Download Unitary Group Actions and Hilbertian Polish

Contemporary Mathematics
Unitary Group Actions and
Hilbertian Polish Metric Spaces
Su Gao
Abstract. We investigate universal actions of the unitary group and universal
equivalence relations induced by these actions. We consider the isometric
equivalence relation for Hilbertian Polish metric spaces and prove that it is
Borel bireducible with the orbit equivalence relation induced by a univeral
action of the unitary group.
1. Introduction
Let H be an infinite-dimensional separable complex Hilbert space and let U (H)
be the group of all unitary tranformations of H. When U (H) is endowed with
the strong operator topology it becomes a Polish group, and we denote it by U ∞
(following [7]). A more standard notation for this topological group is U (H)s .
However, the notation U∞ stresses our interest in the abstract group and in its
arbitrary actions.
In this paper we investigate, collectively, Borel actions of U∞ on standard
Borel spaces and the equivalence relations induced by these actions. The actions
can be compared to each other via their mutual embeddability, and the equivalence
relations via the notion of Borel reducibility. In either context it is well-known that
there are universal elements (c.f. [2]), which are intuitively the most complicated
objects. These universal actions and universal equivalence relations will be the
focus of this paper.
Previously known examples of U∞ actions that generate universal equivalence
relations are complicated and difficult to work with. One of our objectives in this
paper is to identify actions of U∞ that are fairly simple to define but still induce
universal equivalence relations. We show that the natural action of U∞ on F(H),
the space of all closed subsets of H, is such an action. Moreover, for any Borel
action of U∞ on a standard Borel space X, there is a Borel reduction of X into
F(H) which respects the Borelness of invariant sets. In our terminology, F(H) is
faithfully universal. This is not as strong as universal with repect to embeddings
2000 Mathematics Subject Classification. Primary 54H05, 22F05; Secondary 54E35, 03E75.
Key words and phrases. Polish metric spaces, unitary group, Borel action, Borel reducible,
universal action, universal equivalence relation.
This research was supported by NSF grant DMS-0100439.
c
2000
American Mathematical Society
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SU GAO
of actions, but faithfulness is still a significant concept because of its pertinence to
the Topological Vaught Conjecture. We will elaborate on the details in the text
and give definitions as they become relevant.
As an important application we also identify a natural equivalence relation
that is of the same complexity as the universal equivalence relation given by U ∞
actions. This equivalence relation comes from the classification problem of the
so-called Hilbertian Polish metric spaces. A Hilbertian Polish metric space is a
complete metric space isometrically embeddable into H. The classification problem
is to decide whether two given such spaces are isometric and it is essentially an
equivalence relation. We show that this equivalence relation is the most complicated
among all the orbit equivalence relations induced by unitary group actions.
A careful reader will notice that results in this paper are parallel to some of
those in [6]. Instead of treating the Urysohn space and its full isometry group in
[6], here we are dealing with a more familiar space H and the unitary group, a
subgroup of the full isometry group.
The paper is organized as follows. In the next section we give the basic definitions and background results. Then in Section 3 we consider the universal actions
and equivalence relations of the unitary group. In section 4 we consider the classification problem for Hilbertian Polish metric spaces and prove our main result about
the complexity of the classification problem. Throughout the paper we will always
work with the complex Hilbert space. But all results are valid for the real Hilbert
space and, correspondingly, the orthogonal group.
2. Preliminaries
In this section we review the basic results about Polish group actions and
equivalence relations. We also summarize some results on positive definite functions
and closed subgroups of the unitary group. These will be needed in subsequent
sections.
Recall that a topological space is Polish if it is separable and completely metrizable. Likewise, a topological group is called a Polish group if its topology is Polish.
A separable, complete metric space will be called a Polish metric space. For a topological space X, a subset A ⊆ X is called Borel if it is an element of the σ-algebra
of subsets of X generated by the open sets. The Borel structure of X is just the
σ-algebra of all Borel subsets of X. A space with a σ-algebra of subsets is called a
standard Borel space if the σ-algebra is the Borel structure of some Polish topology.
If X and Y are both standard Borel spaces, then a function f : X → Y is called
Borel (measurable) if for any Borel subset B of Y , f −1 (B) is a Borel subset of X.
A subset B of Y is analytic if there are a standard Borel space X, a Borel function
f : X → Y and a Borel subset A of X such that f (A) = B. Basic facts and results
on these concepts can be found in [8].
Let X and Y be standard Borel spaces and E and F be equivalence relations
on X and Y respectively. Then E is Borel reducible to F , denoted E ≤B F , if there
is a Borel function f : X → Y such that, for all x1 , x2 ∈ X,
x1 Ex2 ⇐⇒ f (x1 )F f (x2 ).
In this situation E is intuitively simpler than F , or F is more complicated than E.
If E ≤B F and F ≤B E we say that E and F are Borel bireducible, and intuitively
this means that E and F have the same complexity.
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
3
An important class of equivalence relations is that of orbit equivalence relations
induced by actions of Polish groups. Let G be a Polish group and X a standard
Borel space. We say that G acts on X in a Borel manner, or X is a Borel G-space,
if there is an action a : G × X → X which is a Borel function. Here G × X is
understood to be a standard Borel space since it admits a product Polish topology.
For any x ∈ X, the G-orbit of x in X is the set
[x]G = {g · x | g ∈ G}.
X
The G-orbit equivalence relation EG
is defined by
X
x1 E G
x2 ⇐⇒ ∃g ∈ G(g · x1 = x2 ) ⇐⇒ [x1 ]G = [x2 ]G
X
for all x1 , x2 ∈ X. Since the action mapping a : G × X → X is Borel, EG
is an
X
analytic subset of X × X. However, in general, EG does not have to be Borel.
The notion of Borel reducibility defined above can be viewed as a weak notion of
embeddability of actions and can be used to compare the complexity of all orbit
equivalence relations. But often these comparisons can be done via stronger notions
of embeddability of actions, which we define below.
Let X and Y be Borel G-spaces. A mapping f : X → Y is a G-map if for all
x ∈ X and g ∈ G,
f (g · x) = g · f (x).
If in addition f is an embedding, then f is called a G-embedding. If there is a Borel
X
Y
G-map from X to Y then it follows that EG
≤B EG
.
A Borel G-space X (or the action of G on X) is called universal if for any
Borel G-space Y there is a Borel G-embedding from Y into X. It is known that
for any Polish group G there is a universal Borel G-space ([2], Theorems 2.6.1).
Consequently, if X is a universal Borel G-space then for any Borel G-space Y we
Y
X
X
have EG
≤B EG
, i.e., the orbit equivalence relation EG
is the most complicated
among all G-orbit equivalence relations.
X
In general, if X is a Borel G-space, we say that EG
is universal if any GX
orbit equivalence relation is Borel reducible to EG , i.e., for any Borel G-space Y ,
Y
X
EG
≤B EG
. It is important to note that the Borel G-space X does not need to be
universal in order for the orbit equivalence relation to be universal. To emphasize
the distinction we offer the following intermediate notion.
For a subset A of X, let
[A]G = {g · x | g ∈ G, a ∈ A}.
X
We call A G-invariant if A = [A]G . We say that EG
is faithfully universal if for any
Y
X
Borel G-space Y there is a Borel reduction f : Y → X witnessing EG
≤B EG
such
that, for any invariant Borel subset A of X, we have that [f (A)]G is an invariant
Borel subset of Y . In general, reduction functions satisfying the above property are
called faithful Borel reductions. Faithful Borel reductions were first introduced in [4]
out of model-theoretic motivations and later studied in [5] (there it was called F Sreducibility) for Borel equivalence relations and S∞ -orbit equivalence relations. The
current terminology comes from [7]. The importance of faithful Borel reductions
lies in the fact that they carry down the truth of the Topological Vaught Conjecture
(c.f. [2], [4], and Theorem 3.7 below). For a Polish group G and a Borel G-space
X, the Topological Vaught Conjecture states that, for every invariant Borel subset
A of X, either A contains 2ℵ0 many G-orbits or else A contains only countably
many G-orbits. If Y and X are both Borel G-spaces and there exists a faithful
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SU GAO
reduction from Y into X, then the Topological Vaught Conjecture holds for Y if
it holds for X. The Topological Vaught Conjecture for a Polish group G is the
statement that the Topological Vaught Conjecture holds for all Borel G-spaces. If
X
X is a Borel G-space so that EG
is faithfully universal, then the Topological Vaught
Conjecture for G is equivalent to the instance of it for X. The general Topological
Vaught Conjecture asserts that the Topological Vaught Conjecture holds for all
Polish groups G. This conjecture is still open.
If X and Y are Borel G-spaces and f is a G-embedding from Y into X, then
the image of any invariant Borel subset of Y under f is an invariant Borel subset
X
of X. Thus in particular, if X is a universal Borel G-space then EG
is a faithfully
universal G-orbit equivalence relation. To summarize, the following diagram shows
the implication relations among the three universality notions.
universal G-space
⇓
faithfully universal G-orbit equivalence relation
⇓
universal G-orbit equivalence relation
Counterexamples for the inverse implications can be found in [5]. Our results in
the next section also provide such examples in the context of G being the unitary
group.
Most of the rest of this paper deals with the unitary group U∞ . When there
is no ambiguity about the group in question we shall omit the prefix G- for the
terminology defined above.
We now turn to the unitary group U∞ . We first recall some notation and basic
facts about the Hilbert space H and its unitary group U (H). A linear operator
T : H → H is unitary if
hT u, T vi = hu, vi, for all u, v ∈ H.
The group operation of U (H) is the composition of unitary operators. The above
equation implies that every unitary operator is in fact an isometry of H considered
as a metric space. Moreover, the strong operator topology on U (H) is exactly the
pointwise convergence topology when it is considered as a group of isometries. It
is well-known that the weak operator topology on U (H) coincides with the strong
operator topology, and in the sequel we will use this fact implicitly. More basic
facts about Hilbert spaces and unitary operators can be found in [3]. For more
advanced background on the unitary group see [9].
In the rest of this section we shall try to give a more or less self-contained
account of the results on closed subgroups of U∞ . Most of the following results
belong to the folklore and are hard to attribute. Moreover, some of the results,
even if well-known to experts, can not be found in the literature in the form we
need them.
One of the most useful criteria for deciding whether a Polish group can be topologically embedded into U∞ goes through the concept of positive definite functions.
Recall that a complex-valued function f on a group G is positive definite if for
any n ≥ 1 and arbitrary g1 , . . . , gn ∈ G, c1 , . . . , cn ∈ C,
n
X
i,j=1
f (gj−1 gi )ci cj ≥ 0.
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
5
If G is a group, 1G is its identity element and f is a positive definite function
on G, then among the simplest properties of f are:
(a) f (1G ) ≥ 0,
(b) f (g −1 ) = f (g), and
(c) |f (g)| ≤ f (1G ),
for all g ∈ G. A less obvious property is a result of Schur that the set of all positive
definite functions is closed under multiplication. Proofs of these facts can be found
in, e.g., [1], §3.1.
The following characterization theorem is mainly a folklore. An account close
to what we need here is given in [9], §30. For the convenience of the reader we give
a sketch of the proof afterwards.
Theorem 2.1. Let G be a Polish group and 1G its identity element. Then the
following are equivalent:
(1) G is ismorphic to a (closed) subgroup of U∞ ;
(2) Continuous positive definite functions on G generate the topology of G;
(3) Continuous positive definite functions on G generate a neighborhood basis
of 1G ;
(4) Continuous positive definite functions on G separate 1G and closed sets
not containing 1G ;
(5) There is a continuous positive definite function on G separating 1G and
closed sets not containing 1G .
A few words about the statements before the proof. Let F be the collection of
all continuous positive definite functions on G. A rewording of (2) gives
(20 ) The topology of G is the weakest topology that makes functions in F
continuous.
Should there be any doubt with the terminology used in (3) and (4), here are the
expanded versions with definitions incorporated:
(30 ) For any open set V ⊆ G with 1G ∈ V , there are functions f1 , . . . , fn ∈ F
and open subsets O1 , . . . , On in C such that
1G ∈
0
n
\
i=1
fi−1 (Oi ) ⊆ V.
(4 ) For any closed set F ⊆ G with 1G 6∈ F , there is an f ∈ F such that
sup |f (g)| < f (1G ).
g∈F
It is also worth noting that, since G is separable, it suffices to require in (20 )-(40 )
that there is a countable family F0 ⊆ F satisfying the corresponding properties.
Proof of Theorem 2.1 . It is evident that (2)⇒(3) and (5)⇒(4)⇒(3). We
show that (1)⇒(2), (3)⇒(1) and (3)⇒(4)⇒(5).
(1)⇒(2): Recall that the strong operator topology on U∞ coincides with the
weak operator topology, thus it is the weakest topology to make all the functionals
fx,y (g) = hg(x), yi, x, y ∈ H, continuous. For any v ∈ H, let
fv (g) = hg(v), vi, for all g ∈ U∞ .
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Then fv is a continuous positive definite function on U∞ . The continuity is immediate from the definition of the weak operator topology on U∞ , and the positive
definiteness is established by the standard computation:
n
X
fv (gj−1 gi )ci cj =
i,j=1
i,j=1
=
n
X
i,j=1
n
X
hci gi (v), cj gj (v)i =
*
Now by the polar identity
hgj−1 gi (v), vici cj =
n
X
ci gi (v),
i=1
n
X
n
X
i,j=1
cj gj (v)
j=1
+
hgi (v), gj (v)ici cj
2
n
X
=
ci gi (v) ≥ 0.
i=1
4hg(x), yi = hg(x + y), x + yi − hg(x − y), x − yi
+ihg(x + iy), x + iyi − ihg(x − iy), x − iyi,
any topology of U∞ that makes the demonstrated positive definite functions continuous must make the functionals fx,y (g) = hg(x), yi continuous as well. Thus (2)
is proved.
(3)⇒(1): Fix a countable family {fn }n∈N of continuous positive definite functions on G so that they generate a neighborhood basis of 1G . Without loss of
generality we can assume fn (1G ) ≤ 2−n for all n ∈ N.
Consider the space X of all complex-valued functions on G with finite support,
i.e., functions x : G → C such that for all but finitely many g ∈ G, x(g) = 0. X is
a linear space under addition and scalar multiplication. For x, y ∈ X, let
X X
hx, yi =
fn (h−1 g)x(g)y(h).
g,h∈G n∈N
This sesquilinear form is well-defined since for any g, h ∈ G,
X
X
X
−1 fn (h g) ≤
|fn (h−1 g)| ≤
fn (1G ) < ∞.
n
n
n
Let N = {x ∈ X | hx, xi = 0}. Then N is a linear subspace of X and it is easy to
check that h·, ·i is well-defined on the quotient X/N , making it a pre-Hilbert space.
Let H be the completion of X/N under the induced h·, ·i. Then H is a separable
complex Hilbert space.
We consider the representation of G in U (H) given by the definition that, for
each g ∈ G and x ∈ X,
Tg x(h) = x(g −1 h).
For any g0 ∈ G and x, y ∈ X, we have that
X
hTg0 x, Tg0 yi =
fn (h−1 g)x(g0−1 g)y(g0−1 h)
=
g,h,n
X
g,h,n
fn (h−1 g)x(g)y(h) = hx, yi.
The map g 7→ Tg is obviously a group homomorphism. For any g0 ∈ G and x ∈ X,
we have that
X
X
hTg0 x, xi =
fn (h−1 g)x(g0−1 g)y(h) =
fn (h−1 g0 g)x(g)y(h).
g,h,n
g,h,n
It follows immediately from this and the continuity of fn that g 7→ Tg is continuous.
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
7
We next check that g 7→ Tg is injective. For this it suffices to show that
for g0 6= 1G , Tg0 6= I. Suppose g0 6= 1G . There exists some n ∈ N such that
fn (g0 ) 6= fn (1G ). Consider x0 ∈ X defined by
1, if g = g0 ,
x0 (g) =
0, otherwise.
Then the n-th term of hTg0 x0 − x0 , Tg0 x0 − x0 i is
X
fn (h−1 g)(x0 (g0−1 g) − x0 (g))(x0 (g0−1 h) − x0 (h))
g,h
= 2(fn (1G ) − Refn (g0 )) > 0.
It follows that hTg0 x0 − x0 , Tg0 x0 − x0 i > 0, and therefore Tg0 6= I.
Now to establish that g 7→ Tg is a topological group isomorphic embedding, the
only thing that remains to be checked is that the inverse of g 7→ Tg is continuous.
For this suppose Tgm → Tg∞ , as m → ∞, for group elements gm , g∞ ∈ G. We are
to show that gm → g∞ , as m → ∞. Consider
1, if g = 1G ,
1, if g = g∞ ,
x0 (g) =
and y0 (h) =
0, otherwise,
0, otherwise.
Then hTgm x0 , y0 i → hTg∞ x0 , y0 i as m → ∞. But a straightforward computation
shows that
X
−1
hTgm x0 , y0 i =
fn (g∞
gm ),
n
and
hTg∞ x0 , y0 i =
X
fn (1G ).
n
−1
These imply that for all n ∈ N, fn (g∞
gm ) → fn (1G ) as m → ∞. From our
−1
assumption on {fn } it follows that g∞ gm → 1G , or gm → g∞ , as m → ∞. Thus
(3)⇒(1) is proved.
(3)⇒(4): This is an application of the lemma of Schur (a product of positive
definite functions is positive definite) and other properties of positive definite functions. Suppose F ⊆ G is closed and 1G 6∈ F . Then there are continuous positive
definite functions f1 , . . . , fn and open sets O1 , . . . , On in C such that
n
\
1G ∈
fi−1 (Oi ) ⊆ G \ F.
i=1
Without loss of generality assume that fi (1G ) = 1 for 1 ≤ i ≤ n. Then there are
1 , . . . , n > 0 such that, for any x ∈ F , there is 1 ≤ i ≤ n with
|fi (x)| ≤ 1 − i .
Now the function f = f1 · · · fn is continuous and positive definite on G. Moreover,
f (1G ) = 1 and, letting = min{1 , . . . , n }, we have that for any x ∈ F ,
|f (x)| = |f1 (x)| · · · |fn (x)| ≤ 1 − < 1.
(4)⇒(5): Given a countable family {fn }n∈N of continuous positive definite
functions so that they separate 1G and closed sets not containing 1G , we can first
scale them down so that fn (1G ) ≤ 2−n , for all n ∈ N. Then the function f defined
by
X
f (x) =
fn (x)
n∈N
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SU GAO
is continuous, positive definite on G and separates 1G and closed sets not containing
1G .
Statement (5) can be strengthed further to
(50 ) There is a real-valued continuous positive definite function on G separating
1G and closed sets not containing 1G .
In fact, if f is a complex-valued such function, then f 0 = Ref is as required.
Similarly, functions in statements (3) and (4) can be taken to be real-valued as
well.
An identical proof gives similar characterizations for closed subgroups of the
orthogonal group of the infinite-dimensional separable real Hilbert space. Statements (3)-(5) are unchanged in such a theorem. Thus the closed subgroups of the
unitary group and those of the orthogonal group are the same class of topological
groups. In particular it implies that statement (2) in Theorem 2.1 can be replaced
by one for real-valued continuous positive definite functions as well.
Also clear from the proof is that Theorem 2.1 can be generalized to second
countable topological groups. The metrizability is implied by the existence of
enough many continuous positive definite functions. The completeness requirement
is unnecessary.
As an application of Theorem 2.1, we consider the Banach spaces Lp ([0, 1]) and
lp for 1 ≤ p ≤ 2. The additive groups of these spaces are Polish groups. Concerning
positive definite functions on these groups, the following result of Schoenberg is
well-known.
p
Theorem 2.2 (Schoenberg [11]). For 1 ≤ p ≤ 2, the function e−|x| is positive
definite on R, lp or Lp ([0, 1]).
Thus we have
Corollary 2.3. For 1 ≤ p ≤ 2, the additive groups of lp and Lp ([0, 1]) are
ismorphic to some closed subgroups of U∞ .
p
Proof. The function e−|x| is continuous and obviously it separates the origin
(which is the identity of the additive group) and closed sets not containing it, as
the function is norm dependent.
The formulation of Theorem 2.1 was based on communications with Pestov and
Megrelishvili. Schoenberg’s result and the corollary were brought to our attention
by Megrelishvili.
Now let Iso(H) be the group of all isometries of H (onto itself) endowed with
the topology of pointwise convergence. Then it is easy to see that Iso(H) is a Polish
group and that U∞ is a closed subgroup of Iso(H). In fact it is well-known that
Iso(H) is a semi-direct product of U∞ with the abelian group of translations in H
(i.e., the additive group H). In Section 4 we will need the following theorem of
Uspenskiy.
Theorem 2.4 (Uspenskiy). Iso(H) is isomorphic to a closed subgroup of U ∞ .
Proof. For each v ∈ H define a positive definite function pv by
pv (g) = e−kg(v)−vk
2
for g ∈ Iso(H). Each pv is continuous since Iso(H) has the pointwise convergence
topology. To see that each pv is positive definite, note that for any g1 , . . . , gn ∈
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
9
Iso(H) and c1 , . . . , cn ∈ C,
n
X
pv (gj−1 gi )ci cj
=
i,j=1
=
=
n
X
i,j=1
n
X
i,j=1
n
X
−1
e−kgj
−1
e−kgj
gi (v)−vk2
ci cj
gi (v)−gj−1 gj (v)k2
ci cj
2
e−kgi (v)−gj (v)k ci cj .
i,j=1
2
This last expression is non-negative by Schoenberg’s Theorem 2.2 (e−|x| on R),
Schur’s lemma on products of positive definite functions, and the fact that pointwise
limits of positive definite functions are positive definite.
Finally, it is straightforward to see that {pv | v ∈ H} generates the topology of
Iso(H), and thus by Theorem 2.1 (2) Iso(H) is a closed subgroup of U∞ .
3. Universal actions and universal equivalence relations
For any Polish space X, let F(X) be the space of all closed subsets of X
endowed with the Effros Borel structure (c.f., e.g., 12.B of [8]). Specifically, the
Borel structure on F(X) is generated by sets of the form
{F ∈ F(X) | F ∩ U 6= ∅}
where U is an open subset of X. Then F(X) becomes a standard Borel space. If
moreover a Polish group G acts on X in a Borel manner then the induced action
of G on F(X) is also Borel.
When general results of Becker and Kechris on the existence of universal Borel
G-spaces are applied to the special case of the unitary group U∞ , we obtain the
following realizations of the universal objects ([2], Theorems 2.6.1 and 3.5.3).
Theorem 3.1 (Becker-Kechris). The left translation of U∞ on F(U∞ )N is a
universal Borel U∞ -action. The left translation of U∞ on F(U∞ ) induces a universal U∞ -orbit equivalence relation.
Because of the complicated structure of F(U∞ ) it is desirable to find simpler
realizations of the universal Borel U∞ -space. One approach is to consider the
application actions of U∞ induced by that on H. Kechris and the author [6] defined
a general action for an arbitrary isometry group that resembles the logic action for
S∞ . Since U∞ is naturally a group of isometries, one can deduce the following
result (Corollay 9.3 of [6]).
Theorem 3.2 (Gao-Kechris). For each natural number n > 0, let U∞ act on
F(Hn ) by coordinatewise and pointwise application. Then the product space
Y
F(Hn )
n>0
is a universal Borel U∞ -space.
The following theorem is the main theorem of this section. It shows that, if we
are willing to loosen up the notion of universality, then some significantly simpler
realization of the universal space does exist.
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SU GAO
Theorem 3.3. Let U∞ act on F(H) by pointwise application and let E be the
orbit equivalence relation. Then E is a faithfully universal U∞ -orbit equivalence
relation.
The rest of the section is devoted to the proof of Theorem 3.3. This will be
done in two steps. Denote
Y
X=
F(Hn ) and Y = F(H)N .
n>0
Our first step is to show that there is a faithful Borel reduction from EUX∞ to EUY∞ .
By Theorem 3.2, this implies that EUY∞ is faithfully universal among U∞ -orbit
equivalence relations. Then in the second step, we define a faithful Borel reduction
from Y into F(H), and thus complete the proof.
For each natural number n > 0, we endow Hn with the inner product
n
h(x1 , . . . , xn ), (y1 , . . . , yn )in =
1X
hxi , yi i,
n i=1
for (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ Hn . Thus a norm on Hn is given by
! 21
n
X
1
2
k(x1 , . . . , xn )kn = √
kxi k
,
n i=1
and Hn becomes a Hilbert space unitarily isomorphic to H.
Now for each k ∈ N consider the linear isomorphic embedding
Tk : H 2
~x
k
k+1
−→ H2
7−→ (~x, ~x)
k
Tk is unitary, i.e., it preserves inner products. Hence each H 2 can be regarded as
k+1
a closed subspace of H2
in view of the canonical Tk . We have
k
H ⊂ H2 ⊂ H4 ⊂ · · · ⊂ H2 ⊂ H2
k+1
⊂ ··· .
Now take the direct limit
H∞ =
[
k∈N
k
H2 ,
and let Z be the completion of H∞ . Then Z is unitarily isomorphic to H since H∞
is a pre-Hilbert space with the inner product
h~x, ~y i∞ = h~x, ~yi2k ,
k
where the number k is large enough such that ~x, ~y ∈ H 2 .
Moreover, for ϕ ∈ U (H) and n > 0, let ϕ(n) ∈ U (Hn ) be given by
ϕ(n) (x1 , . . . , xn ) = (ϕ(x1 ), . . . , ϕ(xn )).
Then we have
ϕ = ϕ(1) ⊂ ϕ(2) ⊂ ϕ(4) ⊂ · · · ⊂ ϕ(2
i.e., each ϕ(2
k+1
)
coincides with ϕ(2
k
)
ϕ(∞)
k
k
)
on H2 ⊂ H2
[
k
=
ϕ(2 ) ,
k∈N
⊂ ϕ(2
k+1
k+1
)
⊂ ··· ,
. Therefore we can let
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
11
which is a unitary transformation on H∞ . Let ϕZ be the unique extension of ϕ(∞)
to Z. Then ϕZ ∈ U (Z). Denote
Z
U∞
= { ϕZ ∈ U (Z) | ϕ ∈ U (H) }.
Z
Then U∞
is a closed subgroup of U (Z). The following lemma is in essence a
Z
characterization of U∞
in U (Z).
Lemma 3.4. There is a sequence {Dm }m∈N of closed subsets of Z such that
Z
U∞
= { Φ ∈ U (Z) | ∀m (Φ(Dm ) = Dm ) }.
Proof. It is enough to show that, for each n > 0, there is a sequence {Kn,l }l∈N
of closed subsets of Hn such that
{ ϕ(n) ∈ U (Hn ) | ϕ ∈ U (H) } = { Φ ∈ U (Hn ) | ∀l (Φ(Kn,l ) = Kn,l ) }.
Then the sequence {Dm }m∈N can be taken to be any enumeration of the set
{Kn,l | n, l ∈ N}.
(n)
Fix n > 0 and denote the group on the left side by U∞ . For each tuple
p = (p1 , . . . , pn ) of positive rational numbers, let
Kp = {(x1 , . . . , xn ) ∈ Hn | kxi k ≤ pi , i = 1, . . . , n }.
Each Kp is closed in Hn . Similarly, for each tuple q = (qij )1≤i,j≤n of positive
rational numbers, let
Lq = {(x1 , . . . , xn ) ∈ Hn | ∀1 ≤ i, j ≤ n (kxi − xj k ≤ qij ) }.
Then each Lq is also closed in Hn . Denote
G = { Φ ∈ U (Hn ) | ∀p (Φ(Kp ) = Kp ) and ∀q (Φ(Lq ) = Lq ) }.
(n)
We verify that G = U∞ = {ϕ(n) | ϕ ∈ U∞ }.
It is clear that G consists exactly of Φ ∈ U (H n ) such that
for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ Hn , if Φ(x1 , . . . , xn ) = (y1 , . . . , yn ),
then for all 1 ≤ i, j ≤ n, kxi k = kyi k and kxi − xj k = kyi − yj k.
(n)
Thus immediately U∞ ⊆ G. For the opposite inclusion, assume Φ ∈ G. Note that
Φ(H) = H. We put ϕ = Φ H. By considering Φ ◦ (ϕ(n) )−1 in place of Φ, we
may assume without loss of generality that ϕ is the identity transformation, toward
proving that Φ is also the identity.
Suppose Φ(x1 , . . . , xn ) = (y1 , . . . , yn ) and (x1 , . . . , xn ) 6= (y1 , . . . , yn ). Then
for some 1 ≤ i ≤ n, xi 6= yi . Let i be the least such index and let u = xi . Our
assumption gives that
Φ(u, . . . , u) = (u, . . . , u).
Therefore by linearity of Φ, we have
Φ(x1 − u, . . . , xn − u) = (y1 − u, . . . , yn − u).
Since Φ ∈ G, we must have kxi − uk = 0 = kyi − uk, so yi = u = xi , a contradiction.
To complete the proof of the lemma, we let {Kn,l }l∈N be any enumeration of
the countable set of all Kp and Lq .
12
SU GAO
Now let {Dm }m∈N be a fixed sequence of closed subsets of Z given by Lemma 3.4.
We are ready to define a Borel reduction f : X → Y . In doing so we tacitly identify
Z with H since they are unitarily isomorphic. For a sequence
Y
~ = (Rn )n>0 ∈
R
F(Hn ) = X,
n>0
n
i.e., each Rn being a closed subset of H , we let
~ = (D0 , R1 , D1 , R2 , D2 , . . . , Rn , Dn , . . . ) ∈ F(Z)N .
f (R)
It is clear that f is a Borel function. We claim that f is a faithful reduction from
EUX∞ to EUY∞ .
~ S
~ ∈ X. Note that RE
~ Y S
~ iff there is
To see that it is a reduction, suppose R,
U∞
ϕ ∈ U (H) such that ϕ(Rn ) = Sn for all n > 0. When all Rn and Sn are regarded
as subsets of Z, ϕ(Rn ) = Sn iff ϕZ (Rn ) = Sn . Thus by Lemma 3.4, we have
~ Y S
~
RE
U∞
Z
iff ∃Φ ∈ U∞
∀n > 0 (Φ(Rn ) = Sn )
iff ∃Φ ∈ U (Z) (Φ(Rn ) = Sn for all n > 0 and
Φ(Dm ) = Dm for all m ∈ N)
~ Y f (S).
~
iff f (R)E
U∞
To see that f is faithful, let A be an invariant Borel subset of X. We use the
~ = (R,
~ D),
~ and have that
abbreviation f (R)
~ D)
~ |R
~ ∈ A }.
f (A) = { (R,
Let ~0 be the element of X whose each coordinate is the set {0} ⊂ H. By a theorem
of Miller (c.f., [2], Corollary 2.3.3), the orbit [f (~0)]U∞ in Y is Borel. Now for
~ K)
~ ∈ Y , from
arbitrary (S,
~ K)
~ ∈ [f (A)]U∞ iff (~0, K)
~ ∈ [f (~0)]U∞ and S
~ ∈ A,
(S,
we know that [f (A)]U∞ is Borel. This finishes our first step.
The second step of our proof is fulfilled by the following lemma which is actually
stronger than we need.
Lemma 3.5. There is a Borel U∞ -embedding from Y = F(H)N into F(H).
Proof. For n ∈ N we first define Sn : F(H) → F(H) by
 x
2nx


+
|
x
∈
F
∪



kxk
1 + kxk



{ y ∈ H | kyk = 2n + 1},
if 0 6∈ F ,

Sn (F ) =


2nx
x


+
|
x
∈
F,
x
=
6
0
∪



kxk
1 + kxk


{ y ∈ H | kyk = 2n or 2n + 1}, if 0 ∈ F .
For any F ∈ F(H) and y ∈ Sn (F ), we have 2n ≤ kyk ≤ 2n + 1. Now for F~ =
(F0 , F1 , . . . ) ∈ F(H)N , we let
[
S(F~ ) =
Sn (Fn ).
n∈N
We claim that S(F~ ) is closed. To see this, let {yk }k∈N be a sequence such that for
each k ∈ N, yk ∈ Snk (Fnk ) for some nk , and yk → y∞ as k → ∞ for some y∞ ∈ H.
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
13
Since for n 6= m the distance between Sn (Fn ) and Sm (Fm ) is at least 1, we may
assume that for some fixed n0 , yk ∈ Sn0 (Fn0 ) for all k ∈ N. If for infinitely many
k ∈ N, kyk k = 2n0 + 1, then we have ky∞ k = 2n0 + 1 and so y∞ ∈ Sn0 (Fn0 ). If for
infinitely many k ∈ N, kyk k = 2n0 , then we have that 0 ∈ Fn0 and ky∞ k = 2n0 ,
hence also y∞ ∈ Sn0 (Fn0 ). Otherwise, we may assume that none of the yk has
norm 2n0 or 2n0 + 1. Therefore there are xk ∈ Fn0 such that
yk =
2n0 xk
xk
+
.
kxk k
1 + kxk k
Note that we have from the above equality
kyk k = 2n0 +
and hence
kxk k =
Moreover,
kxk k
,
1 + kxk k
kyk k − 2n0
.
2n0 + 1 − kyk k
−1 !−1
2n0 (2n0 + 1 − kyk k)
kyk k − 2n0
+ 1+
yk .
kyk k − 2n0
2n0 + 1 − kyk k
xk =
Now there are three cases. Case (1): ky∞ k = 2n0 + 1. In this case y∞ ∈
Sn0 (Fn0 ) and there is nothing to prove. Case (2): ky∞ k = 2n0 . In this case we
deduce that
kxk k
kyk k − ky∞ k =
→ 0 as k → ∞.
1 + kxk k
Hence kxk k → 0 and xk → 0. This shows that 0 ∈ Fn0 since Fn0 is closed. Therefore
y∞ ∈ Sn0 (Fn0 ). Case (3): ky∞ k 6= 2n0 or 2n0 + 1. Then by letting k → ∞ in the
last of the displayed formulae in the preceding paragraph we obtain some x∞ such
that xk → x∞ as k → ∞. Thus x∞ ∈ Fn0 and x∞ 6= 0. Moreover, we have that
y∞ =
2n0 x∞
x∞
+
,
kx∞ k
1 + kx∞ k
so y∞ ∈ Sn0 (Fn0 ) as needed.
We verify that S is a Borel U∞ -embedding. It is easy to see that S is Borel
and is an embedding. To see that S respects the group action, it suffices to show
that, for each n ∈ N, T ∈ U (H) and F ∈ F(H),
T (Sn (F )) = Sn (T (F )).
If 0 6∈ F , then 0 6∈ T (F ) and we only need to note that
T (Sn (F ))
= T(
=
2nx
x
+
|x ∈ F
kxk
1 + kxk
∪ { y ∈ H | kyk = 2n + 1})
2nT (x)
T (x)
+
|x ∈ F
kT (x)k
1 + kT (x)k
= Sn (T (F )).
∪ { y ∈ H | kyk = 2n + 1}
14
SU GAO
If 0 ∈ F , the same identity holds except that 0 ∈ T (F ) and the part {y ∈ H | kyk =
2n } must be simultaneously added to all terms. This finishes the proof of the
lemma.
We have thus finished the proof of Theorem 3.3. The following corollary is
immediate from the proof of Lemma 3.5.
Corollary 3.6. Let r2 > r1 ≥ 0 be real numbers. The application action
of U∞ on F({x ∈ H | r1 ≤ kxk ≤ r2 }) induces a faithfully universal U∞ -orbit
equivalence relation. In particular, if B1 (H) is the unit ball of H, then the application action of U∞ on F(B1 (H)) induces a faithfully universal U∞ -orbit equivalence
relation.
We would like to add one more remark about the proof. The reader can compare
the first step of our proof to the argument in [6] for the main theorem there and
readily see that the two proofs have the same structure. In [6] we dealt with the
Urysohn space U instead of H and the isometry group Iso(U) instead of U (H). Our
argument here for the faithful universality of EUY∞ can be repeated for the objects
in [6] to obtain the following improvement.
Theorem 3.7. Let Iso(U) act on F(U)N by coordinatewise and pointwise application. Then the equivalence relation induced by this action is a faithfully universal
orbit equivalence relation for all Borel actions of Polish groups. Consequently, the
Topological Vaught Conjecture for this space is equivalent to the general Topological
Vaught Conjecture.
4. Hilbertian Polish metric spaces
In this section we consider the isometric classification of Hilbertian Polish metric spaces. A Polish metric space is a separable complete metric space. We call a
metric space Hilbertian if it can be isometrically embedded into the Hilbert space
H.
Each Hilbertian Polish metric space has an isometric copy as a closed subset
of H. Conversely, each closed subset of H, with its induced metric from H, is
Hilbertian. Hence the space F(H) can be naturally identified as the space of all
Hilbertian Polish metric spaces.
Note that in the above definition it is equivalent to take the Hilbert space to
be real. In fact, this will simplify some of the computations later in this section, so
we make the convention here that all Hilbert spaces mentioned in this section will
be real. In [11] Schoenberg gave some interesting characterizations of Hilbertian
metric spaces in terms of real-valued positive definite functions and negative definite
functions. To state these characterizations we need to recall the notion of positive
or negative definite functions on a general space X.
A function f : X 2 → R is called positive definite on X if for any x, y ∈ X,
f (x, x) = 0 and f (x, y) = f (y, x), and for any natural number n > 1, arbitrary real
numbers r1 , . . . , rn ∈ R and elements x1 , . . . , xn ∈ X,
X
f (xi , xj )ri rj ≥ 0.
1≤i,j≤n
On the other hand, f is called negative definite on X if for any x, y ∈ X, f (x, x) = 0
and f (x, y) = f (y, x), and for any natural number n > 1, arbitrary real numbers
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
15
r1 , . . . , rn ∈ R and elements x1 , . . . , xn ∈ X,
n
X
X
ri = 0 =⇒
f (xi , yi )ri rj ≤ 0.
i=1
1≤i,j≤n
In case X is a metric space and d is the metric on X, a function g : R → R is called
positive (negative) definite on X if g(d(x, y)) is positive (negative) definite on X.
2
Schoenberg [11] showed that e−λx are positive definite on H for all λ > 0. It
turns out that these functions characterize Hilbertian spaces, in the following sense.
Theorem 4.1 (c.f. [10] and [11]). The following are equivalent for a metric
space (X, d):
(i) X is Hilbertian;
2
(ii) e−λx are positive definite on X for all λ > 0;
2
(iii) e−λx are positive definite on X for a decreasing sequence of positive values
λ converging to 0;
(iv) d2 is negative definite on X;
(v) (Menger) X is separable and for every natural number n > 1, every set
of n + 1 distinct points of X can be isometrically embedded into the space
Rn .
The theorem provides another descriptively simple way of coding Hilbertian
Polish spaces. In general if we code a Polish metric space by specifying the metric
on a countable dense subset, then by Theorem 4.1 (iv), in the space of all codes
for Polish metric spaces the subset of codes for Hilbertian Polish metric spaces is a
closed set. This is another way to see that the space of all Hilbertian Polish metric
spaces is a standard Borel space.
We define the isometry relation ∼
=i for K, L ∈ F(H) by
∼i L ⇐⇒ there is an isometry from K onto L.
K=
Note that ∼
=i is different from the orbit equivalence relation induced by the U∞
action on F(H) considered in the previous section. Rather, it can be identified with
the Iso(H)-orbit equivalence relation on F(H), which we prove below.
Theorem 4.2. The isometry relation ∼
=i for Hilbertian Polish metric spaces
is Borel bireducible with the orbit equivalence relation of the application action of
Iso(H) on F(H).
F (H)
Proof. Let E denote the orbit equivalence relation E
. We need to show
Iso(H)
∼
∼
that (=i ) ≤B E and E ≤B (=i ). Before defining the reductions let us recall a basic
fact of the metric geometry in Hilbert and Euclidean spaces:
If A and B are subsets of Rn , n > 0 or H, and ϕ : A → B
is an isometry from A onto B, then ϕ can be extended to an
isometry from hAi onto hBi, where h·i denote the affine subspace
generated by the set.
We now define the reduction witnessing (∼
=i ) ≤B E. Fix an isometric embedding f of H into H so that f (H)⊥ has infinite dimension. Consider the Borel map
K 7→ f (K) from F(H) into F(H). If K, L ∈ F(H) are not isometric to each other,
then f (K) and f (L) are in no way E-equivalent. On the other hand, if K ∼
=i L,
then f (K) ∼
=i f (L). Let ϕ be an isometry from f (K) onto f (L). Then by the
above fact, ϕ can be extended to an isometry ϕ∗ from hf (K)i onto hf (L)i. By
16
SU GAO
our construction, both hf (K)i⊥ and hf (L)i⊥ have infinite dimension, and therefore
there is an isometry ψ between them. Finally the isometry ϕ∗ ⊕ ψ is an isometry
of the whole space sending f (K) to f (L), and therefore f (K)Ef (L).
In order to establish the converse reduction we first show that E ≤B (∼
=i )×idN ,
where the equivalence relation on the right-hand side is defined on F(H) × N by
∼i ) × idN ⇐⇒ K =
∼i L and m = n.
((K, n), (L, m)) ∈ (=
For this we simply associate with any given K ∈ F(H) the pair (K, n), where
n = 0 if dim(hKi⊥ ) is infinite and n = dim(hKi⊥ ) + 1 otherwise. Note that this is
a Borel map since it is Borel to check if the dimension of hKi⊥ is finite or not. It
is easy to see that this map is a reduction from E to (∼
=i ) × idN , using again the
aforementioned fact and by a similar argument as in the preceding paragraph.
It remains to see that (∼
=i )×idN ≤B (∼
=i ). This requires us to associate with any
pair (K, n), where K is a Hilbertian Polish metric space and n a natural number,
another Hilbertian Polish metric space Kn , so that
K∼
=i L and n = m ⇐⇒ Kn ∼
=i Lm .
For this suppose we are given K and n, and dK is the metric on K. Let K 0 be a
metric space with underlying set K and a new metric d0K defined by
p
d0K (x, y) = 1 − e−dK (x,y) .
To see that d0K is a metric, note that the function 1 − e−t is monotone increasing
for t > 0, which implies that 1 − e−dK (x,y) is a metric, and that the square root
of any metric is a metric. Also note that the transformation from dK to d0K is a
homeomorphism, which implies that d0K is a complete metric iff dK is. Obviously
d0K < 1, and thus the diameter of K 0 is ≤ 1.
We claim that K 0 is a Hilbertian Polish metric space. By Theorem 4.1 it suffices
0 2
to check
Pn that (dK ) is negative definite. Let x1 , . . . , xn ∈ X and r1 , . . . , rn ∈ R
with i=1 ri = 0. Then
n
X
i,j=1
d0K (xi , xj )2 ri rj
=
n
X
(1 − e−dK (xi ,xj ) )ri rj
i,j=1
n
X
= −
e−dK (xi ,xj ) ri rj .
i,j=1
√
By a theorem of Scheonberg (Corollary 1 of [11]) the space (K, dK ) is Hilbertian.
Thus by Theorem 4.1(ii) the function e−dK (x,y) is positive definite on K. Thus the
above displayed formula must be non-positive. This shows that K 0 is Hilbertian.
We then identify K 0 with a closed subset of H so that hK 0 i⊥ has infinite dimension.
We are now ready to define Kn . Let Kn be the union of K 0 with the set of all
points u ∈ H such that the distance between u and hK 0 i is exactly n + 1. It is easy
to see that Kn is a closed subset of H. Moreover, the part Kn − K 0 is a connected
clopen subset of Kn with infinite diameter. We check that this construction works.
Let (K, n) and (L, m) be given, and Kn and Lm are constructed. If K ∼
=i L
then K 0 ∼
=i L0 and therefore if in addition n = m then Kn ∼
=i Lm . Conversely,
suppose Km ∼
=i Ln and ϕ is an isometry witnessing it. Then ϕ(Kn − K 0 ) must be
a connected clopen subset of Lm with infinite diameter, hence it must be Ln − L0 .
It follows that ϕ(K 0 ) = L0 and that n = m. The same ϕ witnesses that K ∼
=i L by
a parallel change of metric on both spaces.
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
17
This finishes our proof that (∼
=i ) × idN ≤B (∼
=i ), and therefore of the theorem.
We can now establish the main theorem of this paper, that is the universality
of these equivalence relations among all U∞ -orbit equivalence relations.
Theorem 4.3. The isometry equivalence relation for Hilbertian Polish metric
spaces is Borel bireducible to a universal U∞ -orbit equivalence relation.
Proof. By a theorem of Mackey (c.f. Theorem 2.3.5 of [8]) and Theorem 2.4,
any Iso(H)-orbit equivalence relation is Borel reducible to some U∞ -orbit equivaF (H)
lence relation. Thus it follows from Theorems 4.2 and 3.3 that (∼
=i ) ≤B EU∞ .
For the converse, we consider the set X = {x ∈ H | 1 ≤ kxk ≤ 2} and the
application action of U∞ on F(X). By Corollary 3.6, the equivalence relation
F (X)
F (X)
EU∞ is a universal U∞ -orbit equivalence relation. We claim that EU∞ ≤B (∼
=i ).
To define the reduction, let S3 = {x ∈ H | kxk = 3} be the sphere of radius 3 in H.
Given any K ∈ F(X), let K 0 = K ∪ {0} ∪ S3 . We check that the map K 7→ K 0
is a required reduction. If K, L ∈ F(X) and there is a unitary transformation
U ∈ U (H) mapping K onto L, then certainly U (0) = 0 and U maps S3 onto S3 ,
therefore in particular K 0 ∼
=i L0 . Conversely, suppose K 0 ∼
=i L0 and this is witnessed
by an isometry ϕ. Then ϕ can be extended to an isometry of H onto H, by the fact
we noted in the proof of Theorem 4.2. Since S3 is a connected clopen subset of K
with diameter 6, its image under ϕ or its extension must be S3 as a subset of L.
From this it follows that ϕ(0) = 0 since the origin can be metrically characterized
as the only point in K with distance 3 to every point of S3 . It then follows that the
extension of ϕ is an isometry of the whole space sending 0 to 0. Thus this extension
must be a unitary transformation and moreover, it sends K onto L.
Since U∞ is naturally a closed subgroup of Iso(H), we have the following
immediate corollary.
Corollary 4.4. The following equivalence relations are pairwise Borel bireducible:
(1)
(2)
(3)
(4)
(5)
(6)
the
the
the
the
the
the
isometry of Hilbertian Polish metric spaces;
universal U∞ -orbit equivalence relation;
universal Iso(H)-orbit equivalence relation;
Iso(H)-orbit equivalence relation on F(H);
N
Iso(H)-orbit equivalence relation on F(H)
;
Q
Iso(H)-orbit equivalence relation on n F(Hn ).
Before closing the paper we would like to explore a bit further the faithfulness
of the universal equivalence
Q relations. A general theorem in [6] (Corollary 9.2 of [6])
implies that the space n F(Hn ) is a universal Borel Iso(H)-space. As the space
improves to F(H)N , we lose this property but retain faithful universality. This is
the content of the theorem to follow. We do not know if there is a faithful reduction
further down to F(H).
Theorem 4.5. The application action of Iso(H) on F(H)N induces a faithfully
universal Iso(H)-orbit equivalence relation.
18
SU GAO
The proof is similar to the first step of that of Theorem 3.3 in the last section.
We again consider the spaces
Y
X=
F(H n ) and Y = F(H)N ,
n>0
X
Y
and show that there is a faithful Borel reduction from EIso
to EIso
.
(H)
(H)
One can repeat the setup of the proof of Theorem 3.3 to the point that the
spaces Hn , H∞ and Z, as well as inner products on them, are defined and each
ϕ ∈ Iso(H) naturally induces ϕ(n) ∈ Iso(Hn ), ϕ(∞) ∈ Iso(H∞ ) and ϕZ ∈ Iso(Z).
Denote
GZ = { ϕZ | ϕ ∈ Iso(H) }.
Then we have a lemma similar to Lemma 3.4 (but with a different proof) below.
Lemma 4.6. There is a sequence {Dm }n∈N of closed subsets of Z such that
GZ = { Φ ∈ Iso(Z) | ∀m (Φ(Dm ) = Dm ) }.
Proof. We again show that for each n > 1 there is a sequence {Km }m∈N of
closed subsets Hn such that
{ ϕ(n) ∈ Iso(Hn ) | ϕ ∈ Iso(H) } = { Φ ∈ Iso(Hn ) | ∀m (Φ(Km ) = Km ) }.
Denote the group on the left side by G(n) . For each tuple q = (qij )1≤i,j≤n of positive
rational numbers, let
Also define
Lq = {(x1 , . . . , xn ) ∈ Hn | ∀1 ≤ i, j ≤ n (kxi − xj k ≤ qij ) }.
G = { Φ ∈ Iso(Hn ) | ∀q (Φ(Lq ) = Lq ) };
in other words, G consists exactly of Φ ∈ Iso(H n ) such that
for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ Hn , if Φ(x1 , . . . , xn ) = (y1 , . . . , yn ),
then for all 1 ≤ i, j ≤ n, kxi − xj k = kyi − yj k.
We verify that G(n) = G.
It is clear that G(n) ⊆ G. For the other inclusion we argue that if Φ ∈ G is the
identity map on H ⊆ Hn then it is identity everywhere on Hn .
Suppose that Φ ∈ G and that for any x ∈ H,
Φ(x, . . . , x) = (x, . . . , x).
(4.1)
In particular, Φ(0, . . . , 0) = (0, . . . , 0) and thus Φ is a unitary transformation of H n .
For any 1 ≤ i ≤ n and x ∈ H, let uxi denote the element of Hn whose i-th coordiate
is x and other coordinates are all 0. We also present uxi as (0, . . . , 0, x, 0, . . . , 0) if i
is understood properly. Since Φ is a linear map, it is enough to show that for any
1 ≤ i ≤ n and x ∈ H, Φ(uxi ) = uxi .
Fix 1 ≤ i ≤ n and 0 6= x ∈ H, and assume
Φ(uxi ) = Φ(0, . . . , 0, x, 0, . . . , 0) = (y1 , . . . , yn ).
Since Φ ∈ G, we have that y1 = · · · = yi−1 = yi+1 = yn . Let y0 denote this common
element. We also have
kyi − y0 k = kxk.
(4.2)
By (5.1) and the unitarity of Φ, we have that for any z ∈ H,
n
X
hx, zi =
hyi , zi.
(4.3)
i=1
UNITARY GROUP ACTIONS AND HILBERTIAN POLISH METRIC SPACES
19
Thus any u orthogonal to x in H is also orthogonal to each yi , 1 ≤ i ≤ n. It follows
that for every 1 ≤ i ≤ n, yi is in the subspace generated by x, and hence is a scalar
multiple of x. Let α, β ∈ R such that y0 = αx and yi = βx. Then (5.2) yields
|α − β| = 1.
(4.4)
Also by the linearity of Φ, our assumption and (5.1), for any z ∈ H,
Φ(−z, . . . , −z, x − z, −z, . . . , −z) = (y0 − z, . . . , y0 − z, yi − z, y0 − z, . . . , y0 − z).
Thus by the unitarity of Φ again, the norms of the vectors on both sides are the
same. Thus
(n − 1)kzk2 + kx − zk2 = (n − 1)kαx − zk2 + kβx − zk2 .
(4.5)
In (5.5) if we plug in z = λx for arbitrary λ, the equation can be simplified to
(n − 1)λ2 + (1 − λ)2 = (n − 1)(α − λ)2 + (β − λ)2
and further to
−2λ + 1 = −2(n − 1)αλ − 2βλ + (n − 1)α2 + β 2 .
Since λ is arbitrary, we must have
and
1 = (n − 1)α + β
(4.6)
1 = (n − 1)α2 + β 2 .
(4.7)
Solving the equation system formed by (5.4), (5.6) and (5.7) we get two sets of
solutions:
α=0
α = 2/n
and
β=1
β = 2/n − 1.
Let vix denote the element of Hn given by the second set of solutions, i.e.,
2
2
2
2
2
vix = ( x, . . . , x, ( − 1)x, x, . . . , x),
n
n
n
n
n
where the term with coefficient 2/n − 1 appears as the i-th coordinate.
The rest of the proof is a verbatim repetition of the part following Lemma 3.4.
Acknowledgement. In the process of developing the results in this paper I
learned a lot from discussions with Bob Kallman, Aleko Kechris, Michael Megrelishvili, Vladimir Pestov and Vladimir Uspenskiy. My heartfelt appreciation goes
to all of them. Moreover, I would like to thank Vladimir Pestov for hosting my research visit to University of Ottawa and Vladimir Uspenskiy for telling me Theorem
2.4 and for his kind permission so that the result can be presented here.
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SU GAO
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Department of Mathematics, PO Box 311430, University of North Texas, Denton,
TX 76203
E-mail address: [email protected]