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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 2, 2008

pp. 197-204

Unitary representability of free abelian

topological groups

Vladimir V. Uspenskij

Abstract. For every Tikhonov space X the free abelian topological
group A(X) and the free locally convex vector space L(X) admit a
topologically faithful unitary representation. For compact spaces X
this is due to Jorge Galindo.

2000 AMS Classification: Primary: 22A25. Secondary 43A35, 43A65,
46B99, 54C65, 54E35, 54H11

Keywords: Unitary representation, free topological group, positive-definite
function, Michael selection theorem

1. Introduction

With every Tikhonov space X one associates the free topological group
F (X), the free abelian topological group A(X), and the free locally convex
vector space L(X). They are characterized by respective universal properties.
For example, L(X) is defined by the following: X is an (algebraic) basis of
L(X), and for every continuous mapping f : X → E, where E is a Hausdorff
locally convex space, the linear extension f̄ : L(X) → E of f is continuous.
There are two versions of L(X), real and complex. There also are versions
of all these free objects for spaces with a distinguished point. We consider
non-pointed spaces.

The unitary group U (H) of a Hilbert space H will be equipped with the
strong operator topology, which is the topology inherited from the Tikhonov
product HH , or, equivalently, the topology of pointwise convergence. We use
the notation Us(H) to indicate this topology. A unitary representation of a
topological group G is a continuous homomorphism f : G → Us(H). Such
a representation is faithful if f is injective, and topologically faithful if f is a
homeomorphic embedding. A topological group is unitarily representable if it
is isomorphic to a topological subgroup of Us(H) for a Hilbert space H (which



198 V. V. Uspenskij

may be non-separable), or, equivalently, if it admits a topologically faithful
unitary representation.

All locally compact groups are unitarily representable. For groups beyond
the class of locally compact groups this is no longer true: there exist abelian
topological groups (even monothetic groups, that is, topologically generated by
one element) for which every unitary representation is trivial (sends the whole
group to the identity), see [1, Theorem 5.1 and Remark 5.2]. Thus one may
wonder what happens in the case of free topological groups: are they unitarily
representable?

In the non-abelian case, this question is open even if X is compact metric,
see [9, Questions 35, 36]. The aim of the present note is to answer the question
in the positive for L(X) and A(X).

Theorem 1.1. For every Tikhonov space X the free locally convex space L(X)
and the free abelian topological group A(X) admit a topologically faithful unitary
representation.

It suffices to consider the case of L(X), since A(X) is isomorphic to the
subgroup of L(X) generated by X [12], [15], see also [5]1. For compact X
(or, more generally, for kω-spaces X) Theorem 1.1 is due to Jorge Galindo [4].
However, it was claimed in an early version of [4] that there exists a metrizable
space X for which the group A(X) is not isomorphic to a subgroup of a unitary
group. This claim was wrong, as Theorem 1.1 shows.

It is known that (the additive group of) the space L1(µ) is unitarily rep-
resentable for every measure space (Ω, µ). (For the reader’s convenience, we
remind the proof in Section 3, see Fact 3.5.) In particular, if µ is the count-
ing measure on a set A, we see that the Banach space l1(A) of summable
sequences is unitarily representable. Since the product of any family of uni-
tarily representable groups is unitarily representable (consider the Hilbert sum
of the spaces of corresponding representations), we see that Theorem 1.1 is a
consequence of the following:

Theorem 1.2. For every Tikhonov space X the free locally convex space L(X)
is isomorphic to a subspace of a power of the Banach space l1(A) for some A.

For A we can take any infinite set such that the cardinality of every discrete
family of non-empty open sets in X does not exceed Card (A).

Theorem 1.1 implies the following result from [5]: every Polish abelian group
is the quotient of a closed abelian subgroup of the unitary group of a separable
Hilbert space (the non-abelian version of the reduction is explained in Section 4,
the argument for the abelian case is the same). It is an open question (A.
Kechris) whether a similar assertion holds for non-abelian Polish groups, see
[8, Section 5.2, Question 16], [9, Question 34].

1The result was stated in [12], but the proof was incomplete. I gave a proof in [15] —
not knowing that I was rediscovering the Wasserstein metric (which is also known under
many other names: Monge – Kantorovich, Kantorovich – Rubinstein, transportation, Earth
Mover’s) and its basic properties, such as the Integer Value Property. The proof is reproduced
in [5], where a historic account is given.



Unitary representability of free abelian topological groups 199

The fine uniformity µX on a Tikhonov space X is the finest uniformity
compatible with the topology. It is generated by the family of all continuous
pseudometrics on X. It is also the uniformity induced on X by the group
uniformity of A(X) or L(X). A fine uniform space is a space of the form
(X, µX ). We note the following corollary of Theorem 1.2 which may be of
some independent interest.

Corollary 1.3. For every Tikhonov space X the fine uniform space (X, µX )
is isomorphic to a uniform subspace of a power of a Hilbert space.

This need not be true for uniform spaces which are not fine: many separa-
ble Banach spaces (for example, c0 or lp for p > 2) do not admit a uniform
embedding in a Hilbert space [3, Chapter 8]. Since the countable power of
an infinite-dimensional Hilbert space H uniformly embeds in H (use the fact
that H uniformly embeds in the unit sphere of itself [3, Corollary 8.11], and
uniformly embed the countable power of the unit sphere into the Hilbert sum
of countably many copies of H), it easily follows that c0 or lp for p > 2 are not
uniformly isomorphic to a subspace of a power of a Hilbert space.

To deduce Corollary 1.3 from Theorem 1.1, note that the left uniformity on
the unitary group Us(H) is induced by the product uniformity of H

H , hence
the same is true for the left uniformity on every unitarily representable group.

For any uniform space X one defines the free abelian group A(X) and free
locally convex space L(X) in an obvious way. The objects A(X) and L(X)
for Tikhonov spaces X considered in this paper are special cases of the same
objects for uniform spaces, corresponding to fine uniform spaces.

Question 1.4 (Megrelishvili). For what uniform spaces X are the groups A(X)
and L(X) unitarily representable? Is it sufficient that X be a uniform subspace
of a product of Hilbert spaces?

We prove Theorem 1.2 in Section 2. The proof depends on two facts: (1)
every Banach space is a quotient of a Banach space of the form l1(A); (2)
every onto continuous linear map between Banach spaces admits a (possibly
non-linear) continuous right inverse. We remind the proof of these facts in
Section 3.

2. Proof of Theorem 1.2

Let X be a Tikhonov space. Let T0 be the topology of the free locally convex
space L(X). Let T1 be the topology on L(X) generated by the linear extensions
of all possible continuous maps of X to spaces of the form l1(A). Theorem 1.2
means that T1 = T0. In order to prove this, it suffices to verify that (L(X), T1)
has the following universal property: for every continuous map f : X → F ,
where F is a Hausdorff LCS, the linear extension of f , say f̄ : L(X) → F ,
is T1-continuous. Since every Hausdorff LCS embeds in a product of Banach
spaces, we may assume that F is a Banach space. Represent F as a quotient
of l1(A) (Fact 3.1), let p : l1(A) → F be linear and onto.



200 V. V. Uspenskij

l1(A)

p

��
X

f
//

g
<<zzzzzzzz
F

s

TT

�

�
) l

1(A)

p

��
L(X)

f̄

//

ḡ
;;wwwwwwww

F

There exists a lift g : X → l1(A) of f , i.e. such a map g for which f = pg.
Indeed, find a non-linear right inverse s : F → l1(A) of p (Fact 3.2), and
set g = sf . Then f = psf = pg. By the definition of T1, the linear map
ḡ : L(X) → l1(A) is T1-continuous. Hence f̄ = pḡ is T1-continuous as well.

3. Basic facts: reminders

Fact 3.1. Every Banach space X is a Banach quotient of a Banach space of
the form l1(A).

By a Banach quotient we mean a Banach space of the form E/F with the
norm ‖x + F ‖ = inf{‖y‖ : y ∈ x + F }.

Proof. Take for A a dense subset of the unit ball of X, and consider the natural
map p : l1(A) → X. �

Fact 3.2 (the Bartle–Graves theorem [13, C.1.2]). Every linear onto map p :
E → F between Banach spaces (or locally convex Fréchet spaces) has a (possibly
non-linear) continuous right inverse s : F → E, i.e. such a map that ps = 1F .

Proof. This is a consequence of Michael’s Selection Theorem for convex-valued
maps [7, 13]. The theorem reads as follows. Suppose X is paracompact, E is a
locally convex Fréchet space, and for every x ∈ X a closed convex non-empty
set Φ(x) ⊂ E is given. Suppose further that Φ is lower semicontinuous: for
every U open in E, the set Φ−1(U ) = {x ∈ X : Φ(x) meets U} is open in X.
Then Φ has a continuous selection: there exists a continuous map s : X → E
such that s(x) ∈ Φ(x) for every x ∈ X.

To prove Fact 3.2, apply Michael’s selection theorem to X = F and Φ
defined by Φ(x) = p−1(x). The lower semicontinuity of Φ follows from the
Open Mapping theorem, according to which p is open.

For various proofs of Michael’s selection theorem, see [13]. In particular, note
a nice reduction of the convex-valued selection theorem to the zero-dimensional
selection theorem [13, A, §3], based on the notion of a Milyutin map. �

For a complex matrix A = (aij ) we denote by A
∗ the matrix (āji). A is

Hermitian if A = A∗. A Hermitian matrix A is positive if all the eigenvalues of
A are ≥ 0 or, equivalently, if A = B2 for some Hermitian B. A complex function
p on a group G is positive-definite, or of positive type, if p(g−1) = p(g) for every
g ∈ G, and for every g1, . . . , gn ∈ G the Hermitian n × n-matrix (p(g

−1
i

gj )) is
positive. If f : G → U (H) is a unitary representation of G, then for every vector
v ∈ H the function p = pv on G defined by p(g) = (gv, v) is positive-definite.
(We denote by (x, y) the scalar product of x, y ∈ H.) Conversely, let p be a



Unitary representability of free abelian topological groups 201

positive-definite function on G. Then p = pv for some unitary representation
f : G → U (H) and some v ∈ H. Indeed, consider the group algebra C[G], equip
it with the scalar product defined by (g, h) = p(h−1g), quotient out the kernel,
and take the completion for H. The regular representation of G on C[G] gives
rise to a unitary representation on H with the required property. (This is the
so-called GNS-construction, see e.g. [2, Theorem C.4.10] for more details.) If
G is a topological group and the function p is continuous, the resulting unitary
representation is continuous as well. These arguments yield the following (see
e.g. [16, Proposition 2.1]):

Fact 3.3. A topological group G is unitarily representable (in other words, is
isomorphic to a subgroup of Us(H) for some Hilbert space H) if and only if for
every neighbourhood U of the neutral element e of G there exist a continuous
positive-definite function p : G → C and a > 0 such that p(e) = 1 and |1 −
p(g)| > a for every g ∈ G \ U .

For a measure space (Ω, µ) we denote by L1(µ) the complex Banach space
of (equivalence classes of) complex integrable functions, and by L1

R
(µ) the real

Banach space of (equivalence classes of) real integrable functions.

Fact 3.4 (Schoenberg [10, 11]). If (Ω, µ) is a measure space and X = L1
R

(µ),
the function x 7→ exp(−‖x‖) on X is positive-definite. In other words, for any
f1, . . . , fn ∈ L

1
R

(µ) the symmetric real matrix (exp(−‖fi − fj‖)) is positive.

Proof. We invoke Bochner’s theorem: positive-definite continuous functions on
Rn (or any locally compact abelian group) are exactly the Fourier transforms

of positive measures. For f ∈ L1(Rn) we define the Fourier transform f̂ by

f̂ (y) =

∫
Rn

f (x) exp (−2πi(x, y)) dx.

Here (x, y) =
∑n

k=1
xkyk for x = (x1, . . . , xn) and y = (y1, . . . , yn).

The positive functions p and q on R defined by p(x) = exp(−|x|) and
q(y) = 2/(1 + 4π2y2) are the Fourier transforms of each other. Hence each of
them is positive-definite. Similarly, the positive functions pn and qn on R

n de-
fined by pn(x1, . . . , xn) = exp(−

∑n
k=1

|xk|) =
∏n

k=1
p(xk) and qn(y1, . . . , yn) =∏n

k=1
q(yk) are positive-definite, being the Fourier transforms of each other. If

m1, . . . , mn are strictly positive masses, the function x 7→ exp(−
∑n

k=1
mk|xk|)

on Rn is positive-definite, since it is the composition of pn and a linear auto-
morphism of Rn. This is exactly Fact 3.4 for finite measure spaces.

The general case easily follows: given finitely many functions f1, . . . , fn ∈
L1

R
(µ), we can approximate them by finite-valued functions. In this way we

see that the symmetric matrix A = (exp(−‖fi − fj‖)) is in the closure of the
set of matrices A′ of the same form arising from finite measure spaces. The
result of the preceding paragraph means that each A′ is positive. Hence A is
positive. �



202 V. V. Uspenskij

As a topological group, L1(µ) is isomorphic to the square of L1
R

(µ). Com-
bining Facts 3.3 and 3.4, we obtain:

Fact 3.5. The additive group of the space L1(µ) is unitarily representable for
every measure space (Ω, µ).

See [3, 6] for more on unitarily and reflexively representable Banach spaces.

4. Open questions

Let us say that a metric space M is of L1-type if it is isometric to a subspace
of the Banach space L1

R
(µ) for some measure space (Ω, µ). A non-abelian

version of Theorems 1.1 and 1.2 might be the following:

Conjecture 4.1. For any Tikhonov space X the free topological group F (X)
is isomorphic to a subgroup of the group of isometries Iso (M ) for some metric
space M of L1-type.

It follows from [16, Theorem 3.1] (apply it to the positive-definite function p
on R used in the proof of Fact 3.4 and defined by p(x) = exp(−|x|)) and Fact 3.4
that for every M ⊂ L1

R
(µ) the group Iso (M ) is unitarily representable. Thus,

if conjecture 4.1 is true, every F (X) is unitarily representable. This would
imply a positive answer to the question of Kechris mentioned in Section 1: is
every Polish group a quotient of a closed subgroup of the unitary group of a
separable Hilbert space? Indeed:

Proposition 4.2. Let P be the space of irrationals. If the group F (P ) is uni-
tarily representable, then every Polish group is a quotient of a closed subgroup
of the unitary group of a separable Hilbert space.

Proof. A topological group is uniformly Lindelöf (or, in another terminology,
ω-bounded) if for every neighbourhood U of the unity the group can be covered
by countably many left (equivalently, right) translates of U . If G is a uniformly
Lindelöf group of isometries of a metric space M , then for every x ∈ M the
orbit Gx is separable (see e.g. the section “Guran’s theorems” in [14]). If G
is a uniformly Lindelöf subgroup of the unitary group Us(H), where H is a
(non-separable) Hilbert space, it easily follows that H is covered by separable
closed G-invariant linear subspaces and therefore G embeds in a product of
unitary groups of separable Hilbert spaces.

If G is a Polish group, there exists a quotient onto map F (P ) → G (because
there exists a continuous open onto map P → G, see Lemma 4.3 below).
The group F (P ), like any separable topological group, is uniformly Lindelöf.
Assume that F (P ) is unitarily representable. Then, as we saw in the first
paragraph of the proof, F (P ) is isomorphic to a topological subgroup of a
power of Us(H), where H is a separable Hilbert space. An easy factorization
argument (see Lemma 4.4) shows that there is a group N lying in a countable
power of Us(H) (and hence isomorphic to a subgroup of Us(H)) such that G
is a quotient of N . The quotient homomorphism N → G can be extended over
the closure of N , so we may assume that N is closed in Us(H). �



Unitary representability of free abelian topological groups 203

The following lemmas were used in the proof above:

Lemma 4.3. For every non-empty Polish space X there exists an open onto
map P → X, where P , as above, is the space of irrationals.

Proof. Consider open covers {Un} of X such that:

• diam U < 2−n for every U ∈ Un;
• each Un is indexed by An = N

n;
• if t ∈ An, then Ut =

⋃
{Us : s ∈ An+1 and t = s|n}

• if s ∈ An+1 and t = s|n, then Us ⊂ Ut.

For every infinite sequence s ∈ NN let xs be the only point in the intersection⋂
Us|n =

⋂
Us|n. Then the map s 7→ xs from N

N (which is homeomorphic to
P ) to X is open and onto. �

Lemma 4.4. Let {Gα : α ∈ A} be a family of topological groups, K a subgroup
of

∏
Gα, H a metrizable topological group, f : K → H a continuous homomor-

phism. Then there exists a countable subset B ⊂ A such that f = g◦pB for some
continuous homomorphism g : KB → H, where pB : K →

∏
{Gα : α ∈ B} is

the projection and KB = pB(K). If f is open and onto, then so is g.

Proof. Let B be a countable base at unity of H. For each U ∈ B pick a finite
set F = FU ⊂ A such that p

−1
F

(V ) ⊂ f −1(U ) for some neighbourhood V of
unity of

∏
{Gα : α ∈ F }. Put B =

⋃
{FU : U ∈ B}. �

5. Acknowledgement

I thank my friends M. Megrelishvili and V. Pestov for their generous help,
which included inspiring discussions and many useful remarks and suggestions
on early versions of this paper.

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Received April 2007

Accepted June 2007

Vladimir Uspenskij (uspensk@math.ohiou.edu)
Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio
45701, USA