This paper appeared in the Journal of the
Nigerian Mathematical Society 11(1992), 63–80.
HOMOMORPHISMS OF MINIMAL AND DISTAL FLOWS
Paul Milnes and John Pym
University of Western Ontario and University of Sheffield
Abstract. Recently we established the existence of Haar measure µ on compact
right topological groups G that satisfy a countability condition, µ being a probability
measure on G that is right invariant, unique as such, and hence also left invariant.
The initial inspiration for the present paper was to reach the same conclusion without
the countability restriction. To do this, we were led to complete the development, that
was initiated by H. Furstenberg and continued by R. Ellis and I. Namioka, of the cycle
of equivalent conditions connecting subgroups of a compact right topological group G,
C ∗ -subalgebras of C(G), joint continuity of translation actions of G on quotient spaces
of G, and almost periodicity of homomorphisms of distal flows. Our study of these
connections led in turn to the conditions which can be imposed on a homomorphism
of minimal flows and are equivalent to its distality. One aspect of the approach
that we offer to equivalence in these matters is reliance on the structure theory for
compact right topological semigroups applied to the enveloping semigroup of a flow.
This allows many of the proofs to be simplified, provides clearer insights (for example,
in interpreting a condition of Ellis, which we now call ‘idempotent preserving’), and
produces in a natural way the groups in the proof that distal homomorphisms arise
from group actions. We are able to add some new conditions to the known list.
In a brief final section, which relies also on Furstenberg’s structure theorem for
distal flows as proved without countability assumptions by Ellis, we establish the
existence of Haar measure µ for all compact right topological groups.
Introduction
Building on the book of W.H. Gottschalk and G.A. Hedlund [10] and H. Furstenberg’s paper [8] (to name but two earlier works), R. Ellis presented in [6] a comprehensive analysis of (discrete) transformation groups, or flows. For any compact
space X, the set X X of all (not necessarily continuous) maps from X to itself is a
compact right topological semigroup under composition. A flow (S, X) consists of
a compact space X and a group S acting on it; that is, there is a homomorphism Ψ
of S into the set of homeomorphisms of X, which forms a subgroup of X X . A key
role in Ellis’s theory is played by the enveloping semigroup ΣX of S, which is the
2000 Mathematics Subject Classification. 54H20, 54H15, 43A07.
Key words and phrases. minimal and distal flow, enveloping semigroup, compact right topological group and semigroup, ideal structure of compact semigroups, distal and almost periodic
flow homomorphism, equicontinuous and jointly continuous action.
This research was supported in part by NSERC grant A7857
Typeset by AMS-TEX
1
closure Ψ(S)− ⊂ X X in the product topology. ΣX is a compact right topological
semigroup with dense topological centre
Λ(ΣX ) := {s ∈ ΣX | t 7→ st, ΣX → ΣX is continuous}
(dense because Ψ(S) ⊂ Λ(ΣX )). The theory of such semigroups is well known
nowadays, but at the time Ellis was writing they had not been considered seriously
and he himself had to develop sufficient theory for his needs. This he did in the
context of the Stone-Čech compactification βS of S and his principal tools were
C ∗ -algebras of functions on βS. Later presentations of the theory (such as Glasner
[9], Veech [17] and Auslander [1]) have for the most part ignored the possibilities
opened up by the establishment of a theory of compact right topological semigroups,
though a start has been made recently by Lawson and Lisan [14]. One of the
main purposes of the present paper is to view flows in the light of their enveloping
semigroups and to show how this illuminates the theory, permitting improvements
in some important theorems about homomorphisms of flows (and more especially
in their proofs).
To illustrate how our approach works, we describe one such result here. Recall
that the flow (S, X) is minimal if no proper closed subset of X is invariant under
S. A map π : (S, Y ) → (S, X) is a (flow) homomorphism if it commutes with
the actions of S on Y and X. In particular, for any minimal left ideal IX of ΣX ,
(S, IX ) is a minimal flow and there is a flow homomorphism from IX to X. Ellis
distinguishes two kinds of homomorphisms of minimal flows as particularly worthy
of study, the distal homomorphisms and the almost periodic homomorphisms. The
various characterizations of distality and almost periodicity are presented in our
Theorems 9 and 10, respectively. We now consider one of the characterisations in
these theorems. If π : Y → X is distal, then there is a further minimal flow (S, Z)
along with homomorphisms θ : Z → X and η : Z → Y such that θ = π ◦ η and θ is a
right topological group homomorphism, which means that there is a compact right
topological group K acting on Z on the right, so that the action of K commutes
with the action of S and the orbits of K are precisely the sets on which θ is constant.
From our perspective the space Z appears naturally as a quotient of IY (a minimal
left ideal of ΣY ), and the group K arises from the algebraic structure of IY ; and
a new fact emerges, namely the homomorphism η is also a right topological group
extension (whose compact right topological group is a subgroup of K). For the more
restricted case of almost periodic flow homomorphisms, the groups constructed are
even topological, and this last conclusion is easily derived from the distal result.
The flow (S, X) is called distal if the enveloping semigroup ΣX ⊂ X X is a
group. Recently, the authors made two improvements to the theory of compact
right topological groups that arise in this way. Given such a group G, there is a
decreasing family {Lξ } of subgroups indexed by the ordinals up to ξ0 , such that
L0 = G and Lξ0 = {e}, and for each ξ ≤ ξ0 , Lξ+1 is normal in Lξ , G/Lξ is
Hausdorff, and the mapping
(sLξ+1 , tLξ+1 ) 7→ sLξ+1 tLξ+1 = stLξ+1 , G/Lξ+1 × Lξ /Lξ+1 → G/Lξ+1
is jointly continuous. This is Namioka’s formulation [16] of Furstenberg’s structure theorem for distal flows [8, or 6; 15.4], and is related to [13; Theorem 1.1],
2
all these results requiring a countability hypothesis. We established in [15] that
the groups Lξ were in fact normal in G. Also, Namioka [16] (and Ellis [6; p.164])
showed that G carries a probability measure invariant under the left action of its
topological centre. Our improved structure theorem for compact right topological
groups enabled us to prove that G had a unique measure (Haar measure) invariant
under all right translations as well as the continuous left translations. However,
Namioka and Ellis knew that their existence theorems for measures did not depend
on countability conditions. It was very difficult to establish the Furstenberg structure theorem without countability hypotheses, but this was eventually achieved by
Ellis [7]. Relying on his result, we are able to conclude from [15] that Haar measure
exists for every compact right topological group with dense topological centre.
Setting and Preliminary Results
Compact Right Topological Groups and the σ-Topology. For notation and
terminology we will follow Namioka [16] and, especially, Berglund et al. [4] as much
as possible. Thus a compact right topological group is a pair (G, τ ), where G is
a group with identity e and τ is a topology on G, so that the right translations
t 7→ ts, G → G are continuous for all s ∈ G. The topological centre Λ(G) of G is
the set
{s ∈ G | t 7→ st, G → G is continuous}.
If Λ(G) = G, then G is called semitopological; if G is also Hausdorff, then G is a
topological group, by Ellis’s famous theorem [5].
Let L be a subgroup of a compact right topological group (G, τ ). Furnished with
the restriction to L of τ (which we also call τ ), L is a compact right topological
group, which we denote by (L, τ ). Similarly, (G/L, τ ) denotes the quotient space
(i.e., the space of left cosets of L) with the quotient topology from τ ; if L0 is a
subgroup of L, then (L/L0 , τ ) is the space of left cosets {tL0 | t ∈ L} with the
subspace-quotient topology from τ.
For a compact, Hausdorff, right topological group (G, τ ), let U denote the set
of open τ -neighbourhoods of e. When Λ(G) is dense in G, the σ-topology of G is
determined by taking S := {U −1 U | U ∈ U} to be a base of σ-neighbourhoods of
e. (The notation σ is to suggest “symmetrized”.) We collect some facts about σ;
see [16] (or [4; Appendix C]) for the details. The reader will note (in (iii)) that σ
is not Hausdorff in the cases that are of interest to us.
1 Theorem. (i) The topology σ is compact, T1 , and coarser than τ.
(ii) (G, σ) is a compact, T1 , semitopological group with continuous inversion.
(iii) σ is not usually Hausdorff; it is Hausdorff if and only if it is the same as
τ , and then (G, τ ) is a topological group.
(iv) For a subgroup L of G, the quotient space (G/L, τ ) is Hausdorff if and only
if L is σ-closed.
Let (G, τ ) be a compact, Hausdorff, right topological group with Λ(G) dense in G
and with σ-topology σ. Let L be a σ-closed subgroup furnished with the restriction
to L of σ (which we also call σ). Set
\
N (L) := {W | W is a σ-closed σ-neighbourhood in L of e}.
We collect some facts about N (L); again, see [16] (or [4]), except for (iv), which is
in [15].
3
2 Theorem. (i) N (L) is a σ-closed normal subgroup of L and
N (L) =
\
{(W ∩ L)(W ∩ L) | W ∈ S} =
\
{(U −1 U ∩ L)(U −1 U ∩ L) | U ∈ U }.
(ii) The quotient topology that L/N (L) gets from τ is the same as that it gets from
σ; in either L/N (L) is a compact, Hausdorff, topological group, and the function
(tN (L), sN (L)) 7→ tsN (L), G/N (L) × L/N (L) → G/N (L)
is continuous for the quotient topologies from τ .
(iii) If Z is a Hausdorff space and f : L → Z is a σ-continuous function, then f
factors through L/N (L): there is a continuous function f : L/N (L) → C such that
f = f ◦ π, where π : L → L/N (L) is the quotient map.
(iv) If L is a normal subgroup of G, so is N (L).
Flows, Flow Homomorphisms, Minimal Flows, and Distal Flows. A flow
(S, X) consists of a compact Hausdorff space X and a group S acting on it, i.e.,
there is a function (s, x) 7→ sx, S × X → X such that ex = x for all x ∈ X, the
function x 7→ sx, X → X is a homeomorphism for all s ∈ S and s(tx) = (st)x for
all s, t ∈ S and x ∈ X. A flow homomorphism or extension π : (S, Y ) → (S, X)
is a continuous function π of Y onto X that commutes with the action of S, i.e.,
π(sy) = s(π(y)) for all s ∈ S and y ∈ Y ; we say that Y is an extension of X,
and that X is a factor of Y . Let Σ = ΣX denote the closure in X X (i.e., the
pointwise closure) of the set of homeomorphisms of X determined by S; Σ is the
enveloping semigroup of (S, X). With the subspace topology from X X , Σ is a
compact, Hausdorff, right topological semigroup and
S ⊂ Λ(Σ) = {T ∈ Σ | T0 7→ T T0 , Σ → Σ is continuous}
[6; 3.2] (or [4; I.6.5]). If π : Y → X is a flow homomorphism, π lifts to a (semigroup)
homomorphism of ΣY onto ΣX , which we also denote by π. (If T : Y → Y is the
limit in ΣY of a net {sα } in S, then for any y ∈ Y we have sα π(y) = π(sα y) →
π(T y), and since π maps Y onto X, it is not hard to conclude that {sα } converges
pointwise on X to a map which it is sensible to denote by π(T ).)
The flow (S, X) is called minimal if Sx := {sx | s ∈ S} is dense in X for
all x ∈ X, and distal if the equality limα sα x = limα sα x0 for x, x0 ∈ X and net
{sα } ⊂ S necessarily implies that x = x0 . A flow (S, X) is minimal if and only if
there exist a point x ∈ X and a minimal idempotent u ∈ Σ such that (Sx)− = X
and ux = x. If the flow (S, X) is minimal and I is a minimal left ideal in Σ, then
Ix = X for any x ∈ X and
(ST )− = {sT | s ∈ S}− = I
for any T ∈ I; in particular, (S, I) with action (s, T ) 7→ sT is a minimal flow. (It
is probably easiest for the reader just to prove these claims; or see [1,6,9,17].) We
quote Ellis’s beautiful characterization of distal flows.
3 Theorem. (Ellis [6]) A f low (S, X) is distal if and only if its enveloping
semigroup Σ is a group, (i.e., a subgroup of the semigroup X X ).
4
The enveloping semigroup of a distal flow is called the Ellis group of the flow.
Let (G, τ ) be a compact, Hausdorff, right topological group with a subgroup S ⊂
Λ(G) that is dense in G. Then, for each s ∈ Λ(G), the left translation λs : t 7→ st,
G → G is continuous; (S, G) is a minimal distal flow with Ellis group (isomorphic
to) G. If L is a σ-closed subgroup of G, the quotient space (G/L, τ ) is Hausdorff
and (S, G/L) is a minimal distal flow, the action being given by λs : tL 7→ stL.
Conversely, every minimal distal flow (S, X) is of this form [16] (or [4]): let G ⊂ X X
be the Ellis group of the flow and take L = {t ∈ G | tx0 = x0 } for some fixed
member x0 ∈ X; then the continuous function r 7→ rx0 , G → X factors through
G/L yielding a homeomorphism π : G/L → X that is also a flow homomorphism.
L is σ-closed, since X is Hausdorff. Thus one sees that a flow automorphism of a
minimal distal flow can be thought of as having the form tL 7→ tt0 L, G/L → G/L
for some fixed t0 ∈ G. Also, a homomorphism of minimal distal flows can always
be taken to be of the form tL0 7→ tL, G/L0 → G/L, where L and L0 are σ-closed
subgroups of G with L0 ⊂ L.
Semigroup Structure and Minimal Flows. Let (S, X) be a minimal flow and
fix a minimal left ideal I in the enveloping semigroup ΣX of S. The structure
theorem for the minimal ideal of a compact semigroup (as in [4; I.3.11], for example)
now tells us that IX has a canonical decomposition as a product IX = EX × GX ,
where EX is the left zero semigroup of all idempotents in IX with product ee0 = e
for all e, e0 ∈ EX , and GX is a subgroup of IX with identity 1, say.
Now for each x ∈ X the stabilizer semigroup Ix := {T ∈ IX | T x = x} is compact and must therefore itself be of the form Ex × Gx , where Ex is a subsemigroup
of EX and Gx is a subgroup of GX . We wish to discuss how these semigroups vary
with x. Fix a member x0 ∈ X; then IX x0 = X, so that every x ∈ X can be written
in the form x = (e, g)x0 for some (e, g) ∈ IX = EX × GX .
4. Lemma. If x = (e, g)x0 , then Gx = gGx0 g −1 .
PROOF. We want to find those elements (f, h) in EX × GX for which
(e, g)x0 = x = (f, h)x = (f, h)(e, g)x0 = (f, hg)x0 .
Now, if (e0 , 1) is an idempotent in Ix0 (which therefore acts as the identity on x0 ),
we find
(e0 , g −1 hg)x0 = (e0 , g −1 )(f, hg)x0 = (e0 , g −1 )(e, g)x0 = (e0 , 1)x0 = x0 .
Thus (e0 , g −1 hg) ∈ Ex0 × Gx0 , whence g −1 Gx g ⊆ Gx0 . By symmetry we get
gGx0 g −1 ⊆ Gx , and hence g −1 Gx g = Gx0 . In particular we see from Lemma 4 that all the points of X of the form (e, 1)x0
(with e ∈ EX ) have the same group Gx0 in their stabilizer semigroups. More
generally, all the stabilizer semigroups Ix with x of the form x = (e, g)x0 for a fixed
g (but an arbitrary idempotent e) have the same group Gx = gGx0 g −1 .
The flow (S, X) is naturally a factor of the flow (S, IX ) under the flow homomorphism ψX : IX → X, T 7→ T x0 . This means that the flow (S, X) can be obtained
by partitioning the set IX into equivalence classes which are the inverse images of
points of X. We shall now determine these.
5
5. Lemma. With the notation of Lemma 4, {(f, h) ∈ IX | (f, h)x0 = x =
(e, g)x0 } is E(e,g)x0 × gGx0 = Ex × gGx0 = Ex × Gx g.
PROOF. This is easy. Indeed, (f, h)x0 = (f, hg −1 )(e, g)x0 , so the set we require is
just {h | hg −1 ∈ gGx0 g −1 }, that is, gGx0 . Let us summarise what we have said in this section. (S, X) is a minimal flow, IX
is a minimal left ideal of its enveloping semigroup ΣX and x0 ∈ X. ψX : T 7→ T x0
is a flow homomorphism of the minimal flow (S, IX ) onto (S, X). Now IX is a
−1
“rectangle” EX × GX . For x ∈ X, the inverse image ψX
(x) is a subrectangle
Ex × Hx , and each Hx is a left coset of a fixed subgroup Gx0 of GX (where “fixed”
means “determined by x0 ”). Any two points of X related by (e, 1)x1 = x2 , where
(e, 1) is an idempotent in IX have the same coset. If we change x0 , and hence the
homomorphism ψX , we change to a subgroup conjugate to the original one. Every
subgroup conjugate to Gx0 occurs, since every element of GX acts on X. The sets
Ex of idempotents we can say nothing about; examples (not included in this paper)
indicate that the situation is complex.
We shall call such a partition of IX into subrectangles Ex × Hx a flow admissible
partition if each Hx is a coset of a group Gx0 called the partition subgroup and the
quotient space is compact and Hausdorff. In the case of a compact right topological
group, there is a criterion for a subgroup to be acceptable as a partition subgroup,
namely that the subgroup be σ-closed (Theorem 1(iv)).
Subgroups of G and C ∗ -subalgebras of C(G). Let (G, τ ) be a compact, Hausdorff, right topological group. C(G) denotes the C ∗ -algebra of continuous C-valued
functions on G. The left (right) translate Lt f (Rt f ) of f ∈ C(G) by t ∈ G is defined
by
Lt f (s) = f (ts) (Rt f (s) = f (st)),
and a subset A ⊂ C(G) is called left (right) translation invariant if Lt f ∈ A (Rt f ∈
A) for all f ∈ A and t ∈ Λ(G) (t ∈ G). Notice the different treatment of ‘left’
and ‘right’ conditions here; the point, of course, is that Lt f is only known to be
continuous for t ∈ Λ(G). For a σ-closed subgroup L of G, let CL (G) be the C ∗ subalgebra of C(G) consisting of functions constant on the cosets of L. CL (G) ∼
=
C(G/L) via the map f 7→ f¯, f¯(tL) = f (t). We also define L# to consist of those
f ∈ C(G) for which f |tL is σ-continuous for all t ∈ G. Equivalently, since (G, σ) is
semitopological,
L# = {f ∈ C(G) | Lt f |L is σ-continuous for all t ∈ G}.
6. Lemma. Let L be a σ-closed subgroup of G.
(i) L# is a left translation invariant C ∗ -subalgebra of C(G), and CL (G) ⊂ L# .
(ii) L# is invariant under right translation by members of the normaliser of L
in G. In particular, L# is invariant under right translation by members of L, and
is right translation invariant if L is a normal subgroup of G.
PROOF. The proof of (i) is easy.
(ii) If u is in the normaliser of L, the mapping ρu : s 7→ su, G → G permutes the
cosets on which the restriction of f is σ-continuous: ρu (tL) = tLu = tuL. Using
the fact that (G, σ) is semitopological again, we see that Ru L# ⊂ L# . 6
For a left translation invariant C ∗ -subalgebra F of C(G), define
F[ := {t ∈ G | f (t) = f (e) for all f ∈ F} = {t ∈ G | Rt f = f for all f ∈ F}.
When F = L# as above, we write F[ = L#
[ .
7. Lemma. Let F be left translation invariant C ∗ -subalgebra of C(G).
(i) F[ is a σ-closed subgroup of G and F = CF[ (G) ∼
= C(G/F[ ). F[ is the largest
subgroup of G such that each f ∈ F is constant on each coset sF[ , s ∈ G.
(ii) The normaliser of F[ contains all members of G by which F is invariant under
right translation. In particular, F[ is a normal subgroup of G if F is right trans#
lation invariant; hence, if F = L# and F[ = L#
[ as above, then L[ is always a
normal subgroup of L, and is a normal subgroup of G if L is.
The mapping F 7→ F[ provides a 1 - 1 correspondence between left translation
invariant C ∗ -subalgebras of C(G) and σ-closed subgroups of G.
PROOF. F[ is clearly a closed subgroup of G. Consider the following obviously
equivalent statements about s and t in G:
1. sF[ = tF[ ;
2. t−1 s ∈ F[ ;
3. f (t−1 s) = f (e) for all f ∈ F;
4. f (rt−1 s) = Lr f (t−1 s) = Lr f (e) = f (r) for all f ∈ F and r ∈ Λ(G);
5. f (rt−1 s) = f (r) for all f ∈ F and r ∈ G; and
6. f (s) = f (t) for all f ∈ F.
We conclude that G/F[ is Hausdorff, which implies that F[ is σ-closed; also F is
isomorphic to a C ∗ -algebra of continuous functions separating points on G/F[ , so
F = CF[ (G) ∼
= C(G/F[ )
by the Stone-Weierstrass Theorem. Of the rest of the details, we mention only that
#
L#
[ ⊂ L, since CL (G) ⊂ L .
8. Proposition. Let L be a σ-closed subgroup of a compact, Hausdorff, right
#
topological group. Then N (L) = L#
[ and L = CN (L) (G).
PROOF. For f ∈ L# , f |L is σ-continuous, and so f |L = f¯ ◦ π for some continuous
f¯: L/N (L) → C, where π : L → L/N (L) is the quotient map (Theorem 2(iii)). This
implies that f is constant on N (L). We conclude that N (L) ⊂ L#
[ . To prove that
#
#
L[ ⊂ N (L), it suffices to show that CN (L) (G) ⊂ L . For then, if s ∈
/ N (L), there
is an f ∈ CN (L) (G) with f (s) 6= f (e). So, let f ∈ CN (L) (G) and t ∈ G. Thinking of
f as a member of C(G/N (L)), we see that on L/N (L) the function
sN (L) 7→ f (tsN (L)) = f (tN (L)sN (L))
is τ -continuous, and hence σ-continuous, since σ = τ on L/N (L) (Theorem2(ii)).
It follows that
s 7→ Lt f (s) = f (ts) (= f (tsN (L)))
#
is σ-continuous on L. Finally, the equality N (L) = L#
[ implies that L = CN (L) (G),
since N (L) is σ-closed (Lemma 5(iii)).
7
Distal Homomorphisms
Let (S, Y ) and (S, X) be minimal flows, and let π : (S, Y ) → (S, X) be a flow
homomorphism, whose lift to a (semigroup) homomorphism of ΣY onto ΣX we also
denote by π. Let IY be a minimal left ideal, so that IX := π(IY ) is a minimal left
ideal of ΣX ; then also π(EY ) = EX and we can choose GX to be π(GY ), so that
IX = π(EY ) × π(GY ). Fix y0 in Y and write x0 = π(y0 ), so flow homomorphisms
ψY : IY → Y and ψX : IX → X are determined. Then the diagram
ψY
IY
↓
Y
π
→ IX
↓
→ X
π
ψX
commutes, which means that the inverse image in IY under π : IY → IX of an equivalence class J ⊂ IX is the union of a set of equivalence classes in IY ; specifically,
if
−1
J = ψX
(x) ⊂ IX and A = π −1 (x) ⊂ Y,
then π −1 (J) = (ψX ◦ π)−1 (x) = (π ◦ ψY )−1 (x) ⊂ IY is the union of the equivalence
classes {ψY−1 (y) | y ∈ A}. These unions of equivalence classes (in IY ) from the
flow homomorphism ψY : IY → Y are just the equivalence classes from the flow
homomorphism
ψX ◦ π = π ◦ ψY : IY → X;
so they form a flow admissible partition of IY with partition subgroup π −1 (Gx0 ) ⊂
GY which contains Gy0 .
In this setting we say that π preserves idempotents if T
Ey = π −1 (Eπ(y) ) for all
y ∈ Y . We define the relativized proximal relation P (π) = {SV∩R(π) | V ∈ N },
where R(π) := {(y, y 0 ) ∈ Y 2 | π(y) = π(y 0 )} is the relation determined by π, and
N is the uniformity of Y , which may be taken to be the set of neighbourhoods of
the diagonal ∆ ⊂ Y 2 , since Y is compact (Kelley [12; Chapter 7, Corollary 30]).
In analogy to the definition of (topological) group extension, which will be given
in the next section, we say that π is a right topological group extension (rtg extension, for short) if there is a compact, Hausdorff, right topological group K acting
continuously on Y on the right so that
(i) π −1 (π(y)) = yK := {yk | k ∈ K} for all y ∈ Y, and
(ii) s(yk) = (sy)k for all s ∈ S, y ∈ Y and k ∈ K.
The continuity of the action means that each map y 7→ yk, Y → Y is continuous;
some maps k 7→ yk, K → Y may fail to be continuous. Note that we can assume,
by replacing K with a quotient group of K if necessary, that K acts freely on Y ,
i.e., yk = y for some y ∈ Y implies that k is the identity of K.
Except for the first one (idempotent preserving), these definitions appear explicitly in at least one of Ellis [6], Glasner [9], Veech [17], Auslander [1], and elsewhere.
We call a homomorphism π of minimal flows distal if it satisfies one (hence all) of
the equivalent conditions of the next theorem. This adjective describes condition
(i) quite accurately. However, in a C ∗ -algebraic setting, Ellis [6] has used it in
connection with conditions analogous to (iii) and (iv).
We wish to point out that our condition (vi) provides a stronger characterization
of distal homomorphism than (v), the similar one currently in the literature (e.g.,
8
in [6]), in that both homomorphisms θ and η are rtg extensions (rather than just
θ).
9. Theorem. The following assertions about a homomorphism π : (S, Y ) → (S, X)
of minimal flows are equivalent:
(i) {(y, y 0 ) ∈ Y 2 | π(y) = π(y 0 ) and limα sα y = limα sα y 0
for some net {sα } ⊂ S} = ∆.
(ii) {(y, y 0 ) ∈ Y 2 | π(y) = π(y 0 ) and T y = T y 0 for some T ∈ ΣY } = ∆.
(iii) P (π) = ∆.
(iv) π preserves idempotents.
(v) There exist a minimal flow (S, Z) and extensions θ : Z → X and η : Z → Y
such that θ is an rtg extension and θ = π ◦ η.
(vi) There exist a minimal flow (S, Z) and rtg extensions θ : Z → X and η : Z →
Y
such that θ = π ◦ η.
PROOF. The equivalence of (i) and (ii) is obvious, and their equivalence to (iii)
is almost as obvious. We omit these proofs. Also, (vi) clearly implies (v).
(ii) implies (iv). Let (e, g) and (e0 , g) be two points of IY = EY × GY with the
same second coordinate g, and which lie in the same equivalence class for the flow
homomorphism
ψ X ◦ π = π ◦ ψ Y : EY × G Y → X
(notation as above). Set y := ψY (e, g) = (e, g)y0 and y 0 := ψY (e0 , g) = (e0 , g)y0 , so
that π(y) = π(y 0 ). If we now take T = (e, 1), we find that
T y = (e, 1)(e, g)y0 = (e, g)y0 and T y 0 = (e, 1)(e0 , g)y0 = (e, g)y0 = T y.
From hypothesis (ii), we deduce that y = y 0 . This says that (e, g) and (e0 , g) belong
to the same equivalence class for ψY . Thus π is idempotent preserving.
(v) implies (ii). Suppose that T y = T y 0 for some y, y 0 ∈ Y with π(y) = π(y 0 )
and T ∈ ΣY . Find z, z 0 ∈ Z such that η(z) = y and η(z 0 ) = y 0 . Then θ(z) = θ(z 0 ),
since θ = π ◦ η. Let k be the member of KX , the compact right topological group
of the rtg extension θ, such that z 0 = zk, and let T1 ∈ ΣZ be such that η(T1 ) = T ,
so that
η(T1 z) = T y = T y 0 = η(T1 z 0 ) = η((T1 z)k)
and also
η(T2 T1 z) = η(T2 )η(T1 z) = η(T2 )η(T1 zk) = η((T2 T1 z)k)
for all T2 ∈ ΣZ . This means that η(z0 ) = η(z0 k) for all z0 ∈ Z, since (S, Z) is
minimal, and in particular
y = η(z) = η(zk) = η(z 0 ) = y 0 ,
as required.
(iv) implies (vi). We start with π : Y → X which preserves idempotents, and
suppose that we have a fixed element y0 ∈ Y which gives an admissible partition of
IY into sets of the form E(e,g)y0 × gGy0 (as in Lemma 5). From x0 := π(y0 ) ∈ X,
we get a second, coarser, admissible partition of IY , namely that coming from the
homomorphism π ◦ ψY = ψX ◦ π : IY → X (as mentioned at the beginning of this
9
section). Its partition group is π −1 (Gx0 ), which contains Gy0 . Its idempotent sets
are the same as those of the first partition, since π preserves idempotents.
To get the minimal flow (S, Z) that we want, we need another admissible partition of IY , one that has the same idempotent sets as the other two, but whose
partition group is contained in Gy0 and is normal in π −1 (Gx0 ) (and a fortiori in
Gy0 ).
\
H := {hGy0 h−1 | h ∈ π −1 (Gx0 )}
is the obvious choice. If we accept for the moment that the sets
E(e,g)y0 × gH, (e, g) ∈ EY × GY = IY ,
form an admissible partition with quotient space Z, then (S, Z) is a minimal flow,
being a homomorphic image of (S, IY ). The extensions θ : Z → X and η : Z → Y
are defined by
θ(E(e,g)y0 × gH) = E(e,g)y0 × gπ −1 (Gx0 ) and η(E(e,g)y0 × gH) = E(e,g)y0 × gGy0 ;
remember that π is idempotent preserving, so that E(e,g)y0 can be identified with
E(e,g)x0 . Clearly, θ = π ◦ η. We must show that θ and η are rtg extensions. Their
groups KY and KX are Gy0 /H and π −1 (Gx0 )/H, respectively. The action of KX
on Z, for example, is defined for
(z, k) = (E(e,g)y0 × gH, g 0 H) ∈ Z × KX
by
(z, k) 7→ ρk z := (E(e,g)y0 × gH)(g 0 H) = E(e,g)y0 × gHg 0 H = E(e,g)y0 × gg 0 H
(g 0 being in π −1 (Gx0 ), of which H is a normal subgroup), so θ−1 (θ(z)) = zKX .
Also, the action is effective.
Let us show that this third partition with H as partition group is an admissible
partition, i.e., that Z is Hausdorff. (The use of product spaces in the proof is much
as in Veech [17; pp. 808-809].) Recall from the paragraph following Lemma 5 that
hGy0 h−1 is an admissible partition group for IY for any h ∈ GY . Indeed, if we fix
any e ∈ Ey0 and let µh be the map T 7→ T ((e, h)y0 ) from IY to Y , then µh is a flow
homomorphism and the associated partition of IY is the one with partition group
hGy0 h−1 (and the same idempotent sets as before). For each h ∈ π −1 (Gx0 ) write
Yh := Y , and put
W :=
Y
Y
{Yh | h ∈ π −1 (Gx0 )} and µ :=
{µh | h ∈ π −1 (Gx0 )}.
Then µ : IY → W is continuous, and putting w = ((e, h)y0 )h ∈ W , we have
µ(T ) = T w = T
w if and only if T (e, h)y0 = (e, h)y0 for all h ∈ π −1 (Gx0 ), that
is, T ∈ Ey0 × h hGy0 h−1 (since E(e,h)y0 = Ey0 for all h ∈ π −1 (Gx0 ) because
π preserves idempotents). This shows that the partition does have a Hausdorff
quotient; indeed, Z is homeomorphic to
Z 0 := µ(IY ) = {(T (e, h)y0 )h ∈ W | T ∈ IY }.
10
The right action of KX = π −1 (Gx0 )/H on Z that was described above takes the
following form on Z 0 ∼
= Z:
(z 0 , k) = ((T (e, h)y0 )h , g 0 H) 7→ (T (e, hg 0 )y0 )h
(which is well defined, since H ⊂ Gy0 ).
We have yet to establish the topological properties of KY and KX . We give the
details for KX . Put z0 = Ey0 × H ∈ Z, so that θ(z0 ) = x0 , and define ψZ : IY → Z
by ψZ (T ) = T z0 . For g 0 ∈ π −1 (Gx0 ), set k := g 0 H ∈ π −1 (Gx0 )/H = KX . Since
ΣY is right topological, ρg0 : T 7→ T g 0 , IY → IY is continuous. We can then deduce
the continuity of the right action of KX on Z (i.e., the continuity of ρk : Z → Z
defined two paragraphs above) from the commutativity of the following diagram.
ψZ
IY
↓
Z
ρg 0
→
→
ρk
IY
↓
Z
ψZ
Finally, the map k 7→ z0 k is bijective, so z0 KX can be identified with KX . It
also equals θ−1 (x0 ) and so is compact. Thus KX has a natural topology in which
it is a compact right topological group. This completes the proof that (iv) implies
(vi). Almost Periodic Homomorphisms
Let Y and X be minimal flows and let Y be an extension of X, i.e., there is a flow
homomorphism π : (S, Y ) → (S, X). We give some more definitions that appear in
Ellis [6], Glasner [9], Veech [17], Auslander [1], and elsewhere. π is called a group
extension if it is an rtg extension for which the group K is a compact topological
group and the action of K on G is jointly continuous. As for the definition of an rtg
extension, K can be assumed to act freely, i.e., yk = y for some y ∈ Y implies that
k is the identity of K. π is called equicontinuous if, for every U in the uniformity
N = NY of Y , there is a V ∈ N such that (sy, sy 0 ) ∈ U for all s ∈ S whenever
(y, y 0 ) ∈ V and π(y) = π(y 0 ) (for short, S(V ∩ R(π)) = SV ∩ R(π) ⊂ U, where R(π)
is the relation determined
by π, R(π) := {(y, y 0 ) ∈ Y × Y | π(y) = π(y 0 )}). We
T
also define Q(π) := {(SV ∩ R(π))− | V ∈ N }. Clearly, Q(π) ⊃ P (π) (which was
defined in the last section). Note that (y, y 0 ) ∈ Q(π) if and only if there exist nets
{yα } and {yα0 } in Y and {sα } in S such that yα → y, yα0 → y 0 , π(yα ) = π(yα0 ) for
all α, and both nets {sα yα } and {sα yα0 } converge to the same limit (which, in the
sequel, we shall denote by y1 ).
A homomorphism π of minimal flows is called almost periodic if it satisfies one
(hence all) of conditions (i) - (v) ((i) - (ix) in setting (b)) of the next theorem.
We wish to point out that our condition (v) provides a stronger characterization of
almost periodic homomorphism than (iv), the similar one currently in the literature
(e.g., in [6]), in that both homomorphisms θ and η are group extensions (rather
than just θ). Conditions like (vi) and (vii) appear in Ellis [6]. Condition (viii) was
derived from Knapp [13]; see also [11; p. 116].
10. Theorem. (a) Let π : (S, Y ) → (S, X) be a homomorphism of minimal flows.
Then (i) - (v) of the assertions which follow are equivalent.
11
(b) Suppose that the flows (S, Y ) and (S, X) are also distal. We may assume that
Y = G/L0 and X = G/L, where (G, τ ) = S − ⊂ Y Y is the Ellis group of (S, Y ), σ
is the σ-topology of (G, τ ), L0 and L are σ-closed subgroups of G with L0 ⊂ L, and
π is given by sL0 7→ sL. Then (i) - (ix) of the following assertions are equivalent.
(i) π is equicontinuous.
(ii) Q(π) = ∆.
(iii) π is distal and there exists an x0 ∈ X such that S|π−1 (x0 ) is equicontinuous,
i.e., the set of functions
{y 7→ sy, π −1 (x0 ) → Y | s ∈ S}
is equicontinuous.
(iv) There exist a minimal flow (S, Z) and extensions θ : Z → X and η : Z → Y
such that θ is a group extension and θ = π ◦ η.
(v) There exist a minimal flow (S, Z) and group extensions θ : Z → X and
η : Z → Y such that θ = π ◦ η .
(vi) L# = CN (L) (G) ⊂ CL0 (G).
0
(vii) L#
[ = N (L) ⊂ L .
(viii) The function (s, tL0 ) 7→ stL0 , G × L/L0 → G/L0 is jointly continuous.
(ix) S|L/L0 is equicontinuous, i.e., the set of functions
{tL0 7→ stL0 , L/L0 → G/L0 | s ∈ S}
is equicontinuous.
PROOF. Certain implications among these conditions are easy to see. Trivially,
(v) implies (iv), and by Theorem 9(iii), (i) and (ii) imply (iii). In setting (b), the
equivalence of (viii) and (ix) is a basic fact about equicontinuity [4; I.6.15]). Also,
the equalities in (vi) and (vii), which do not involve L0 , were proved in Proposition
8 and are not at stake here, so that the equivalence of (vi) and (vii) is obvious. We
now complete the proof of the theorem.
(a): (i) is equivalent to (ii) (much as in [1; p. 97]). Suppose π is equicontinuous
and (y 0 , y) ∈ Q(π). Let U be a member of the uniformity N = NY of Y and
choose V ∈ N so that SV ∩ R(π) ⊂ U. Then (y 0 , y) ∈ (SV ∩ R(π))− ⊂ U− . Since
T
{U− | U ∈ N } = ∆, we conclude that Q(π) = ∆. Conversely, let U be an open
member of N with complement CU ⊂ Y × Y . Then
\
φ = ∆ ∩ CU = {(SV ∩ R(π))− | V ∈ N } ∩ CU.
By the finite intersection property, since N is directed, there is a V ∈ N such that
SV ∩ R(π) ⊂ (SV ∩ R(π))− ⊂ U,
as required.
(iii) implies (v). Since π is a distal homomorphism, we have from Theorem 9(v) a
minimal flow (S, Z) and rtg extensions θ : Z → X and η : Z → Y , along with their
compact right topological groups KX and KY , so that θ = π ◦η. The equicontinuity
of S|A for one pre-image A := π −1 (x0 ) ⊂ Y implies the equicontinuity of ΣY |A ,
and in particular of π −1 (Gx0 )|A . Now we continue on from the deliberations of the
last three paragraphs of the proof of Theorem 9. Dealing with Z 0 , as in the first
12
of these paragraphs, we see from the continuity of h|A for each h ∈ π −1 (Gx0 ) that
the function
k (= g 0 H) 7→ (T (e, hg 0 )y0 )h∈π−1 (Gx0 )
is continuous, since (g 0 H)y0 = g 0 y0 ∈ A for all g 0 ∈ KX = π −1 (Gx0 )/H; so the right
action of KX on Z 0 ∼
= Z is separately continuous, and hence jointly continuous.
Finally, consider the topology of KX to be that from the compact set z0 KX . The
equicontinuity shows that, for each k = g 0 H, the map λk : k 0 7→ kk 0 , KX → KX ,
is continuous, i.e., KX is semitopological and hence topological. Thus θ is a group
extension, and similarly one shows that η is a group extension.
(iv) implies (ii). Let K be the compact topological group of the extension θ:
K acts equicontinuously on Z on the right, θ−1 (θ(z)) = zK for all z ∈ Z, and
(sz)k = s(zk) for all s ∈ S, z ∈ Z and k ∈ K. We must show that Q(π) = ∆. So,
suppose that (y, y 0 ) ∈ Q(π), i.e., there exist nets {yα } and {yα0 } in Z and {sα } in
S, so that yα → y, yα0 → y 0 , π(yα ) = π(yα0 ) for all α, and both nets {sα yα } and
{sα yα0 } converge to the same limit y1 , say. We must show that y = y 0 . For each
α, pick zα ∈ Z so that η(zα ) = yα ; since π(yα ) = π(yα0 ) and θ = π ◦ η, pick also
kα ∈ K so that η(zα kα ) = yα0 . Without loss of generality, we may asssume that
zα → z0 and sα zα → z1 in Z and kα → k in K, so (sα zα )kα = sα (zα kα ) → z1 k. It
follows that
η(z1 ) = lim η(sα zα ) = lim sα yα = y1 = lim sα yα0 = lim η(sα zα kα ) = η(z1 k),
α
α
α
α
and hence, as in the proof that (v) implies (ii) in Theorem 9, that η(z) = η(zk) for
all z ∈ Z. But {η(zα )} converges both to y and to η(z0 ), and {η(zα kα )} converges
both to y 0 and to η(z0 k). Thus y = y 0 , since η(z0 ) = η(z0 k), which completes the
proof that (iv) implies (ii).
(b): (ii) implies (vii). Following the proof of Proposition 14.18 in Ellis [6] rather
closely, we let s ∈ N (L) (⊂ L) and must show that s ∈ L0 . So, let U ∈ U, the set of
open τ -neighbourhoods in G of the identity. Then s ∈ (U −1 U ∩L)(U −1 U ∩L), N (L)
being the intersection over U ∈ U of such sets. Since (U −1 U ∩ L)−1 = U −1 U ∩ L,
we conclude that the intersection
(U −1 U ∩ L)s ∩ (U −1 U ∩ L) = U −1 U s ∩ U −1 U ∩ L
is non-void. Let rU be a member of this intersection; hence there exist sU , tU ∈ U
−1
such that sU rU ∈ U s and tU rU ∈ U . Thus (tU s−1
U )sU = tU and (tU sU )sU rU =
tU rU are in U ; since G is right topological and S is dense in G, we can replace
tU s−1
U with a member uU ∈ S and still have uU sU , uU sU rU ∈ U . Now, using U as
a directed set, we have net convergence in G as follows:
sU → e, sU rU → s, uU sU → e, and uU sU rU → e.
Passing to G/L0 , we have
sU L0 → L0 , sU rU L0 → sL0 , uU (sU L0 ) → L0 , and uU (sU rU L0 ) → L0 .
Since each rU ∈ L,
π(sU rU L0 ) = sU rU L = sU L = π(sU L0 ),
13
and we conclude that (sL0 , L0 ) ∈ Q(π). But Q(π) = ∆, so sL0 = L0 and s ∈ L0 , as
required.
(vii) implies (viii). We have
(sN (L), tN (L)) 7→ stN (L), G/N (L) × L/N (L) → G/N (L)
is jointly continuous (Theorem 2(ii)). It follows that
(s, tN (L)) 7→ stN (L), G × L/N (L) → G/N (L)
is jointly continuous, and then that
(s, tL0 ) 7→ stL0 , G × L/L0 → G/L0
is jointly continuous.
(ix) implies (iii). All homomorphisms of distal flows preserve idempotents (since
there is only one idempotent), so π is distal and satisfies the first requirement of
(iii). The explicit hypothesis of (ix) states that π satisfies the second requirement
of (iii).
Remarks. 1. The theorem shows that, in setting (b), N (L) is the smallest among
σ-closed subgroups L1 of L such that
(s, tL1 ) 7→ stL1 , G × L/L1 → G/L1
is jointly continuous.
2. Our efforts in the presentation of a proof of Theorem 10 have been directed
towards completeness, brevity, and also ease of exposition. Some specific implications not needed for our presentation have attractive direct proofs for which all the
ingredients have been assembled. We mention two of these.
(vii) implies (iv). The required group extension is
θ : s0 N (L) 7→ s0 L, G/N (L) → G/L.
Its compact topological group is L/N (L), whose right action on G/N (L)
(tN (L), rN (L)) 7→ tN (L)rN (L) = trN (L), G/N (L) × L/N (L) → G/N (L)
is jointly continuous by Theorem 2(ii) and commutes with the action of S on
G/N (L). The extension η : G/N (L) → G/L0 is given by η(s0 N (L)) = s0 L0 , so
that θ = π ◦ η.
(viii) implies (vii). The proof that (ii) implies (vii) requires little modification
to yield this implication. Start in the same way with s ∈ N (L), and use the joint
continuity and distality in place of Q(π) = ∆ to conclude that s ∈ L0 .
3. Since all homomorphisms of minimal distal flows are distal, the distal hypothesis does not appear explicitly in (ix), as it does in (iii). As one ought to
expect, the distality cannot be dropped from (iii) in setting (a) (that is, not without changing to an inequivalent condition). Here is an example of a homomorphism
of minimal flows, where the group acts equicontinuously on every pre-image, yet
the homomorphism is not distal.
14
Example. For the homomorphism π : (S, Y ) → (S, X), S = Z and Y = {±1} × T.
We define a topology τ for Y by saying a typical basic neighbourhood of (1, eia )
and (−1, eib ), where a < b, is a set of the form
B := {(1, eia ), (−1, eib )} ∪ {(, eiν ) | = ±1, a < ν < b}.
(The compact Hausdorff space (Y, τ ) has been used in [3,4,15] and elsewhere.) The
action of Z on Y is given by
(n, (, w)) 7→ (, ein w);
(Z, Y ) is a minimal flow. For X, we just identify each pair of points (±1, w), w ∈ T,
except when w ∈ D := {ein | n ∈ Z}. Thus X = (T \ D) ∪ {(±1, w) | w ∈ D}.
Here are typical basic neighbourhoods
for (1, eia ) and (−1, eib ), where eia , eib ∈ D with a < b :
{(1, eia ), (−1, eib )} ∪ {(±1, eiν ) | eiν ∈ D, a < ν < b} ∪ {eiν ∈ T \ D | a < ν < b};
and for eia ∈ T \ D :
{eiν ∈ T \ D | a − δ < ν < a + δ} ∪ {(±1, eiν ) | eiν ∈ D, a − δ < ν < a + δ},
where δ > 0. (There may be some ambiguity in the notation here. In an attempt
to clarify it, we point out that, when we say eiν ∈ D with a < ν < b, we mean that
there exist n, k ∈ Z such that ν = n + 2πk and a < n + 2πk < b.) The action of Z
on X is given by
n : w 7→ ein w, w ∈ T \ D, and n : (±1, w) 7→ (±1, ein w), w ∈ D.
Note that ein w ∈ D if and only if w ∈ D. Then (Z, X) is also a minimal flow.
Define a flow homomorphism π : Y → X by π(±1, w) = w for w ∈ T \ D, and
π(±1, w) = (±1, w) for w ∈ D. Then each pre-image consists of one or two points,
so of course Z restricted to it is equicontinuous. One sees the lack of distality for
π merely from the fact that pre-images can have different cardinalities, and also
by noting that, for any w ∈ T \ D ⊂ X, if nj (mod 2π) & 0 as j → ∞, then nj
maps the two points of the pre-image (±1, w) ∈ Y to (±1, einj w), which converge
to (1, w) in Y as j → ∞.
It is interesting to note that, for this example, Y is just (isomorphic in an obvious
way to) IX , the unique minimal left ideal in the enveloping semigroup ΣX of X,
and π is just ψX : IX → X, the notation being that of the paragraph before Lemma
5. In this setting, IY is also just IX , and contains two idempotents. To see that π
does not preserve idempotents, observe that E(±1,w) is a singleton for any w ∈ T,
while, for any w ∈ T \ D, π −1 (Eπ(±1,w) ) contains both idempotents of IY ∼
= IX .
Structure Theorem and Haar Measure
for Compact Right Topological Groups
Let (G, τ ) be a compact, Hausdorff, right topological group. A strong normal
system for G is a family {Lξ | ξ ≤ ξ0 } of subgroups of G indexed by the set of
ordinals less than or equal to an ordinal ξ0 and satisfying
15
(i) each Lξ is a σ-closed normal subgroup of G, L0 = G and Lξ0 = {e};
(ii) for ξ < ξ0 , Lξ ⊃ Lξ+1 and the function
(sLξ+1 , tLξ+1 ) 7→ stLξ+1 , G/Lξ+1 × Lξ /Lξ+1 → G/Lξ+1
is continuous for the quotient topologies from
T τ ; and
(iii) for each limit ordinal ξ ≤ ξ0 , Lξ = η<ξ Lη .
It follows that each G/Lξ is a group and, in the quotient topology of τ , is compact,
Hausdorff, and right topological with Λ(G)/Lξ ⊂ Λ(G/Lξ ). (See 1.3.7-8 in [4], for
example.) Hence, each Lξ /Lξ+1 is a compact, Hausdorff, topological group.
11 Theorem. Let (G1 , τ ) be compact, Hausdorff, right topological group with dense
topological centre Λ(G1 ). Let G be a closed subgroup of G1 . Then G has a strong
normal system of subgroups.
PROOF. As in [15], it suffices to deal with the case where G1 = G. Then, still
as in [15], the result is proved by transfinite induction:
L0 = G, Lξ+1 := N (Lξ ) (= L#
[ ), and Lξ =
\
Lν for limit ordinals ξ.
ν<ξ
A cardinality argument shows that we must have Lξ+1 = Lξ sooner or later. In
[15] (following [16]), the countability condition is used to show that Lξ = {e}
when Lξ+1 = Lξ . Here, we use Furstenberg’s structure theorem for distal flows as
established without countability restrictions by Ellis [7]. Suppose Lξ 6= {e}. Then
ψ : s 7→ sLξ , G → G/Lξ , is a homomorphism of minimal distal flows (for which we
take the acting group S to be Λ(G)). Proposition 1.10 in [7] tells us that ψ factors
through a flow (S, Z) such that the homomorphism π : (S, Z) → (S, G/Lξ ) is almost
periodic and not 1 - 1. Since (S, Z) is a flow homomorphic image of (S, G), we may
take Z to be of the form G/L00 for some σ-closed subgroup L00 of G. By Theorem
10(vii), we have N (Lξ ) ⊂ L00 6⊆ Lξ , as required.
A probability measure µ on G is called left invariant if it is invariant under the
continuous left translations, i.e., if µ(sB) = µ(B) for all s ∈ Λ(G) and all Borel sets
B ⊂ G. (It follows from a remark at the end of [2] that a compact right topological
group need not have a unique left invariant probability measure.) We call µ right
invariant and a Haar measure if it is invariant under all right translations, i.e., if
µ(Bs) = µ(B) for all s ∈ G and all Borel sets B ⊂ G. We showed in [15] that, for
any compact right topological group that has a strong normal system (and also for
one that does not), the conclusions of the next theorem hold.
12 Theorem. Let G be as in Theorem 11. Then Haar measure exists on G. It is
the unique right invariant probability measure on G; it is also left invariant.
Bibliography.
1. J. Auslander, Minimal Flows and Their Extensions, North - Holland, Amsterdam, 1988.
2. L. Auslander and F. Hahn, Real functions coming from flows on compact spaces and concepts
of almost periodicity, Trans. Amer. Math. Soc. 106 (1963), 415–426.
3. J.W. Baker and P. Milnes, The ideal structure of the Stone - Čech compactification of a group,
Math. Proc. Camb. Phil. Soc. 82 (1977), 401–409.
16
4. J.F. Berglund, H.D. Junghenn and P. Milnes, Analysis on Semigroups: Function Spaces,
Compactifactions, Representations, Wiley, New York, 1989.
5. R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119–126.
6. R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.
7. R. Ellis, The Furstenberg structure theorem, Pacific J. Math. 76 (1978), 345–349.
8. H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515.
9. S. Glasner, Proximal Flows, Lecture Notes in Mathematics 517, Springer - Verlag, New York,
1976.
10. W.H. Gottschalk and G.A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloquium
Publication vol. 36, Providence, R.I., 1955.
11. H.D. Junghenn and R.D. Pandian, Existence and structure theorems for semigroup compactifications, Semigroup Forum 28 (1984), 109–122.
12. J.L. Kelly, General Topology, Van Nostrand, Princeton, N.J., 1955.
13. A.W. Knapp, Distal functions on groups, Trans. Amer. Math. Soc. 128 (1967), 1–40.
14. J.D. Lawson and A. Lisan, Transitive dynamical systems: a semigroup approach, (preprint).
15. P. Milnes and J. Pym, Haar measure for compact right topological groups, Proc. Amer. Math.
Soc., (to appear).
16. I. Namioka, Right topological groups, distal flows and a fixed point theorem, Math. Systems
Theory 6 (1972), 193–209.
17. W.A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), 775–830.
London, Ontario N6A 5B7, Canada and Sheffield S3 7RH, England
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