Normed groups

140 Pages·2009·0.62 MB·English

Normed groups: dichotomy and duality
N. H. Bingham
Mathematics Department, Imperial College London,
South Kensington, London SW7 2AZ
A. J. Ostaszewski
Mathematics Department, London School of Economics,
Houghton Street, London WC2A 2AE
May 2008 BOst-Normed Groups-12-Lip.tex
CDAM Research Report LSE-CDAM-2008-10
Abstract
Thekeyvehicleoftherecentdevelopmentofatopologicaltheoryof
regularvariationbasedontopologicaldynamics[BOst13],andembrac-
ing its classical univariate counterpart (cf. [BGT]) as well as fragmen-
tary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]),
are groups with a right-invariant metric carrying (cid:135)ows. Following the
vector paradigm, they are best seen as normed groups. That concept
only occasionally appears explicitly in the literature despite its fre-
quent disguised presence, and despite a respectable lineage traceable
back to the Pettis closed-graph theorem, to the Birkho⁄-Kakutani
metrization theorem and further back still to Banach(cid:146)s ThØorie des
opØrations linØaires. We collect together known salient features and
develop their theory including Steinhaus theory uni(cid:133)ed by the Cate-
goryEmbeddingTheorem[BOst11],theassociatedthemesofsubaddi-
tivity and convexity, and a topological duality inherent to topological
dynamics. We study the latter both for its independent interest and
as a foundation for topological regular variation.
Classi(cid:133)cation: 26A03
Keywords: multivariate regular variation, topological dynamics,
(cid:135)ows,convexity,subadditivity,quasi-isometry,Souslin-graphtheorem,
automatic continuity, density topology, Lipschitz norm.
1
Contents
1 Introduction 2
2 Metric versus normed groups 4
3 Normed versus topological groups 29
4 Subadditivity 55
5 Theorems of Steinhaus type and Dichotomy 60
6 The Kestelman-Borwein-Ditor Theorem: a bitopological ap-
proach 70
7 The Subgroup Theorem 80
8 The Semigroup Theorem 82
9 Convexity 86
10 Automatic continuity: the Jones-Kominek Theorem 97
11 Duality in normed groups 111
12 Divergence in the bounded subgroup 119
13 References 126
1 Introduction
Group norms, which behave like the usual vector norms except that scaling
is restricted to the basic scalars of group theory (the units 1 in an abelian
(cid:6)
context and the exponents 1 in the non-commutative context), have played
(cid:6)
a part in the early development of topological group theory. Although ubiq-
uitous, they lack a clear and uni(cid:133)ed exposition. This lack is our motivation
here, since they o⁄er the right context for the recent theory of topological reg-
ular variation. This extends the classical theory (for which see, e.g. [BGT])
from the real line to metrizable topological groups. Normed groups are just
2
groups carrying a right-invariant metric. The basic metrization theorem for
groups, the Birkho⁄-Kakutani Theorem of 1936 ([Bir], [Kak], see [Kel], Ch.6
Problems N-R, [Klee], [Bour] Part 2, Section 4.1, and [ArMa], compare also
[Eng] Exercise 8.1.G and Th. 8.1.21), is usually stated as asserting that
a (cid:133)rst-countable Hausdor⁄ group has a right-invariant metric. It is prop-
erly speaking a (cid:145)normability(cid:146)theorem in the style of Kolmogorov(cid:146)s Theorem
([Kol], or [Ru-FA2], Th. 1.39; in this connection see also [Jam], where strong
forms of connectedness are used in an abelian setting to generate norms), as
we shall see below. Indeed the metric construction in [Kak] is reminiscent
of the more familiar construction of a Minkowski functional (for which see
[Ru-FA2] Sect. 1.33), but is implicitly a supremum norm (cid:150)as de(cid:133)ned below;
in Rudin(cid:146)s derivation of the metric (for a topological vector space setting,
[Ru-FA2] Th. 1.24) this norm is explicit. Early use by A. D. Michal and
his collaborators was in providing a canonical setting for di⁄erential calculus
(see the review [Mich] and as instance [JMW]) and included the noteworthy
generalization of the implicit function theorem by Bartle [Bart] (see Section
6). Innamethegroupnormmakesanexplicitappearancein1950in[Pet1]in
the course of his classic closed-graph theorem (in connection with Banach(cid:146)s
closed-graph theorem and the Banach-Kuratowski category dichotomy for
groups). It reappears in the group context in 1963 under the name (cid:145)length
function(cid:146), motivated by word length, in the work of R. C. Lyndon [Lyn2] (cf.
[LynSch]) on Nielsen(cid:146)s Subgroup Theorem, that a subgroup of a free group
is a free group. (Earlier related usage for function spaces is in [EH].) The
latter name is conventional in geometric group theory despite the parallel
usage in algebra (cf. [Far]) and the recent work on norm extension (from a
normal subgroup) of B(cid:246)kamp [Bo].
When a group is topologically complete and also abelian, then it admits
a metric which is bi-invariant, i.e. is both right- and left-invariant, as [Klee]
showed in solving a problemof Banach. This context is of signi(cid:133)cance for the
calculus of regular variation (in the study of products of regularly varying
functions with range a normed group) (cid:150)see [BOst15].
Fresh interest in metric groups dates back to the seminal work of Milnor
[Mil] in 1968 on the metric properties of the fundamental group of a manifold
and is key to the global study of manifolds initiated by Gromov [Gr1], [Gr2]
in the 1980s (and we will see quasi-isometries in the duality theory of normed
groups), for which see [BH] and also [Far] for an early account; [PeSp] con-
tains a variety of generalizations and their uses in interpolation theory (but
the context is abelian groups).
3
The very recent [CSC] (see Sect. 2.1.1, Embedding quasi-normed groups
into Banach spaces) employs norms in considering Ulam(cid:146)s problem (see [Ul])
on the global approximation of nearly additive functions by additive func-
tions. This is a topic related to regular variation, where the weaker concept
of asymptotic additivity is the key. Recall the classical de(cid:133)nition of a regu-
larly varying function, namely a function h : R R for which the limit
!
@ h(t) := lim h(tx)h(x) 1 (1)
R (cid:0)
x
!1
exists everywhere; for f Baire, the limit function is a continuous homomor-
phism (i.e. a multiplicative function). Following the pioneering study of
[BajKar] launching a general (i.e., topological) theory of regular variation,
[BOst13] has re-interpreted (1), by replacing x with x ; for
j j ! 1 jj jj ! 1
functions h : X H; with tx being the image of x under a T-(cid:135)ow on X
!
(de(cid:133)ned in Section 4), and with X;T;H all groups with right-invariant met-
ric (right because of the division on the right) (cid:150)i.e. normed groups (making
@h a di⁄erential at in(cid:133)nity, in Michal(cid:146)s sense [Mi]). In concrete applications
X
the groups may be the familiar Banach groups of functional analyis, the as-
sociated (cid:135)ows either the ubiquitous domain translations of Fourier transform
theory or convolutions from the related contexts of abstract harmonic analy-
sis(e.g. Wiener(cid:146)sTauberiantheorysorelevanttoclassicalregularvariation(cid:150)
see e.g. [BGT, Ch. 4]). In all of these one is guaranteed right-invariant met-
rics. Likewiseinthefoundationsofregularvariationthe(cid:133)rsttoolisthegroup
(X) of bounded self-homeomorphisms of the group X under a supremum
H
metric (and acting transitively on X); the metric is again right-invariant and
hence a group norm. It is thus natural, in view of the applications and the
Birkho⁄-Kakutani Theorem, to demand right-invariance.
We show in Section 4 and 6 that normed groups o⁄er a natural setting
for subadditivity and for (mid-point) convexity.
2 Metric versus normed groups
This section is devoted to group-norms and their associated metrics. We
collect here some pertinent information (some of which is scattered in the
literature). Acentraltoolforapplicationsistheconstructionofthesubgroup
of bounded homeomorphisms of a given group of self-homeomorphisms of
G
a topological group X; the subgroup possesses a guaranteed right-invariant
metric. Thisisthearchetypalexampleofthesymbiosisofnormsandmetrics,
4
and it bears repetition that, in applications just as here, it is helpful to work
simultaneously with a right-invariant metric and its associated group norm.
We say that the group X is normed if it has a group-norm as de(cid:133)ned
below (cf. [DDD]).
De(cid:133)nition. We say that : X R+ is a group-norm if the following
jj(cid:1)jj !
properties hold:
(i) Subadditivity (Triangle inequality): xy x + y ;
jj jj (cid:20) jj jj jj jj
(ii) Positivity: x > 0 for x = e;
jj jj 6
(iii) Inversion (Symmetry): x 1 = x :
(cid:0)
jj jj jj jj
If (i) holds we speak of a group semi-norm; if (i) and (iii) and e = 0
jj jj
holds one speaks of a pseudo-norm (cf. [Pet1]); if (i) and (ii) holds we speak
of a group pre-norm (see [Low] for a full vocabulary).
We say that a group pre-norm, and so also a group-norm, is abelian, or
more precisely cyclically permutable, if
(iv) Abelian norm (cyclic permutation): xy = yx for all x;y:
jj jj jj jj
Other properties we wish to refer to are :
(i) for all x;y : xy K( x + y );
K
jj jj (cid:20) jj jj jj jj
(i) for all x;y : xy max x ; y :
ult
jj jj (cid:20) fjj jj jj jjg
Remarks 1
1. Mutatis mutandis this is just the usual vector norm, but with scal-
ing restricted to the units 1: The notation and language thus mimick the
(cid:6)
vector space counterparts, with subgroups playing the role of subspaces; for
example, for a symmetric, subbadditive p : X R+; the set x : p(x) = 0
! f g
is a subgroup. Indeed the analysis of Baire subadditive functions (see Sec-
tion 4) is naturally connected with norms, via regular variation. That is why
normed groups occur naturally in regular variation theory.
2. When (i) , for some constant K; replaces (i), one speaks of quasi-
K
norms (see [CSC], cf. (cid:145)distance spaces(cid:146)[Rach] for a metric analogue). When
(i) replaces (i) one speaks of an ultra-norm, or non-Archimedean norm.
ult
3. Note that (i) implies joint continuity of multiplication, while (iii) im-
pliescontinuityofinversion,butineachcaseonlyattheidentity,e ,amatter
X
we return to in Section 3. (Montgomery [Mon1] shows that joint continuity
is implied by separate continuity when the group is locally complete.) See
belowforthestrongernotionofuniformcontinuityinvokedintheUniformity
Theorem of Conjugacy.
4. Abeliangroupswithorderednormsmayalsobeconsidered, cf. [JMW].
5
Remarks 2
Subadditivity implies that e 0 and this together with symmetry
jj jj (cid:21)
implies that x 0; since e = xx 1 2 x ; thus a group norm
(cid:0)
jj jj (cid:21) jj jj jj jj (cid:20) jj jj
cannot take negative values. Subadditivity also implies that xn n x ;
jj jj (cid:20) jj jj
for natural n: The norm is said to be 2-homogeneous if x2 = 2 x ; see
jj jj jj jj
[CSC] Prop. 4.12 (Ch. IV.3 p.38) for a proof that if a normed group is
amenable or weakly commutative (de(cid:133)ned in [CSC] to mean that, for given
x;y; there is m of the form 2n; for some natural number n; with (xy)m =
xmym), then it is embeddable as a subgroup of a Banach space. In the
case of an abelian group 2-homogeneity corresponds to sublinearity, and here
Berz(cid:146)sTheoremcharacterizesthenorm(see[Berz]and[BOst5]). Theabelian
property implies only that xyz = zxy = yzx ; hence the alternative
jj jj jj jj jj jj
name of (cid:145)cyclically permutable(cid:146). Harding [H], in the context of quantum
logics, uses this condition to guarantee that the group operations are jointly
continuous (cf. Theorem 2 below) and calls this a strong norm. See [Kel] Ch.
6 Problem O (which notes that a locally compact group with abelian norm
has a bi-invariant Haar measure). We note that when X is a locally compact
group continuity of the inverse follows from the continuity of multiplication
(see [Ell]). The literature concerning when joint continuity of (x;y) xy
!
follows from separate continuity reaches back to Namioka [Nam] (see e.g.
[Bou], [HT], [CaMo]).
Convention. For a variety of purposes and for the sake of clarity, when
wedealwithametrizablegroupX ifweassumeametricdX onX isright/left
invariant we will write dX or dX; omitting the superscript and perhaps the
R L
subscript if context permits.
Remarks 3
For X a metrizable group with right-invariant metric dX and identity e ;
X
the canonical example of a group-norm is identi(cid:133)ed in Proposition 2.3 below
as
x := dX(x;e ):
X
jj jj
Remarks 4
If f : R+ R+ is increasing, subadditive with f(0) = 0; then
!
x := f( x )
jjj jjj jj jj
is also a group-norm. See [BOst5] for recent work on Baire (i.e., having the
Baire property) subadditive functions. These will appear in Section 3.
6
We begin with two key de(cid:133)nitions.
De(cid:133)nition and notation. For X a metric space with metric dX and
(cid:25) : X X a bijection the (cid:25)-permutation metric is de(cid:133)ned by
!
dX(x;y) := dX((cid:25)(x);(cid:25)(y)):
(cid:25)
When X is a group we will also say that dX is the (cid:25)-conjugate of dX: We
(cid:25)
write
x := dX((cid:25)(x);(cid:25)(e));
(cid:25)
jj jj
and for d any metric on X
Bd(x) := y : d(x;y) < r ;
r f g
suppressing the superscript for d = dX; however, for d = dX we adopt the
(cid:25)
briefer notation
B(cid:25)(x) := y : dX(x;y) < r :
r f (cid:25) g
Following [BePe] Auth(X) denotes the group of auto-homeomorphisms of X
under composition, but without a topological structure. We denote by id
X
the identity map id (x) = x on X:
X
Examples A. Let X be a group with metric dX: The following permu-
tation metrics arise naturally in this study.
1. With(cid:25)(x) = x 1 werefertothe(cid:25)-permutationmetricastheinvolution-
(cid:0)
conjugate, or just the conjugate, metric and write
d~X(x;y) = dX(x;y) = dX(x 1;y 1); so that x = x = x 1 :
(cid:25) (cid:0) (cid:0) jj jj(cid:25) jj jj jj (cid:0) jj
2. With (cid:25)(x) = (cid:13) (x) := gxg 1; the inner automorphism, we have (drop-
g (cid:0)
ping the additional subscript, when context permits):
dX(x;y) = dX(gxg 1;gyg 1); so that x = gxg 1 :
(cid:13) (cid:0) (cid:0) jj jj(cid:13) jj (cid:0) jj
3. With (cid:25)(x) = (cid:21) (x) := gx; the left shift by g, we refer to the (cid:25)-
g
permutation metric as the g-conjugate metric, and we write
dX(x;y) = dX(gx;gy):
g
If dX is right-invariant, cancellation on the right gives
dX(gxg 1;gyg 1) = dX(gx;gy); i.e. dX(x;y) = dX(x;y) and x = gxg 1 :
(cid:0) (cid:0) (cid:13) g jj jjg jj (cid:0) jj
7
For dX right-invariant, (cid:25)(x) = (cid:26) (x) := xg; the right shift by g, gives nothing
g
new:
dX(x;y) = dX(xg;yg) = dX(x;y):
(cid:25)
But, for dX left-invariant, we have
x = g 1xg :
(cid:25) (cid:0)
jj jj jj jj
4 (Topological permutation). For (cid:25) Auth(X); i.e. a homeomor-
2
phism and x (cid:133)xed, note that for any " > 0 there is (cid:14) = (cid:14)(") > 0 such
that
d (x;y) = d((cid:25)(x);(cid:25)(y)) < ",
(cid:25)
provided d(x;y) < (cid:14); i.e
B (x) B(cid:25)(x):
(cid:14) (cid:26) "
Take (cid:24) = (cid:25)(x) and write (cid:17) = (cid:25)(y); there is (cid:22) > 0 such that
d(x;y) = d ((cid:24);(cid:17)) = d((cid:25) 1((cid:24));(cid:25) 1((cid:17))) < ",
(cid:25) 1 (cid:0) (cid:0)
(cid:0)
provided d (x;y) = d((cid:25)(x);(cid:25)(y)) = d((cid:24);(cid:17)) < (cid:22); i.e.
(cid:25)
B(cid:25)(x) B (x):
(cid:22) (cid:26) "
Thus the topology generated by d is the same as that generated by d: This
(cid:25)
observation applies to all the previous examples provided the permutations
are homeomorphisms (e.g. if X is a topological group under dX): Note that
for dX right-invariant
x = (cid:25)(x)(cid:25)(e) 1 :
(cid:25) (cid:0)
jj jj jj jj
5. For g Auth(X);h X; the bijection (cid:25)(x) = g((cid:26) (x)) = g(xh) is a
h
2 2
homeomorphism provided right-shifts are continuous. We refer to this as the
shifted g-h-permutation metric
dX (x;y) = dX(g(xh);g(yh));
g-h
which has the associated g-hshifted norm
x = dX(g(xh);g(h)):
g-h
jj jj
6 (Equivalent Bounded norm). Set d(cid:22)(x;y) = min dX(x;y);1 : Then
(cid:22) f g
d is an equivalent metric (cf. [Eng] Th. 4.1.3, p. 250). We refer to
x(cid:22):= d(cid:22)(x;e) = min dX(x;e);1 = min x ;1 ;
jj jj f g fjj jj g
8
as the equivalent bounded norm.
7. For = Auth(X) the evaluation pseudo-metric at x on is given by
A A
d (f;g) = dX(f(x);g(x));
Ax
and so
f = d (f;id) = dX(f(x);x)
jj jjx Ax
is a pseudo-norm.
De(cid:133)nition (Re(cid:133)nements). 1 (cf. [GJ] Ch. 15.3 which works with
pseudometrics.) Let (cid:1) = dX : i I be a family of metrics on a group X.
f i 2 g
The weak (Tychonov) (cid:1)-re(cid:133)nement topology on X is de(cid:133)ned by reference to
the local base at x obtained by a (cid:133)nite intersections of "-balls about x :
Bi(x); for F (cid:133)nite, i.e. Bi1(x) ::: Bin(x); if F = i ;:::;i ;
" " \ \ " f 1 ng
i F
\2
where
Bi(x) := y X : dX(x;y) < " :
" f 2 i g
2. The strong (cid:1)-re(cid:133)nement topology onX is de(cid:133)nedbyreference to the local
base at x obtained by a full intersections of "-balls about x :
Bd(x):
"
d (cid:1)
\2
Clearly
Bd(x) Bi(x); for F (cid:133)nite,
" (cid:26) "
d (cid:1) i F
\2 \2
hence the name. We will usually be concerned with a family (cid:1) of conjugate
metrics. We note the following, which is immediate from the de(cid:133)nition. (For
(ii) see the special case in [dGMc] Lemma 2.1, [Ru-FA2] Ch. I 1.38(c), or
[Eng] Th. 4.2.2 p. 259, which uses a sum in place of a supremum, and
identify X with the diagonal of (X;d); see also [GJ] Ch. 15.)
d (cid:1)
2
Q
Proposition 2.1. (i) The strong (cid:1)-re(cid:133)nement topology is generated by
the metric
dX(x;y) = sup dX(x;y) : i I :
(cid:1) f i 2 g
9
(ii) The weak (cid:1)-re(cid:133)nement topology for (cid:1) a countable family of metrics
indexed by I = N is generated by the metric
dX(x;y)
dX(x;y) = sup2 i i :
(cid:1) (cid:0) 1+dX(x;y)
i I i
2
Examples B. 1. For X a group we may take (cid:1) = dX : z X to
f z 2 g
obtain
dX(x;y) = sup dX(zx;zy) : z X ;
(cid:1) f 2 g
and if dX is right-invariant
x = sup zxz 1 :
(cid:1) (cid:0)
jj jj jj jj
z
2. For X a topological group we may take (cid:1) = dX : h Auth(X) ; to
f h 2 g
obtain
dX(x;y) = sup dX(h(x);h(y)) : h Auth(X) :
(cid:1) f 2 g
3. In the case = Auth(X) we may take (cid:1) = d : x X ; the
A f Ax 2 g
evaluation pseudo-metrics, to obtain
d (f;g) = supd (f;g) = supdX(f(x);g(x)); and
A(cid:1) Ax
x x
f = supd (f;id ) = supdX(f(x);x):
jj jj(cid:1) Ax X
x x
In Proposition 2.12 we will show that the strong (cid:1)-re(cid:133)nement topology re-
stricted to the subgroup (X) := f : f < is the topology
(cid:1)
H f 2 A jj jj 1g
of uniform convergence. The weak (cid:1)-re(cid:133)nement topology here is just the
topology of pointwise convergence.
The following result illustrates the kind of use we will make of re(cid:133)nement.
Proposition 2.2 (Symmetrization re(cid:133)nement) If x is a group pre-
j j
norm, then the symmetrization re(cid:133)nement
x := max x ; x 1
(cid:0)
jj jj fj j j jg
is a group-norm
10