APPENDIX C
Weakly almost periodic functions
We develop here the basic theory of weakly almost periodic functions. This provides
us with a canonical mechanism for averaging the matrix coefficients of a unitary representation of a group G in a G-invariant way. The definition of weak mixing for p.m.p.
actions of general groups is based on this averaging, which can also be used to characterize ergodicity via an abstract ergodic theorem (Theorem 1.19).
In Section C.3 the space WAP(G) of weakly almost periodic functions is introduced
and is shown to be a unital C∗ -algebra. The latter requires heavy use of the EberleinSmulian theorem, which gives a sequential characterization of relative weak compactness
for subsets of a Banach space. We prove Eberlein-Smulian theorem in Section C.1 following the standard argument of Whitley [147]. The Eberlein-Smulian theorem is also
an ingredient in our proof in Section C.2 of Grothendieck’s double limit criterion [66],
which is needed, by way of Proposition C.6, to establish Theorem C.13. In Section C.4
we prove the Ryll-Nardzewski fixed point theorem [123] following the geometric argument in [64], which is based on [4] (for other approaches see [55, 107]). This is then used
in Section C.5 to show the existence of a unique two-sided invariant mean on WAP(G).
Other references for the material of this appendix are [24, 57].
C.1. The Eberlein-Smulian theorem
As one can see using the finite intersection property, a subset A of a metric space is
relatively compact if and only if each sequence in A has a convergent subsequence. This
characterization of relative compactness fails more generally in Hausdorff spaces, but it
does hold for the weak topology in arbitrary Banach spaces, which is the main content
of the Eberlein-Smulian theorem, recorded as Theorem C.2 below. We will apply the
Eberlein-Smulian theorem repeatedly in Section C.3, where it is critical to be able to test
for the relative weak compactness of orbits in ℓ∞ (G) in a way that is not only sequential
but also involves only sequences that reside in the orbit itself and not merely in its closure,
which is impossible to get a direct handle on.
When the Banach space is separable, the following lemma reduces the problem to
the case of metric spaces, for which we have the desired sequential characterization of
223
224
C. WEAKLY ALMOST PERIODIC FUNCTIONS
relative compactness as noted above. We will use this lemma more generally to establish
the implication (i)⇒(ii) in Theorem C.2.
L EMMA C.1. Let E be a separable Banach space and A a weakly compact subset of
E. Then the weak topology on A is metrizable.
P ROOF. Take a countable dense sequence {xn } in the unit ball of E. By the HahnBanach theorem, for every n there is a ϕn in the unit ball of E ∗ such that |ϕn (x)| = kxk.
Then
∞
X
d(x, y) =
2−n |ϕn (x − y)|
n=1
defines a metric on E. Since A is weakly compact, its images in C under the functionals in E ∗ are all bounded, and so A is bounded by the uniform boundedness principle.
Consequently the identity map from A with the weak topology to A with the topology
induced by d is continuous and hence is a homeomorphism by the compactness of the
domain.
The strategy for establishing the difficult implication (iii)⇒(i) below is to view the
given Banach space E as a subspace of its second dual E ∗∗ and use the weak∗ compactness
of the bounded subsets of E ∗∗ to produce convergent subnets. The limits of these subnets
are then shown to lie in E itself, where they are in fact weak limits, since the weak∗
topology on E ∗∗ restricts to the weak topology on E.
T HEOREM C.2 (Eberlein-Smulian). For a subset A of a Banach space E, the following are equivalent:
(i) A is relatively weakly compact,
(ii) every sequence in A has a weakly convergent subsequence,
(iii) every sequence in A has a weak cluster point.
P ROOF. (i)⇒(ii). Let {xn } be a sequence in A. Set E0 = span{xn : n ∈ N}. Since
E0 is norm closed it is also weakly closed by the Hahn-Banach theorem, and so by (i) the
set A ∩ E0 is relatively weakly compact in the weak topology of the separable Banach
space E0 . Since compactness and sequential compactness are the same for subsets of a
metric space, it follows by Lemma C.1 that {xn } has a subsequence that converges weakly
in E0 and hence also weakly in E, yielding (ii).
(ii)⇒(iii). Trivial.
(iii)⇒(i). Condition (iii) implies that the images of A in C under the functionals in
∗
E are all bounded and so A is bounded by the uniform boundedness principle. It follows
wk∗
of the image of A under
by the Banach-Alaoglu theorem that the weak∗ closure ρ(A)
∗∗
∗
the canonical embedding ρ : E → E is weak compact. To obtain (i) it thus suffices to
C.2. GROTHENDIECK’S DOUBLE LIMIT CRITERION
225
wk∗
⊆ ρ(E), as this implies that every net in ρ(A) has a weakly convergent
show that ρ(A)
subnet inside the Banach space ρ(E).
wk∗
So let v ∈ ρ(A) . For n ∈ N we will recursively choose xn ∈ A and finite sets Ωn
in the unit sphere of E ∗ such that
S
(a) |(v − ρ(xn ))(ϕ)| < 1/n for all ϕ ∈ nk=1 Ωk , and
(b) kwk/2 ≤ maxϕ∈Ωn |w(ϕ)| ≤ kwk for all w in the linear span of {v} ∪ {ρ(xk ) :
1 ≤ k ≤ n − 1}.
We begin by choosing any x1 ∈ A and any ϕ in the unit sphere of E ∗ such that |v(ϕ)| ≥
kvk/2, and setting Ω1 = {ϕ}. Assuming now that for some n > 1 we have chosen
x1 , . . . , xn−1 and Ω1 , . . . , Ωn−1 , we can find a finite Ωn satisfying (b) because the linear
span in that condition is finite-dimensional and hence has totally bounded unit sphere, and
S
then an xn satisfying (a) because nk=1 Ωk is finite.
By (iii) the sequence {xn } has a weak cluster point y. To complete the proof we will
verify that v = ρ(y). It follows from (b) that if w is an element in the closed linear span
S
of {v} ∪ {ρ(xn ) : n ∈ N} which satisfies w(ϕ) = 0 for all ϕ ∈ ∞
k=1 Ωk , then w = 0.
Sn
But v − ρ(y) is such an element, since for all n and ϕ ∈ k=1 Ωk we have
|(v − ρ(y))(ϕ)| ≤ |(v − ρ(xn ))(ϕ)| + |ϕ(xn − y)| <
1
+ |ϕ(xn − y)|.
n
We thus obtain (i).
C.2. Grothendieck’s double limit criterion
In order to prove Proposition C.6, which will in turn be used to prove Theorem C.13,
we will need Theorem C.4 below, which is Grothendieck’s double limit characterization
of relative weak compactness for bounded subsets of C(X), where X is a compact Hausdorff space. It is important here that X not be assumed to be metrizable, as we will take
it to be the Stone-Cech compactification of the group G in the proof of Proposition C.6.
L EMMA C.3. Let X be a compact Hausdorff space and B a bounded subset of C(X).
Then B is weakly compact if and only if it is pointwise compact.
P ROOF. The forward direction is immediate from the fact that pointwise topology
is weaker than the weak topology. To establish the reverse direction, suppose that B is
pointwise compact. By Theorem C.2 it suffices to show that B is weakly sequentially
compact.
Let {fn } be a sequence in B, and let us first prove that it has a pointwise convergent
subsequence. Note that the inclusion A ֒→ C(X) of the unital C∗ -subalgebra of C(X)
generated by the functions fn corresponds functorially via the Gelfand representation to
the quotient map X → Y where Y is the spectrum of A. Since A is separable, the compact
226
C. WEAKLY ALMOST PERIODIC FUNCTIONS
space Y is metrizable, and so we may assume that X is metrizable and hence separable by
replacing it with Y . Fix a countable dense subset X0 of X. By pointwise compactness,
the sequence {fn } has a pointwise cluster point f ∈ B. By a diagonal procedure we
can then construct a subsequence {fnj } such that fnj (x) → f (x) for all x ∈ X0 . As
the functions in B are continuous, the topology on B of convergence at each point in X0
is Hausdorff and hence must coincide with the pointwise topology, since B is pointwise
compact. Therefore fnj → f pointwise.
R
R
Finally, by the Lebesgue dominated convergence theorem we have fnj dµ → f dµ
for all Radon measures µ on X, which are the same as the bounded linear functionals on
C(X) by the Riesz representation theorem. We conclude that B is weakly sequentially
compact, finishing the proof.
T HEOREM C.4. Let X be a compact Hausdorff space and X0 a dense subset of X.
Let A be a norm-bounded subset of C(X). Then A is relatively weakly compact if and
only if
lim lim fn (xm ) = lim lim fn (xm )
n
m
m
n
for all sequences {fn } in A and {xm } in X0 for which all of these limits exist.
P ROOF. Suppose first that A is relatively weakly compact. Let {fn } be a sequence in
A and {xm } a sequence in X0 such that all of the limits in the expressions limn limm fn (xm )
and limm limn fn (xm ) exist. Then {fn } has a weak cluster point f . Taking a cluster point
x of {xm }, we then have
lim lim fn (xm ) = lim f (xm ) = f (x) = lim fn (x) = lim lim fn (xm ).
m
n
m
n
n
m
Conversely, suppose that the double limit criterion in the theorem statement holds and
let us show that A is relatively weakly compact. By Lemma C.3 it suffices to prove that
A is relatively pointwise compact. This amounts to showing that the pointwise closure C
of A in the space of all functions on X contains only continuous functions. Suppose that
this is not the case. Then we can find an f ∈ C, an x ∈ X, and a δ > 0 such that every
neighbourhood of x contains an x′ ∈ X0 satisfying |f (x) − f (x′ )| ≥ δ. We recursively
construct sequences {fn } in A and {xn } in X0 by first picking any f1 ∈ A and then for
every n ≥ 1, having chosen f1 , . . . , fn and x1 , . . . xn−1 , taking an xn ∈ X0 such that
(i) |fm (x) − fm (xn )| < 1/n for all m = 1, . . . , n, and
(ii) |f (x) − f (xn )| ≥ δ,
and then an fn+1 ∈ A such that
(i) |fn+1 (xm ) − f (xm )| < 1/n for all m = 1, . . . , n, and
(ii) |fn+1 (x) − f (x)| < 1/n.
C.3. THE C∗ -ALGEBRA OF WEAKLY ALMOST PERIODIC FUNCTIONS
227
Then limn limm fn (xm ) = limn fn (x) = f (x), while limn fn (xm ) = f (xm ) for each m.
Since A is norm bounded there is a compact subset of C which contains the image of every
function in C, and so we can take a subsequence {xmj } such that {f (xmj )} converges to
some z ∈ C for which |z − f (x)| ≥ δ. Then
lim lim fn (xmj ) = f (x) 6= z = lim lim fn (xmj ),
n
j
j
n
in contradiction to the double limit criterion. Therefore every function in C must be
continuous, as desired.
C.3. The C∗ -algebra of weakly almost periodic functions
For f ∈ ℓ∞ (G) and s ∈ G we write sf and f s for the functions in ℓ∞ (G) given by
t 7→ f (s−1 t) and t 7→ f (ts−1 ), respectively. Then (s, f ) 7→ sf and (s, f ) 7→ f s define the
left action and right action of G on ℓ∞ (G). The use of s−1 in the definitions means that
the axioms of left and right actions are satisfied in a way that matches our terminology,
which reflects the order of multiplication.
A function f ∈ ℓ∞ (G) is said to be weakly almost periodic if the weak closure of its
left G-orbit Gf = {sf : s ∈ G} is weakly compact. We write WAP(G) for the set of all
weakly almost periodic functions in ℓ∞ (G).
P ROPOSITION C.5. WAP(G) is a unital sub-C ∗ -algebra of ℓ∞ (G) which is both left
and right invariant.
P ROOF. It is immediate from the definition that WAP(G) is left invariant, contains
the constant functions, and is closed under conjugation. Since the map h 7→ ht on ℓ∞ (G)
is bounded and linear and hence weakly continuous, for all f ∈ ℓ∞ (G) and t ∈ G we
have Gf t = Gf t, so that WAP(G) is right invariant. We note furthermore that WAP(G)
is a linear subspace, for if λ ∈ C and f, g ∈ WAP(G) then the map (k, h) 7→ λk + h from
wk
wk
the product Gf × Gg of weak orbit closures to E is weakly continuous and thus its
image, which contains the orbit of λf + g, is weakly compact.
Next we argue that WAP(G) is norm closed. By Theorem C.2 it suffices to show,
given a function f in the norm closure of WAP(G) and a sequence {sk } in G, that {sk f }
has a weakly convergent subsequence. Take a sequence {fn } in WAP(G) which converges in norm to f . Starting with {sk } and using Theorem C.2, we recursively extract
subsequences and then take the diagonal so as to construct a sequence {tk } in G such
that for each n the sequence {tk fn }k converges weakly to some hn ∈ ℓ∞ (G). Then the
sequence {hn } is Cauchy and hence converges to some h ∈ ℓ∞ (G). Now given a ϕ in
the unit ball of ℓ∞ (G)∗ and an ε > 0 we can take an n such that kf − fn k < ε/3 and
khn − hk < ε/3 and then a k such that |ϕ(tk fn − hn )| < ε/3, in which case
|ϕ(tk f − h)| ≤ |ϕ(tk (f − fn ))| + |ϕ(tk fn − hn )| + |ϕ(hn − h)| < ε,
228
C. WEAKLY ALMOST PERIODIC FUNCTIONS
which shows that tk f → h weakly as k → ∞. Thus WAP(G) is closed.
Finally, to show that WAP(G) is closed under multiplication, let f, g ∈ WAP(G).
Using the Gelfand representation we will now regard ℓ∞ (G) as C(βG), in particular
when speaking of pointwise convergence. By Theorem C.2 the weak closures of Gf and
Gg are weakly sequentially compact. Thus, by considering point masses on βG, given a
sequence in G we can find a subsequence {sn } such that {sn f } converges pointwise, and
then a subsequence {snk } of {sn } such that {snk g} converges pointwise. Then {snk (f g)}
convergesR pointwise to some
R h ∈ C(βG). By the Lebesgue dominated convergence
theorem, snk (f g) dµ → h dµ for all finite Radon measures µ on βG, and since the
space of such measures corresponds to C(X)∗ by the Riesz representation theorem this
means that snk (f g) → h weakly. It follows by Theorem C.2 that the weak closure of
G(f g) is weakly compact, so that f g ∈ WAP(G).
Our next proposition will be useful in the proof of Theorem C.12. It says that being
weakly almost periodic, as we have defined it in terms of the left action, is equivalent to
being “right” weakly almost periodic.
P ROPOSITION C.6. Let f ∈ ℓ∞ (G). Then f ∈ WAP(G) if and only if the weak
closure of its right G-orbit f G = {f s : s ∈ G} is weakly compact.
P ROOF. We view ℓ∞ (G) as C(βG) via the canonical identification, where βG is the
Stone-Cech compactification of G. If f ∈ C(βG) and {sn } and {tm } are sequences in
−1
G, then (sn f )(tm ) = (f t−1
m )(sn ) for all n and m, and so the double limit criterion in
Theorem C.4, as applied using the dense set G ⊆ βG, holds for Gf if and only if it holds
for f G. It thus follows by Theorem C.4 that Gf is relatively weakly compact if and only
if f G is relatively weakly compact, yielding the result.
Next we turn to the question of what type of functions lie in WAP(G) besides the
constants.
P ROPOSITION C.7. Functions in C0 (G) are weakly almost periodic.
P ROOF. Let f ∈ C0 (G). If U is an open neighbourhood of 0 in ℓ∞ (G) for the weak
topology, then it must contain sf for all but finitely many s ∈ G, as a simple approximation argument shows using the canonical identification C0 (G)∗ ∼
= ℓ1 (G). It follows
that every open cover of Gf ∪ {0} for the weak topology admits a finite subcover, so that
Gf ∪ {0} is weakly compact. Hence f is weakly almost periodic.
For a unitary representation π : G → B(H) and ξ, ζ ∈ H we write fξ,ζ for the
function on G defined by s 7→ hπ(s)ξ, ζi. Such a function is called a matrix coefficient.
When ξ = ζ it is of positive type, which for a function f : G → C means that for for all
C.4. THE RYLL-NARDZEWSKI FIXED POINT THEOREM
229
s1 , . . . , sn ∈ G and λ1 , . . . , λn ∈ C one has
n X
n
X
λi λj f (s−1
j si ) ≥ 0,
i=1 j=1
i.e., the n × n matrix (f (s−1
j si ))i,j is positive. In fact every function of positive type
arises as a matrix coefficient of the form fξ,ξ under some unitary representation: Given
a function f : G → C of positive type, we define hδs , δt i = f (t−1 s) on characteristic
functions of singleton subsets of G and then linearly extend to define a possibly degenerate
inner product on the space V of finitely supported functions G → C, with positivity
following from the positive type condition. Factoring out from V the vectors of zero
length and then completing yields a Hilbert space H, and we get a unitary representation
of G on H by extending the action G y V given by sg(t) = g(s−1 t) for g ∈ V and
s, t ∈ G. Then f is realized as fδe ,δe .
P ROPOSITION C.8. Matrix coefficients are weakly almost periodic.
P ROOF. Let fξ,ζ be a matrix coefficient associated to some unitary representation π :
G → B(H). Define the bounded conjugate linear map Tξ : H → ℓ∞ (G) by Tξ η(s) =
hπ(s)ξ, ηi for s ∈ G. Since the set of vectors in H of norm at most kζk is weakly
compact, the set π(G)ζ is relatively weakly compact. Since ϕ 7→ ϕ ◦ Tξ defines a map
from ℓ∞ (G)∗ to H∗ , it follows that the image of π(G)ζ under Tξ is relatively weakly
compact. But this image is precisely the left orbit of fξ,ζ .
P ROPOSITION C.9. Let G y (X, µ) be a p.m.p. action, and let A and B be measurable subsets of X. Then the function
s 7→ µ(sA ∩ B) − µ(A)µ(B)
on G is weakly almost periodic.
P ROOF. Using the Koopman representation for the action, the function in the proposition statement can be expressed as the matrix coefficient fξ,ζ where ξ = 1A − µ(A)1X
and ζ = 1B − µ(B)1X . By Proposition C.8 this function is weakly almost periodic. C.4. The Ryll-Nardzewski fixed point theorem
For a subset A of a Banach space we write co(A) for its convex hull. It is a standard
fact that the following sets coincide:
(i) the intersection of all closed convex sets containing A,
(ii) the norm closure of co(A),
(iii) the weak closure of co(A).
230
C. WEAKLY ALMOST PERIODIC FUNCTIONS
We denote this common set by co(A) and refer to it as the closed convex hull of A.
The equivalence of (ii) and (iii) is Mazur’s theorem and is a consequence of the HahnBanach theorem. It permits to say closed convex set with the understanding that it is meant
equivalently in the norm and weak topologies.
The concepts and arguments in this section all apply more generally to locally convex
topological vector spaces. The following lemma is moreover true if we replace the weak
topology on a Banach space with any locally convex topology on a vector space. However,
since our application in Section C.5 involves Banach spaces, we will frame everything in
that context.
L EMMA C.10. Let E be a Banach space and let A be a weakly closed subset of E
such that co(A) is weakly compact. Then ext co(A) ⊆ A.
P ROOF. Let y ∈ ext co(A). It suffices to show, given a finite set Ω ⊆ E ∗ and ε > 0,
that y ∈ A + U where U = {x ∈ E : |ϕ(x)| < ε for all ϕ ∈ Ω}. Set V = {x ∈ E :
|ϕ(x)| < ε/2 for all ϕ ∈ Ω}. Since co(A) is weakly compact we can find x1 , . . . , xn ∈ E
S
such that A ⊆ ni=1 (xi + V ). Setting Ai = A ∩ (xi + V ) we have
[
n
co(A) ⊆ co
co(Ai )
i=1
since the operation of taking closed convex hulls preserves inclusions and the set on the
P
right side is clearly closed. Thus we can express y as a finite convex combination i λi yi
where yi ∈ co(Ai ). Since each of the sets co(Ai ) is included in co(A), we must have
y = yi for some i by extremeness, so that y ∈ xi + V ⊆ A + U .
The following lemma says that we can always shave off a subset of small diameter
from a norm separable weakly compact convex set so that the remaining part is still closed
and convex.
L EMMA C.11. Let E be a Banach space and let K be a nonempty weakly compact
convex subset of E which is norm separable. Let δ > 0. Then there is a closed convex
subset L of K such that L 6= K and diam(K\L) ≤ δ.
P ROOF. Set B = {x ∈ E : kxk ≤ δ/4} and write D for the weak closure of ext K.
Note that D is nonempty since K is the closed convex hull of its extreme points by the
Krein-Milman theorem. Since K is norm separable, D has a countable norm dense subset
D0 . Then D is equal to the union of the weakly closed sets D ∩ (B + y) for y ∈ D0 , and
so by the Baire category theorem there exists a particular y ∈ D0 such that D ∩ (B + y)
has nonempty interior in the weak topology. Thus we can find a weakly open set U ⊆ E
such that D ∩ U is nonempty and contained in B + y.
C.4. THE RYLL-NARDZEWSKI FIXED POINT THEOREM
231
Write K1 for the closed convex hull of D\U and K2 for the closed convex hull of
D ∩ U . Define
L = {λx1 + (1 − λ)x2 : x1 ∈ K1 , x2 ∈ K2 , and δ/(4 diam(K)) ≤ λ ≤ 1},
which is a convex and weakly compact subset of K. We claim that L 6= K. Suppose
to the contrary that L = K. Then every extreme point of K must lie in K1 since λ is
not permitted to be zero in the definition of L. Consequently ext K ⊆ ext K1 . Since
ext K1 ⊆ D\U by Lemma C.10, it follows that D = ext K ⊆ D\U , contradicting the
nonemptiness of D ∩ U . Thus L 6= K.
Now let x ∈ K\L. Since K is the closed convex hull of its extreme points by the
Krein-Milman theorem and co(K1 ∪ K2 ) is closed by the weak compactness of K1 and
K2 , we have K = co(K1 ∪ K2 ). We can thus write x = λx1 + (1 − λ)x2 for some
x1 ∈ K1 , x2 ∈ K2 , and 0 ≤ λ < δ/(4 diam(K)), in which case
δ
kx − x2 k = |λ|kx1 − x2 k < .
4
Since diam(K2 ) ≤ δ/2 by our choice of B, we conclude by the triangle inequality that
diam(K\L) ≤ δ.
Given a metric space (X, d), we say that an action G y X is distal if for all distinct
x, y ∈ X there is an δ > 0 such that d(sx, sy) ≥ δ for all s ∈ G. When X is compact
this notion is independent of the choice of compatible metric. Note however that in the
following theorem the metric topology will typically not be compact.
T HEOREM C.12 (Ryll-Nardzewski). Let E be a Banach space. Let K be a nonempty
weakly compact convex subset of E, G a group, and G y K a norm distal affine action
which is continuous for either the norm or weak topology. Then there is a G-fixed point
in K.
P ROOF. We may assume that G is countable, for if the result holds in this case then
the collection of fixed point sets of the countable subsets of G has the finite intersection
property and hence has nonempty intersection by compactness. We may also assume that
E is separable, as we can take any x ∈ K and replace E with span(Gx) and K by the
closed convex hull of Gx, which is weakly compact in span(Gx) since the weak topology
of a closed subspace of E is the same as the relativization of the weak topology of E by
the Hahn-Banach theorem.
Since the collection of nonempty G-invariant closed convex subsets of K is closed
under the operation of intersecting a subcollection which is totally ordered by inclusion,
by Zorn’s lemma there is a nonempty G-invariant closed convex set K0 ⊆ K which is
minimal with respect to these properties. To complete the proof we will argue that K0 is
a singleton. Suppose to the contrary that there exist distinct x, y ∈ K0 . Then by distality
232
C. WEAKLY ALMOST PERIODIC FUNCTIONS
there is a δ > 0 such that inf s∈G ksx − syk > δ. By Lemma C.11 there is a closed convex
set L ⊆ K0 such that L 6= K0 and diam(K0 \L) ≤ δ/2. Since the closed convex hull of
the orbit of (x + y)/2 is G-invariant by our continuity hypothesis on the action, it cannot
be included in L because of our choice of K0 , and so we can find an s ∈ G such that
s((x + y)/2) ∈ K0 \L. Then sx and sy cannot both lie in the convex set L. By relabeling
if necessary we may assume that sx ∈ K0 \L. Then
ksx − syk = 2ksx − s((x + y)/2)k ≤ δ,
a contradiction. We conclude that K0 is a singleton.
C.5. The two-sided invariant mean on weakly almost periodic functions
A mean on a unital sub-C∗ -algebra A of ℓ∞ (G) is a unital positive linear functional
on A, i.e., a linear functional σ : A → C such that σ(1) = 1 and σ(f ) ≥ 0 whenever
f ≥ 0. On general unital C∗ -algebras such functionals are called states and they are
automatically bounded with norm one. If A is invariant under the left action of G, then
the mean σ is said to be left invariant if σ(sf ) = σ(f ) for all f ∈ A and s ∈ G. If A is
invariant under the right action of G, then σ is said to be right invariant if σ(f s) = σ(f )
for all f ∈ A and s ∈ G.
T HEOREM C.13. On WAP(G) there exist a unique left invariant mean and a unique
right invariant mean, and they coincide.
P ROOF. Let f ∈ WAP(G). By the Krein-Smulian theorem [?, Thm. V.6.4][24, Thm.
12.1], the closed convex hull of a weakly compact subset of a Banach space is itself
wk
weakly compact, and thus since Gf is weakly compact so is co(Gf ). Thus by Theowk
rem C.12 the left action of G on co(Gf ) has a fixed point λ. As the set f G is weakly
compact by Proposition C.6, the same argument shows that there is a fixed point ρ for
the right action of G on co(f G). Note that λ and ρ must be constant functions. Take a
net {Lα } of convex combinations of left translation operators f 7→ sf and a net {Rβ }
of convex combinations of right translation operators f 7→ f s such that Lα f → λ and
Rβ f → ρ. Given an ε > 0, take particular α and β such that kLα f − λk < ε/2 and
kRβ f − ρk < ε/2. Since Lβ Rα f = Rα Lβ f due to the fact that the left and right actions
commute, we get
kλ − ρk = kRβ λ − Lα ρk ≤ kRβ (λ − Lα f )k + kLα (Rβ f − ρ)k < ε.
It follows that λ = ρ and that λ and ρ are unique. Identifying C with the space of
constant functions on G, we thereby obtain a map m : WAP(G) → C which is evidently
homogeneous and two-sided invariant, with m(1) = 1 and m(f ) ≥ 0 whenever f ≥ 0.
C.5. THE TWO-SIDED INVARIANT MEAN ON WEAKLY ALMOST PERIODIC FUNCTIONS
233
We will next show, given f, g ∈ WAP(G), that m(f + g) = m(f ) + m(g), which
will establish the linearity of m. Let ε > 0. Take a convex combination L of left translation operators such that kLf − m(f )k < ε/2. Since co(G(Lg)) ⊆ co(Gg) and m(Lg)
and m(g) are the unique constant functions in co(G(Lg)) and co(Gg), respectively, we
must have m(Lg) = m(g). We can thus find a convex combination L′ of left translation
operators such that kL′ (Lg) − m(g)k < ε/2, in which case
kL′ L(f + g) − (m(f ) + m(g))k ≤ kL′ (Lf − m(f ))k + kL′ Lg − m(g)k < ε.
Since m(f + g) is the unique fixed point in co(G(f + g)) for the left action, we conclude
that m(f + g) = m(f ) + m(g).
To see that m is the unique left invariant mean on WAP(G), let m̃ be another left
invariant mean. By the first paragraph, for every f ∈ WAP(G) and ε > 0 we can find a
convex combination L of left translation operators such that kLf − m(f )k < ε, and since
m̃(Lf ) = m̃(f ) we have
|m̃(f ) − m(f )| = |m̃(Lf − m(f ))| ≤ kLf − m(f )k < ε.
Therefore m̃ = m. The same argument shows that m is also the unique right invariant
mean on WAP(G).
The invariant mean in Theorem C.13 will always be written m.
Following C ∗ -algebraic convention, we say that a function f ∈ ℓ∞ (G) is positive
if f = |f |. We round out this section by giving a characterization of those positive
functions f ∈ WAP(G) satisfying m(f ) = 0 that tells us on how large a subset of G such
a function must almost vanish. First we isolate the following fact contained in the proof
of Theorem C.13.
L EMMA C.14. For every f ∈ WAP(G) and ε > 0 there is a finite convex combination
g of left translates of f such that kg − m(f )k < ε.
D EFINITION C.15. A set K ⊆ G is syndetic if there exists a finite set F ⊆ G such
that F K = G. The set K is thickly syndetic if for every nonempty finite set F ⊆ G the
T
set s∈F sK is syndetic.
P ROPOSITION C.16. A positive function f ∈ WAP(G) satisfies m(f ) = 0 if and only
if for every ε > 0 the set f −1 [0, ε) is thickly syndetic.
P ROOF. Suppose first that m(f ) = 0. Let ε > 0 and let F be a nonempty finite subset
P
of G. Set g = s∈F sf . Noting that m(g) = 0, by Lemma C.14 there is a finite convex
P
combination h = t∈E λt (tg) of left translates of g such that khk < ε. Thus for every
P
u ∈ G there is a t ∈ E such that s∈F f (s−1 t−1 u) < ε and hence f (s−1 t−1 u) < ε for
234
C. WEAKLY ALMOST PERIODIC FUNCTIONS
T
S
every s ∈ F , which shows that the set K = f −1 [0, ε) satisfies G = t∈E t( s∈F sK) and
thus is thickly syndetic.
Conversely, suppose that the set f −1 [0, ε) is thickly syndetic for every ε > 0. By
P
Lemma C.14, given an ε > 0 we can find a finite convex combination g = s∈F λs (sf )
T
of left translates of f such that kg − m(f )k < ε/2. By assumption, s∈F sf −1 [0, ε/2) is
syndetic and hence contains at least one element t. Then f (s−1 t) < ε/2 for all s ∈ F , so
that g(t) < ε/2 and hence
m(f ) ≤ | m(f ) − g(t)| + |g(t)| < ε.
We conclude that m(f ) = 0.
We observe finally that for amenable G the mean m can be expressed concretely as a
limit of averages across any Følner sequence.
P ROPOSITION C.17. Suppose that G is amenable, and let {Fn } be a Følner sequence
for G. Then for every f ∈ WAP(G) one has
1 X
f (s).
m(f ) = lim
n→∞ |Fn |
s∈F
n
P ROOF. For any 1 ≤ n1 < n2 < . . . the sequence of unital positive linear functionals
P
f 7→ limn→∞ |Fni |−1 s∈Fn f (s) on WAP(G) has a weak∗ cluster point by compactness,
i
and every such cluster point is left G-invariant and hence coincides with m by uniqueness.
It follows that for every f ∈ WAP(G) the limit in the proposition statement exists and is
equal to m(f ).