k .Proof. Note that since the subspaces V i are mutually orthogonal, it is enoughto prove the lemma for k = 1. Assume then V = ℓ and write v = v sothat π H ( v ) = 0 (i.e. v H ). We need to show that there exists ε > v ′ ∈ S ℓ (0 ,
1) with || v ′ − v || < ε there exists v ′′ ∈ S ℓ (0 ,
1) with v ′′ − v ∈ H and v ′′ ≡ v v ′ . The latter is equivalent to || v ′′ − v || = || v ′ − v || (since v ′ , v ′′ have the same norm). Since v H , there exists w ∈ S ℓ (0 ,
1) suchthat w = v and w − v ∈ H . Let ε = || w − v || . Write S = S ℓ (0 , ∩ ( H + v )and note that S = S ℓ (0 , ∩ A for some infinite-dimensional closed affinesubspace A of ℓ . Hence, S is homeomorphic to the sphere S ℓ (0 ,
1) and thusis connected. By the intermediate-value theorem, the function f : S → R given by f ( s ) = || s − v || assumes all values between 0 and ε on S , andso for every v ′ ∈ S ℓ (0 ,
1) with || v ′ − v || < ε there exists v ′′ ∈ S with || v ′′ − v || = || v ′ − v || . This ends the proof. (cid:3) Lemma 10.6.
Suppose ¯ v = ( v , . . . , v k ) is an orthonormal tuple in ℓ and V , . . . , V k ⊆ ℓ are closed infinite-dimensional subspaces such that v i ∈ V i and V i ⊥ V j for i = j . Write T = { ¯ w = ( w , . . . , w k ) : ¯ w ≡ ¯ v ∧ ∀ i ≤ k w i ∈ V i } . Then T (2 , ε ) -generates an open set, for every ε > .Proof. Fix ε >
0. Find ¯ v ′ = ( v ′ , . . . , v ′ k ) in ℓ such that • ¯ v ′ ≡ ¯ v • v ′ i ⊥ v j for every i = j • for every i ≤ k we have π V j ( v i ) = 0 for every j = i .For each i, j ≤ k write v ′ ij for π V j ( v ′ i ) and note that if i = j , then v ′ ij ⊥ v j .Find δ > i ≤ k the following holds: for everysequence ( v ′′ j : j = i, j ≤ k ) of vectors in V i such that || v ′′ j − v ′ ji || < δ thereexists ˜ v ∈ V i with || ˜ v || = 1, ˜ v ⊥ v ′′ j for every j = i and || ˜ v − v i || < ε/ δ < ε/
2. Write U = { ¯ v ′′ = ( v ′′ , . . . , v ′′ k ) : ¯ v ′′ ≡ ¯ v ∧ d ℓ (¯ v ′′ , ¯ v ′ ) < δ } . Claim 10.7.
For every ¯ v ′′ ∈ U there are ϕ , ϕ ∈ O ( ℓ ) such that ϕ ϕ (¯ v ) = ¯ v ′′ and ϕ (¯ v ) , ϕ (¯ v ) ∈ T , as well as d ℓ ( ϕ (¯ v ) , ¯ v ) < ε and d ℓ ( ϕ (¯ v ) , ¯ v ) < ε .Proof. Fix ¯ v ′′ in U . Note that, by the choice of δ , for each i ≤ k there exists˜ v i ∈ V i such that || ˜ v i || = 1, || ˜ v i − v i || < ε/ v i ⊥ π V i ( v ′′ j ) for every j = i .Now, for every i ≤ k find w i ∈ V i such that( † ) qftp( w i /v i ) = qftp( v ′′ i / ˜ v i ) . Such vectors w i exist since each V i is isomorphic to ℓ . Now, ( † ) implies thatqftp( w i v i ) = qftp( v ′′ i ˜ v i )for each i ≤ k and hence the map ψ i such that ψ i : w i v ′′ i , ψ i : v i ˜ v i is a partial automorphism of ℓ for each i ≤ k . Now, since for i = j both thedomains and ranges of ψ i and ψ j are pairwise orthogonal, the map S i ≤ k ψ i is a partial automorphism of ℓ . Extend S i ≤ k ψ i to ϕ ∈ O ( ℓ ). Find also ϕ ∈ O ( ℓ ) such that ϕ : v i w i for each i ≤ k .Note that since || ˜ v i − v i || < ε/ || v ′′ i − v i || < ε/
2, we have || v ′′ i − ˜ v i || <ε and hence ( † ) implies that || w i − v i || < ε for each i ≤ k . Therefore, d ℓ ( ϕ (¯ v ) , ¯ v ) < ε . Also d ℓ ( ϕ (¯ v ) , ¯ v ) < ε , as well as ϕ (¯ v ) ∈ T and ϕ (¯ v ) ∈ T . As we clearly have ϕ ϕ (¯ v ) = ¯ v ′′ , this proves the claim. (cid:3) Claim 10.7 clearly means that T (2 , ε )-generates an open set, so this endsthe proof. (cid:3) UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 33
Lemma 10.8.
Suppose ¯ v = ( v , . . . , v k ) is an orthonormal tuple in ℓ andlet H ⊆ ℓ be an infinite-dimensional closed subspace such that the vectors π H ( v ) , . . . , π H ( v k ) are linearly independent. Write N = { ¯ w = ( w , . . . , w k ) : ¯ w ≡ ¯ v ∧ ∀ i ≤ k w i − v i ∈ H } . Then there exists ε > such that N is -relatively ε -saturated over ¯ v .Proof. Write w j = π H ( v i ) for each i ≤ k . Claim 10.9.
There exist w ′ , . . . , w ′ k ∈ H such that w ′ i ⊥ w ′ j and w ′ i ⊥ w j for i = j ≤ k and w ′ i w i for every i ≤ k .Proof. Inductively on i ≤ k construct w ′ i ∈ H such that w ′ i w i and w ′ i ⊥ w j for j = i and w ′ i ⊥ w ′ j for j < i as well as w , w ′ , . . . , w i , w ′ i , w i +1 , . . . , w k are linearly independent. Suppose w ′ , . . . , w ′ i − have been constructed. Let W i = { w , w ′ , . . . , w i − , w ′ i − , w i +1 , . . . , w k } ⊥ ∩ H and note that since w i / ∈ span( w , w ′ , . . . , w i − , w ′ i − , w i +1 , . . . , w k ), we havethat W ′ i = W i ∩ { w i } ⊥ is a proper subspace of W i . Also, W ′′ i = W i ∩ span { w , w ′ , . . . , w i , w ′ i , w i +1 , . . . , w k } is a proper subspace of W i since W i is infinite-dimensional. Now, W ′ i ∪ W ′′ i do not cover W i , so find w ′ i ∈ W i \ ( W ′ i ∪ W ′′ i ) and note that it is as needed. (cid:3) Using Claim 10.9, find closed infinite-dimensional subspaces V i for i ≤ k such that for each i = j ≤ k we have • v i ∈ V i and V i ⊥ V j , • H ∩ V i is infinite-dimensional, • π H ∩ V i ( v i ) = 0.Find ε > T = { ¯ w = ( w , . . . , w k ) : ¯ w ≡ ¯ v ∧ ∀ i ≤ k w i ∈ V i } . Then N is T -relatively ε -saturated by Lemma 10.5 and T (2 , ε )-generatesan open set, by Lemma 10.6. This ends the proof. (cid:3) Definition 10.10.
Say that a sequence of k -tuples ¯ a n in ℓ is strongly lin-early independent if there is a sequence of infinite-dimensional closed sub-spaces V n ⊆ ℓ such that • V n ⊥ V m for n = m , • ¯ a m ⊥ V n for n = m , • the projections of the elements of ¯ a n to V n are linearly independent. Lemma 10.11. If p is a quantifier-free type of an orthonormal tuple in ℓ ,then any strongly linearly independent sequence in p ( ℓ ) is -weakly isolated. Proof.
Suppose ¯ a n = ( a n , . . . , a nk ) is strongly linearly independent in p . Notethat a n , . . . , a nk form an orthonormal tuple. Let N n = { ¯ v = ( v , . . . , v k ) ∈ p ( ℓ ) : ∀ i ≤ k v i − a ni ∈ V n } . We claim that there are ε n > N n and ε n witnesses that ¯ a n is 2-weakly isolated. For each n find ε n > v = ¯ a n . Then N n is 2-relatively ε n -saturated over ¯ a n .Suppose now that ¯ b n = ( b n , . . . , b nk ) ∈ N n are such that qftp(¯ b n ) =qftp(¯ a n ) for each n ∈ N and d ℓ (¯ b n , ¯ a n ) < ε n . Then b ni − a ni ∈ V n foreach i ≤ k . Find ϕ n ∈ O ( V n ) such that ϕ n ( π V n (¯ a n )) = π V n (¯ b n ) and let ϕ ∈ O ( ℓ ) be such that ϕ extends all the ϕ n and is equal to the identity onthe orthogonal complement of the union of V n ’s. Then ϕ (¯ a n ) = ¯ b n for each n ∈ N . This ends the proof. (cid:3) Proposition 10.12.
The Hilbert space ℓ admits -weakly isolated sequences.Proof. Suppose p is a quantifier-free k -type of a tuple ¯ a = ( a , . . . , a k ) in ℓ .First note that we can assume that the elements of ¯ a form an orthonormalset. Otherwise, one can consider a tuple which is an orthonormal basis forthe space spanned by ¯ a and work with a quantifier-free m -type q for some m ≤ n . Then, for every m -tuple ¯ b ∈ q ( M ) there is a unique tuple ¯ b ′ ∈ p ( M )such that the linear spans of ¯ b and ¯ b ′ are the same and • if (¯ b n : n ∈ N ) is isolated in q , then (¯ b ′ n : n ∈ N ) is isolated in p , • the map ¯ b ¯ b ′ is a homeomorphism of q ( M ) and p ( M ).In fact, for simplicity of notation, assume that k = 1 (the argument forarbitrary k is analogous).Suppose now that Z ⊆ p ( ℓ ) is nonmeager. Restricting to an open subsetof p ( ℓ ) if neccessary, we can assume that Z is nonmeager in every nonemptyopen subset of ℓ .Write Gr( ℓ ) for the space of all closed subspaces of ℓ and Gr( ℓ , ∞ ) forthe space of infinite-dimensional closed subspaces of ℓ . The topology onGr( ℓ ) is induced from the strong operator topology via the map V π V .Write d Gr for a compatible metric on Gr( ℓ ). Note that there is a sequenceof functions ρ n : Gr( ℓ , ∞ ) → (0 , ∞ ) such that whenever W n ∈ Gr( ℓ , ∞ )is a decreasing sequence of infinite-dimensional closed subspaces of ℓ and d Gr ( W n , W n +1 ) < ρ n +1 ( W n ), then T n W n is also infinite-dimensional.By induction on n ∈ N find vectors a n ∈ Z , positive reals δ n and pairwiseorthogonal infinite-dimensional closed subspaces W n ⊆ ℓ such that:(i) W ⊕ . . . ⊕ W n is co-infinite dimensional,(ii) π W n ( a n ) = 0.(iii) || π W n ∩ span( a n +1 ) ⊥ ( a n ) || = δ n ,(iv) if m < n , then a m ⊥ W n (v) if m < n , then we have || π W m ∩ ( S ni = m +1 span( a i )) ⊥ ( a m ) || > δ m , UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 35 (vi) if m < n and ε = ρ n ( W m ∩ ( S n − i = m +1 span( a i )) ⊥ ), then d Gr ( W m ∩ ( n [ i = m +1 span( a i )) ⊥ , W m ∩ ( n − [ i = m +1 span( a i )) ⊥ ) < ε. After this is done, put V n = W n ∩ ( S m>n span( a m )) ⊥ . Note that V n areinfinite-dimensional by (vi) and mutually orthogonal given that W n are mu-tually orthogonal. Also, (iv) and the definition of V n imply that if n = m ,then a m ⊥ V n . The projection of a n onto V n is nonzero by the condition (v)and hence ¯ a n is strongly linearly independent, as witnessed by V n and hence2-weakly isolated by Lemma 10.11.To perform the induction step, suppose a , . . . , a n and W , . . . , W n as wellas δ , . . . , δ n − are chosen. Using the fact that a proper subspace of ℓ ismeager as well as the assumption that Z is nonmeager in any nonemptyopen set, find a n +1 ∈ Z which does not belong to span( S ni =1 W i ∪ { a i } ) and( †† ) a n +1 W n ∩ span( π W n ( a n )) ⊥ and a n +1 is so close to a n that for m < n we have || π W m ∩ ( S ni = m +1 span( a i )) ⊥ ( a m ) || > δ m and for every m < n , writing ε nm = ρ n +1 ( W m ∩ ( S ni = m +1 span( a i )) ⊥ ) we have d Gr ( W m ∩ ( n +1 [ i = m +1 span( a i )) ⊥ , W m ∩ ( n [ i = m +1 span( a i )) ⊥ ) < ε nm . This implies that (v) and (vi) are satisfied at the induction step.Note that the projection of a n +1 to (span( S ni =1 W i ∪ { a i } )) ⊥ is nonzero.Find an infinite-dimensional closed space W n +1 such that • W n +1 is orthogonal to span( S ni =1 W i ∪ { a i } ), • the projection of a n +1 onto W n +1 is nonzero, • W ⊕ . . . ⊕ W n +1 is co-infinite dimensional.This gives (i), (ii) and (iv). Finally, we claim that π W n ∩ span( a n +1 ) ⊥ ( a n ) isnonzero. Indeed, otherwise a n ⊥ W n ∩ span( a n +1 ) ⊥ and so π W n ( a n ) ⊥ W n ∩ span( a n +1 ) ⊥ . But then, since a n +1 / ∈ W ⊥ n (by ( †† )), we have that W n ∩ (span( π W n ( a n ))) ⊥ = W n ∩ (span( a n +1 )) ⊥ and so a n +1 ⊥ W n ∩ (span( π W n ( a n ))) ⊥ , which contradicts ( †† ). Let then δ n = || π W n ∩ span( a n +1 ) ⊥ ( a n ) || >
0. This ends the construction and the proof. (cid:3)
Finally, we verify that the stronger property discussed in Section 7 holdsfor the Hilbert space. Say that a sequence of tuples ¯ a n ∈ ℓ is a proper orthog-onal sequence if the subspaces spanned by different ¯ a n are pairwise ortogonaland the orthogonal complement of their union is infinite-dimensional. Claim 10.13.
Any proper orthogonal sequence in ℓ is independent.Proof. Let ¯ a n be a proper orthogonal sequence in the quantifier-free typeof a given ¯ a and let H n be a sequence of orthogonal infinite-dimensionalsubspaces of the orthogonal complement of the space spanned by the vectorsin all ¯ a n ’s. Write H ′ n for the space spanned by H n and ¯ a n and note that thesubspaces H ′ n witness that the sequence ¯ a n is independent. (cid:3) Lemma 10.14.
The Hilbert space ℓ admits independent sequences.Proof. Fix k ∈ N and ¯ a = ( a , . . . , a k ) ∈ U k . Let (¯ s n : n ∈ N ) be a sequenceof finite tuples and without loss of generality assume that ¯ s n is a subtupleof ¯ s n +1 . We need to find an independent sequence ¯ a n in the quantifier-freetype of ¯ a such that ¯ a n ≡ ¯ s n ¯ a n +1 . Find the sequence ¯ a n of tuples as well asadditional vectors v n so that • The elements of ¯ a n are orthogonal to all elements of ¯ s n , to all ele-ments of ¯ a i ’s for i < n as well as to v n • v n +1 is orthogonal to all elements of ¯ a i ’s for i ≤ n • ¯ a n ≡ ¯ s n ¯ a n +1 The sequence is easy to construct using the fact that if ¯ b and ¯ s are twotuples whose elements are pairwise orthogonal, then the orbit of ¯ b withrespect to the stabilizer of ¯ s contains vectors orthogonal to any finite tuple.The sequence is then proper orthogonal, and hence independent by Claim10.13. (cid:3) Questions
There are still many natural examples of automorphism groups for whichthe automatic continuity (and even the uniqueness of Polish group topology)is open. Here we list some of them.
Question 11.1.
Does the group of automorphisms of the Cuntz algebra O have the automatic continuity property? Question 11.2.
Does the group of automorphisms of the hyperfinite II factor have the automatic continuity property? Question 11.3.
Does the group of linear isometries of the Gurariˇı spacehave the automatic continuity property?Finally, the problem of uniqueness of separable topology for the groupIso( U ) remains open. For other groups considered in this paper, uniquenessof separable topology follows from the combination of automatic continu-ity property and minimality (or even total minimality which says that anyHausdorff quotient of the group is minimal). For the unitary group this has UTOMATIC CONTINUITY FOR ISOMETRY GROUPS 37 been proved by Stojanov [50] and for the group Aut([0 , , λ ) by Glasner [15](see also [6] for a recent general framework for these kind of results). Question 11.4.
Is the group Iso( U ) minimal? Acknowledgement.
Part of this work has been done during the author’sstay at the University Paris 7 in the academic year 2013 /
14. The authoris grateful to Zo´e Chatzidakis, Amador Mart´ın-Pizarro and Todor Tsankovfor many useful comments. The author also wishes to thank Ita¨ı Ben Yaa-cov, Alexander Kechris and Julien Melleray for inspiring discussions at theWorkshop on Homogeneous Structures during the special semester at theHausdorff Institute in Mathematics in the fall 2013. The author would alsolike to thank Piotr Przytycki for valuable discussions and Maciej Malicki foruseful comments.
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