averaging sequences and modulated ergodic theorems for weakly almost periodic group representations

A V E R A G I N G S E Q U E N C E S A N D M O D U L A T E D E R G O D I C

T H E O R E M S F O R W E A K L Y A L M O S T P E R I O D I C G R O U P

R E P R E S E N T A T I O N S

By

MICHAEL LIN* AND ARKADY TEMPELMAN

Abstract. Let T be a weakly almost periodic (WAP) representation of a locally compact a-compact group G by linear operators in a Banach space X, and let M = M(T) he its ergodic projection onto the space of fixed points (i.e., M x is the unique fixed point in the closed convex hull of the orbit of x). A sequence of probabilities {#~} is said to average T [weakly] if f T(t)xdtzn converges [weakly] to M(T)x for each x E X. We call {#~} [weakly] unitarily averaging i f it averages [weakly] every unitary representation in a Hilbert space, and [weakly] WAPR-averaging i f it averages [weakly] every WAP representation. We investigate some of the relationships of these notions, and connect them with properties of the regular representation (by translations) in the space WAP(G).

Theorem. Let {#,~} be a WAPR-averaging sequence, and f measurable on G with lim supn f I f V d ~ < oofor some v > 1. Then f f ( t )T( t )x dlzn (t) converges strongly for every weakly almost periodic representation T(t) in a Banach space, i f and only i f f f(t)g(t) dt~(t) converges for every g E AP( G).

When {tL,~} is only unitarily averaging, a similar modulated ergodic theorem is obtained (by completely different methods) for unitary representations, and the limit is identified. A sequence {#~} of probabilities on G is called pointwise averaging i f for any action {0~ : t E G} in a a-finite measure space (f~, ~-, m) and for every ~ E L1 (m) the sequence fG ~(O~w) dp~(t) converges for m-almost all w E f~ to a ftmcfion

which is invariant with respect to {0t}. We show that a pointwise averaging sequence averages every almost periodic representation, obtain a pointwise modulated ergodic theorem for f a Besicovitch function, and prove a generalized Wiener Wintner Theorem.

1 A v e r a g e a b l e r e p r e s e n t a t i o n s a n d a v e r a g i n g s e q u e n c e s

Let G be a locally compact or-compact group with unit element e and right Haar

measure A. Consider a continuous bounded representation T(t) of G in a Banach space X (we always assume T(e) = I). If # is a regular probability measure on

G we say that the Bochner integral Tux = f T(t)x d#(t) is an average of x with

*Research partially supported by the Israel Science Foundation.

237 JOURNAL D'ANALYSE MATHEMATIQUE, Vol. 77 (1999)

238 M. LIN AND A. TEMPELMAN

respect to T; any T-invariant limit of averages y = lira f T(t)x dl~ is called a mean value of x with respect to T. The representation T is said to be averageable, i f

every element x in X possesses a unique mean value. Since any average o f x is

in ~ { T ( t ) x : t e G}, the closed convex hull of the orbit o f x, averageability o f

T means that for every x E X there is a unique fixed point in ~{T( t ) x : t E G}, denoted by m(x).

Let us recall the following facts about averageable representations (e.g., [T-4],

w

1. The mean value M(x) is T-invariant: T(t)M(x) -- M(x), Vt E T, by

definition.

2. M(T(t)x) = M(x), Vt E T, by equality o f the orbits.

3. IIM(x)[[ <_ (suptec [IT(t)ll)llxil.

4. Ergodic decomposition: i f T is an averageable representation, then

(1.1) X = {y : T(t)y = y Vt E G} @ closed span{x - T(t)x : x e X, t e G}.

5. Denote by I the first summand in (1.1) - the subspace o f all T-invariant

elements, and by D the second summand. Then for any x, its projection on I along

D coincides with M(x). Thus, M(x) = 0 i f and only i f x E D.

R e m a r k The subspaces I and D are defined for every representation. I f G is

amenable we always have I n D = {0}, but in general we may have I C/D r {0}.

D e f i n i t i o n . I f T is an averageable representation, we say that a sequence o f

probabilities {#,~} is a T-averaging sequence, i f M(x) = lira f T(t)x dtz~ for each

x E X .

Def in i t i on , Let T be a continuous bounded representation in a Banach space

X, and let ~- be some topology in X dominated by the norm topology. We say that

the sequence of measures {/~} is T T-ergodic i f f (T(t) - T(ts) )x dish(t) -~ 0, x e X, s E G; i f ~- is the norm topology we say that {/~,~} is T-ergodic, This means

that the sequence f T(t)d~(t) is a right-ergodic sequence for {T(t) : t 6 G} [K,

p. 75]. Note that the definition of ergodicity applies also to non-averageable

representations.

The following proposition is a generalization o f Theorem 3.1.1 in [T-4].

Proposition 1.1. Let T be an averageable representation. A sequence {#~}

is ~- T-ergodic i f and only i f it is "r T-averaging, i.e.,

f T(t) du.(t) = e X.

AVERAGING SEQUENCES AND MODULATED ERGODIC THEOREMS 239

Proof . Let x = M(x) + z where z C D. It is obvious that f T ( t ) M ( x ) d#~(t) = M(x).

The relation r-lim~-~oof T(t)zd#~(t) = 0, for every z = y - T(s)y, is

equivalent to ~- T-ergodicity of {#n} (since T(t)z = T(t)y - T(ts)y). Using the

usual approximation technique we easily prove that "r T-ergodicity is equivalent to

~--lim~,~oo fT ( t ) zd#~( t ) = 0 for each z E D. This implies our statement.

R e m a r k s . 1. For an averageable representation, T-averageability of {#,~} can

be checked without knowledge of the mean value operator M.

2. Let {pn} be a sequence of probabilities, and T an averageable representation,

x E X. I f f T ( t ) x d#~(t) converges weakly in X to a fixed point, then the limit is

M(x), since M(x) is the unique fixed point in ~{T( t ) x } (which is weakly closed).

A sequence of measurable sets {A~} is a (righO Folner sequence if, for every

t E G ,

tim A(AnAA'~t) -- O.

If {A,,} is a Felner sequence, then the sequence {#n} with

1 - 1 A .

is a norm ergodic sequence for any representation (and hence norm-averaging for

any averageable representation), since I[#n - #,~ * 5tll --,~--.oo 0 for every t c G.

Existence of Folner sequences is one of the characterizations of amenable locally

compact (y-compact groups. Using FMner sequences we obtain that for G amenable,

every bounded continuous representation for which the ergodic decomposition (1.1)

holds is averageable. Ergodic sequences for the representation by translations in

LI(G, ;~) were studied in [L-W]. Existence of such sequences implies amenability

of G. It was shown in [L-W] that such sequences are norm ergodic for every

bounded continuous representation, and we shall call them just ergodic sequences.

Below we give some important classes of averageable representations and of

corresponding averaging sequences.

I. Representa t ions in reflexive B a n a c h spaces. Every bounded contin-

uous representation of G in a reflexive Banach space is averageable (see WAP

representations below).

IL Trans lat ions in the spaces AP(G) and WAP(G). A bounded continuous

function f E C(G) is (weakly) almostperiodie if the set {St * f (s) = f(st) : t E G} is (weakly) conditionally compact in C(G). The set of (weakly) almost periodic


functions is denoted by (W)AP(G). AP(G) and WAP(G) are closed subspaces of the space UCBe(G) ofteft uniformly continuous bounded functions [E].

For a regular probability # on G, the convolution operator defined by # �9

f(t) = f f(ts) d#(s) is a Markov operator, having ,~ as a ~r-finite invariant measure.

The right translation operators (Ss * f)(t) = f(ts) yield representations of G in Lp(G, )~), 1 _< p _< 0% and in C(G), UCBl(G), AP(G) o r WAP(G). Except for

C(G) and Lo~(G), the representation is strongly continuous in each of the other

function spaces, and the convolution operator is an average of the representation.

Von-Neumann [N] proved that the translation representation 5 in the space AP(G) is averageable (existence of the mean value for almost periodic functions). The averageability of 5 in WAP(G) was proved by Ry[1-Nardzewski [RN-1].

Thus, 5-ergodic sequences in (W)AP(G) are averaging sequences, and it is

natural to consider three types of 5-averaging sequences in these function spaces:

(i) Norm 6-averaging sequences (called AP- and WAP-averaging sequences):

II#~ * ( f - 6t * f)[[ ~ 0. Under this condition we have the "uniform" mean value

theorem:

!ira sup[ f f ( s t ) d ~ ( t ) - M(f)[ = 0 Vf 6 (W)AP(G). n c ~ s E G J

(ii) Weakly AP- and WAP-averaging sequences: #,~ �9 ( f - 6t * f ) --~ 0 in the

weak topology. (iii) Pointwise AP- and WAP-averaging sequences: #~ �9 (f - 5t * f) --, 0

pointw/se.

I II . W e a k l y a lmos t pe r iod ic r ep re sen t a t i ons . A bounded continuous

operator representation of G in a Banaeh space X is called (weakly) almost periodic if Or(x) = {T(t)x : t E G} is (weakly) conditionally compact for all x E X.

Thus, any bounded continuous representation in a reflexive space is weakly

almost periodic. Clearly, {y : Or(y) is (weakly) conditionally compact} is a

closed invariant subspace, and the restriction of the representation to this subspace

is (weakly) almost periodic. Ryll-Nardzewski [RN-1 ] proved that if T(t) is a weakly almost periodic repre-

sentation of G, then it is averageable. Sequences which are (weakly) T-averaging for every weakly almost periodic representation T will be called (weakly) WAPR- averaging sequences. Every (weakly) WAPR-averaging sequence is obviously (weakly) WAP-averaging. By [L-W] and Proposition 1.1, any ergodic sequence

{~,~} is WAPR-averaging, i.e., for any WAP representation T we have

lim fa T(t)x d#n(t) = M(x).


We now exhibit WAPR-averaging sequences which are not ergodic. Let # be an

adapted probability on G (i.e., the closed subgroup generated by its support is G),

and define #,~ = ~ }--]~-1 #J. For every weakly almost periodic representation T(t) 1 n j we have that T u x = g ~ j = l T~x converges strongly for any x E X to the ergodic

projection on the fixed points of T u (see [T-4], Ch. 3, Theorem 5.1), which are

exactly the common fixed points [RN-2]. Thus, {#~} is WAPR-averaging, but not

ergodic if G is not amenable. An example in [G1] (or [D]) shows G amenable with

an adapted # with/2 (defined by #(A) = #(A -1) ) having a non-constant bounded

harmonic function, which means that {#,~} is not ergodic (e.g., see [L-W]). Thus

we have non-ergodic WAPR-averaging sequences also on amenable groups.

Let us note that weakly WAP-averaging sequences need not be norm WAP-

averaging: let G be non-compact with WAP(G) = Const ~ Co(G) (e.g., G is

non-compact simple analytic with finite center [V]), and let #~ = 6~,~ for 8,~ ~ oo.

Then {tz,~} is weakly WAP-averaging, but not WAP-averaging.

T h e o r e m 1.2. The following conditions are equivalent for a sequence o f

probabilities {#n}:

(i) {#~} is" weakly WAP-averaging. (ii) {/z~} is pointwise WAP-averaging. (iii) {/z~} is weakly WAPR-averaging.

Proof . Clearly (iii)=, (i), and (i)=~(ii). (ii)~(iii): Let T be a weakly almost periodic representation. Fix z E X. Since

T , x C -~6{T(t)x : t E G}, the sequence {T,, x} is weakly conditionally compact.

Fix now also x* E X*. Then f(t) := (z*,T(t)z) is in WAP(G) [Bu, p. 36]. By (ii)

f T(t)xdlz~(t)) = f f(t)dl~,~(t) -~ M f . <x*,T, x) (x*,

Since { T u x } is weakly sequentially compact, the above shows that it converges

weakly to an element Ax E X , with (x*, Ax} = M ( (x* , T(. )x) ) for any x* E X *. For

x = ( I - T(s))y and x* 6 X* we have f(t) : ( 9 - 6~ .g)(t) where 9(t) = (x*, T(t)y), so M(f ) = 0, and thus for such x we have Ax = 0. Hence {#~} is weakly

T-averaging.

The existence of the invariant mean for WAP(G) [RN-1],[RN-2] allows us to

obtain the following ]acobs-DeLeeuw-Glicksberg decomposition [DL-G,

Theorem 4.11], which will be used below.

T h e o r e m 1.3. Let T(t) be a WAP continuous representation of a locally compact group G in a Banach space X. Let Xo be the set o f x E X such that 0 is in


the weak closure of Or(x), and let Xp be the closed subspace of X generated by the finite-dimensional T-invariant subspaces of X. Then Xo and Xp are closed T-invariant subspaees with 32o N Xp = {0}, and X = Xp | Xo.

In the decomposition of WAP(G) (with respect to translations), WAP(G)o is

characterized by f E WAP(G)o if and only if M(If[ ) = 0 (e.g., [Bu]). Since

the mean M is positive, M(lf] ~) < [I]IIL-1M(lfl) = 0 for f e WAP(G)o and 1 < r < c~. The proof that f(t) = {x*, T(t)x) is in WAP(G) for x E X, x* E X*

[Bu, p.36] yields that f E WAP(G)o for x E 32o. Hence

(1.2) M(I@*,T(.)x}[ ~)=0 f o r z E X o , x * E X * , l < _ r < ~ .

IV. Uni tary representations. A sequence of probabilities {/~} on G is

called (weakly) unitarily averaging if for every continuous unitary representation

T in a Hilbert space H we have T u x ~ Px (weakly) for every x E H, where P

is the orthogonal projection on the common fixed points. It is easy to see that this

projection corresponds to the ergodic decomposition (1.1). The Cesaro averages

of t~ adapted are WAPR-averaging, hence unitarily averaging. This answers a

question in [M-Pa] about existence of unitarily ergodic sequences. In separable groups, Losert and Rindler [Lo-Ri] had earlier constructed sequences {tj} such

that #n = g ~ j = l tj is unitarily averaging. We denote by B~(G) the closed subspace of C(G) spanned by the coefficient

functions of all continuous unitary representations, i.e., B,, (G) is generated by the

fimctions of the form f(t) = (T(t)x, y) for some continuous unitary representation

T(-) in a Hilbert space H, x, y E H. The space Bz(G), introduced in [M-Pa], is the closed subspace of B~(G) spanned by the coefficient functions of all continuous irreducible unitary representations. We then have AP(G) C Bx(G) C B~(G) C WAP(G). By the Gelfand--Raikov Theorem, for every positive definite function f

there is a unitary representation U in a Hilbert space H and a vector x E H such that

f(t) = {U(t)x, x) ([H-Ro, (32.3)], so the polarization identity yields that the space B~(G) is generated by the positive definite functions. Hence Bu(G) is a closed self-

adjoint algebra [H-Ro, (32.10)]. It was proved in [C] that AP(G) | Co (G) c B~(G), and equality does not hold for G nilpotent non-compact. It follows that for G

Abelian non-compact~ B~(G) (which is the closure ofthe Fourier-Stieltjes algebra) is different from B~(G), which in this case equals AP(G). For G discrete or

nilpotent, WAP(G)/B~(G) contains a subspace isometric to ~ [C].

T h e o r e m 1.4. The following conditions are equivalent for a sequence {~,~}

of probabilities on a locally compact a-compact group G: (i) {~n } is unitarily averaging.


(ii) I1#~ * ( f - 6 t * f)Hc~ ~ Oforevery t E G, f E Bu(G).

(iii) * f - M( f ) I I~ ~ 0 v f e Bu(G).

(iv) H/2~ �9 #n * f - M(/)[I~ --* o Vf e B~(G).

(v) [zn �9 #,~ �9 f - M ( f ) --+ Opointwise V f E B~(G).

(vi) fz~ �9 #,~ �9 ( f - at * f ) --+ O pointwise gt E G, f E B~(G).

P r o o f . ( i )~(i i ) : We assume that {#n} is unitarily averaging. Let T(t) be a

continuous unitary representation in a Hilbert space H, and f ( t ) = {T(t)x, y} for

some x, y E H. It is enough to prove (ii) for such functions f . Then

<Tu,~x,T*(t)y).

Since (6, * f ) (Q = f ( t s ) = <T(ts)x, y), we have

= f = (Tu.T(s)x ,T*( t )y} . # n , ( 6 s , f ) ( t )

Hence for t E G we have

I#~ * ( f - 68 * S)(t)l = I ( T ~ ( I - T(s))x ,T*(t)Y)l <- [IT~.( I - T(s))xll [lYll -* 0

since T~,~ converges strongly to the projection on the fixed points corresponding to

(1.1). (ii)=~(iii): By the decomposition (1.1) applied to the representation by transla-

tions in B~(G), we have f - M ( f ) E clm U~ev {g - 6~g : g E B,~(G)}. Since M ( f )

is a constant function, (iii) follows from (ii).

The implications ( i i i ) ~ ( i v ) ~ ( v ) ~ ( v i ) are clear.

(vi)=~(i): Let T(t) be a unitary representation in H, and let x = ( I - T(s ) )y w k h

y E H. Put f ( t ) = (T(t)x, x) and 9(t) = (T(t)y, x). Then f = g - 58 * g, so by (vi),

#,~ * #n * f --* 0 pointwise. Now

f (#~ , f ) ( s -~) d#,~(s) = fz~ * #n * f(e) --~ O. a l

R e m a r k . The equivalence o f (i) and (iii) follows also from L e m m a 4.1.1 o f

[T-4].

T h e o r e m 1.5. The fol lowing conditions are equivalent f o r a sequence {#~}

o f probabilities on a locally compact a-compact group G:

(i) {#,~} is weakly unitarily averaging.

(ii) ~ �9 ( f - 6t * f ) --* 0 weakly for every t E G, f E B~(G).

(iii) #~ �9 ( f - 6t * f ) ~ O pointwise for every f E B~(G), t E G.

(iv) f fd#~ ~ M ( f ) f o r a n y f E B~(G).


Proof . Clearly (ii)=~(iii) and (iii)r by Proposition 1.1.

(i)=~(ii): We assume that {#,~} is weakly unitarily averaging. Let T( t ) be a

continuous unitary representation in a Hilbert space H, and f~,v(t) = (T( t )x , y} for

x, y C H. Let r E B,,(G)*. For y fixed, r is linear in x, so there is a zy c H

with

r = (x, zy} Vx e H.

Fix now x, y C H, and let f = f~,y. It is enough to prove (ii) for such functions f .

It was shown in (i)~(ii) of Theorem t .4 that for t e G we have

I~,~ * ( f - 5s * f ) ( t ) = (T ,~ ( I - T ( s ) ) x , T * ( t ) y ) = ( T ( t ) T , ~ ( I - T ( s ) ) x , y ) .

We therefore have, using (i),

r * ( f - 5s * f ) ) = ( T ~ ( I - T(s ) )x, zy) ---* O.

(iv)=~(i) is similar to (ii)~(iii) in Theorem t.2: Let T( t ) be a unitary represen-

tation in H, and let x = ( I - T ( s ) ) y with y E H. For z E H, put f ( t ) = (r(t)x, z) and g(t) = (T( t )y , z}. Then f = g - 6s * g, so M ( f ) = 0, and by (iv), f f d # ~ --, O.

N o w

(Tu z , z} = ( f T ( t ) x d ~ n ( t ) , z ) = f (T(t)z,z)d.n(t) = f f ( t ) d p n ( t ) --~ 0.

R e m a r k s . 1. If we put in Theorems 1.4 and 1.5 the space Bz(G) instead of

B~,(G), the same proofs yield the equivalence with the convergence T,~ --, P for

every irreducible unitary representation.

2. When G is separable, the integral decomposition allows us to obtain (weak)

unitary ergodicity from the corresponding (weak) convergence T,,, ~ P for every

irreducible unitary representation (see [T-4]). Thus, it was shown in [M-Pal that

in separable groups, {/~,~} is weakly unitarily averaging if and only if

f fd#,~ ---* M ( f ) V f e Bz(G). (*)

In [%4, Theorem 4.8.2] it was shown that in the separable case, {#,~} is unitarily

averaging if and only if

(**) [1#~ * f - M(f)[Io~ --+ o Vf E Bz(G).

In both cases, the separability assumption has been recently removed by Lau and

Losert [La-Lo]. Conditions (iv)-(vi) of Theorem 1.4 are new even for the separable

c a s e .


3. Necessary and sufficient conditions for {#~} to be weakly unitarily averaging

are given in [L-W-2].

V. A l m o s t pe r iod ic r ep re sen t a t i ons . Almost periodic (AP) representations

and AP-averaging sequences were defined earlier. We have the following result.

T h e o r e m 1.6. The following are equivalent for a sequence of probabilities

(i) {#~} is pointwise AP-averaging. (ii) {#~} is weakly AP-averaging. (iii) {#n} is AP-averaging. (iv) {#~} is T-averaging for everyunitary representation T in a finite-dimensional

Hilbert space. (v) {#~} is T-averaging for every almost periodic representation T.

Proof. Clearly (v)~(iv) and (ii i)~(ii)~(i). (iv)~(iii) follows from the Approximation Theorem - the closed subspace

generated by the coefficient functions of finite-dimensional unitary representations

is AP(G). Assume (i). Let T be an almost periodic representation in X. The proof of

(ii)~(iii) of Theorem 1.2 shows that for any x E X we have T , x ---, M(x) weakly.

Since Or(x) is conditionally compact, its closed convex hull is compact, so the

sequence { T , x } converges strongly to M(x). Hence (v) holds.

Proposition 1.7. Every weakly unitarily averaging sequence {#~} is AP-

averaging.

Proof. By the Approximation Theorem, AP(G) C BI(G), so we have #~ * f --* M(f) weakly for any f E AP(G). We now apply the previous Theorem.

R e m a r k s . 1. The converse of the previous proposition is false: Let G be

a non-compact semi-simple analytic group with finite center. By [V], AP(G) contains only constants, and WAP(G) = AP(G) ~ C0(G); hence by [C] also B~(G) = AP(G) �9 Co(G). Let #~ = 5e. It is AP-averaging, but not weakly

unitarily averaging (no convergence to 0 on Co (G)). 2. For G Abelian, an AP-averaging sequence {#~} satisfies limn f 7(t)d#~(t) =

0 for every non-zero character, so it is (norm) unitarity averaging by [B1-Ei].

2 M o d u l a t e d ergod ic t h e o r e m s

Let {#n} be an averaging sequence for a class of averageable representations

of G. By definition, f T(t)x d~(t) converges strongly for every T in the class.


We are interested here in "modulating" the representation - i.e., find functions

f(t) on G such that f f(t)T(t)x d~, converges strongly for every T in the class.

This problem was first studied, for Abelian groups and unitary representations,

in IT-3] (together with pointwise convergence for actions), using modulation by

Besicovitch functions (with respect to "strongly" ergodic sequences). Besicovitch

sequences on R or N (with respect to Cesaro averages) were independently used (a

year later) in [RN-3], where the current term "weighted" ergodic theorems (rather

than "modulated") originated. Ergodic sequences and WAP representations on

groups were considered in [L-O]. Since these papers use ergodic sequences, their

results are valid only in amenable groups. The sequences we shall use will be

assumed at most WAPR-averaging, so the results will apply also in non-amenable

groups. For our norm results we can consider a wider class of"weights".

For {#.} an AP-averaging sequence of probabilities on G and 1 < r < c~ we

define (see also IT-3], [L-O])

W,,{u~} = {f measurable: limsupn__.oo f I f r d/zn < oo}.

f r For 1 < r < oo, l[ [Ir,{t,~} liInsup,~oo f Ill r d#~ < oo defines a seminorrn on

~,{~,~}. We will call this seminorm "the r-seminorm relative to {#,~}". Clearly all

bounded measurable functions are in Wr,{u,d, and W,,{u.~} c Wq,{~} for 1 < q < r.

For notational convenience we define Woo,{~} = Loo,{~,,~}.

P r o p o s i t i o n 2.1. Let {/z~} be an AP-averaging sequence of probabilities on G, and let f be in WI,{~,~}. Then the following are equivalent:

(i) f f(t)g(t) d#~(t) converges for every g E AP(G). (ii) For every bounded continuous representation T(t) in a finite-dimensional

Banach space, f f(t)T(t)x d#,~(t) converges strongly. (iii) For every continuous irreducible unitary representation U(t) in a finite-

dimensional Hilbert space, f f( t)U(t)x d#,~ (t) converges weakly.

Proof . (i)~(ii): Let T(t) be a continuous bounded representation in a finite-

dimensional Banach space X, and fix x E X. For x* E X*, the function g(t) = (x*, T(t)x) is almost periodic (since bounded representations in finite-dimensional

spaces are almost periodic). The given convergence yields that f f(t)T(t)x d#,~ (t) converges weakly, and hence strongly since X is finite-dimensional.

(ii)=~(iii) is obvious.

(iii)=~(i) follows from the Approximation Theorem - AP(G) is the closed

subspace generated by the coefficient functions of continuous finite-dimensional

irreducible unitary representations.


R e m a r k s . 1. Let f �9 (W)AP(G). Since (W)AP(G) is an algebra, f satisfies

condition (i) of the previous proposition for any weakly (W)AP-averaging sequence

{#,~}, with limit M(fg) (independently of the averaging sequence). Hence the limit obtained in Proposition 2.1 depends in this case only on the representation (and on

the functions f and g), but not on the averaging sequence.

2. The limits in (iii) are used for defining general "Fourier coefficient operators"

for f - see Section 4.

T h e o r e m 2.2. Let {#n} be a WAPR-averaging sequence of probabilities on G, and let f be in the II " Hl,{u~} -el~ (in WI,{~}) ofUr>l Wr,{~}. Then the following are equivalent:

(i) For every weakly almost periodic representation T(t) in a Banach space, f f(t)T(t)z d#n(t) converges weakly.

(ii) For every weakly almost periodic representation T(t) in a Banach space, f f(t)T(t)z dUN (t) converges strongly.

(iii) For every irreducible unitary representation U(t) in a finite-dimensional Hilbert space, f f(~)U(t)x d#~(t) converges strongly.

(iv) f f(t)9(t) d#,~(t) converges for every 9 �9 AP(G).

Proof . (iv) implies (iii) by the previous proposition. (i)~(iv): We look at

the representation by right translations in AP(G), which is almost periodic. By

(i), f f(t)~, �9 g d#~(t) converges weakly, hence pointw/se. The convergence at the

identity e is (iv). We now have to prove only (iii)=>(ii). Fix a WAP representation T(t) in

a Banach space X, and let X = Xp ~ X0 be its Jaeobs--DeLeeuw--Glicksberg

decomposition. Let X' be a finite-dimensional T-invariant subspace. Since a

bounded representation in a finite-dimensional space is equivalent to a unitary one (e.g., [Ly, p. 84]), it is equivalent to a direct sum of finitely many irreducible unitary

representations. Hence (iii) implies that f f(t)T(t)z d/z~ (t) converges strongly for

x E X t. Since Xp is the closed linear manifold generated by the finite-dimensional T-invariant subspaces, we have the required convergence on a linear manifold

dense in Xp. Since

s.p II f <- .up IIT ;t)ll .:p f IS(')l '..(') < oo, t6G

we have the convergence on all of Xp. h remains to prove the convergence on Xo,

and since it is an invariant subspace, we may and do assume X = Xo. By a change

of norm, we may and do assume that IIT(Q II = I for every t (so inver~bility implies

that each T(t) is an isome~y).

248 M. L/N AND A. TEMPELMAN

Let K be the closed unit bali of X*, which is compact in the weak-* topology.

Define R(t) on C(K) by R(t)r = r for r E C(K) and x* E K. Then

R(t)r E C(K) since T*(t) is a continuous map of K for fixed t E G. We then have

R(ts)r = r = r

= = R ( t ) n ( s ) r

Let C = {r E C(K) : t --* R(t)r is continuous}. It is clearly a closed subspace

of C(K) containing the constants, and self-adjoint. For each x E X, the function

r := (x*, x} is continuous on K, and by the continuity o f the representation T

it is in C. Hence C separates the points of K. Since R(t) is multiplicative, C is an

algebra, so equals C(K). Hence the representation R is continuous.

We now show that the representation R is WAR Let A be the set of functions

r E C(K) with {R(t)r : t E G} conditionally weakly compact . . ,4 is obviously

linear and self-adjoint, and contains the constants. A separates points, since it

contains all functions ex defined for x E X by ex(x*) = (x*, x}. Let r r E A and {tj} an infinite sequence. By the Eberlein-Shmulian Theorem, there exists a

subsequence {tj, } with R(tj,)r converging weakly to a continuous function r

(i = 1, 2). Hence R(tj~)r converge pointwise to r and by Lebesgue's Theorem,

R( t~)( r162 converges weakly to the continuous fi.mcfion r162 Hence A is an

algebra. Finally, let en E A with lien - r --* 0. Let {t~} be a sequence. Using the diagonal process, we can find a subsequence {sj} c {t~}, such that for each n

there is r E C(K) with limj R(sj)r = r pointwise. Since R(sj) are isometrics,

{r is a Cauchy sequence, so converges to some r ~ C(K). Now

R ( s j ) r - r = R ( s j ) ( r - r + R ( s j ) r - + ( r - r

shows that R(sj)r --~ r pointwise, hence weakly. Thus, r E A, so A = C(K) by

the Stone-Weierstrass Theorem, and the representation R is WAP.

Fix x E Xo. By definition, 0 is in the weak closure of Or(x), so by weak

compactness there is a sequence {t3. } with T(t~)x --* 0 weakly. Define r E C(K) by r = [(x*,x)l. Then R(t j)r ~ 0 pointwise on K, so r is in Xo(R), so in

the ergodic decomposition (1.1) r E D(R). Since {~n} is WAPR-averaging, we

conclude that

f ](x*,T(t)x)ld#n = I I f R(tDed#~lloo --* O. sup l l~* l l< t J d

(The novelty is the uniform convergence on K. Pointwise convergence on K

follows from (1.2) and Proposition 1.1, and requires {#,~} to be only weakly WAP-

averaging.)

AVERAGING SEQUENCES AND MODULATED ERGODIC THEOREMS 2 4 9

Let 1 < r < oc, let q = r / ( r - 1) be the dual index, and let f E W~,{~}. Then

II f f(t)T(t)xdl~.[] = sup I f f(t)(x*,T(t)x}dl~] II~*ll<_a

<_ sup f If(t)l I(x*,r(t)x)ld.~ IIz*ll<l

< sup ( f If(t)l~d~=) ~/~ ( f [(~*,T(t)x)[~e..)'/~ --II~*(I<l a

_< sup ( [ If(t)l~d#.) 1/~ ii.ll<.-,)/.( i I1~*11<1 J

--~0.

Hence limn II f f(t)T(t)xd#~(t)H = 0 for f E W~,(..). Approximation in the

11 " [[1,(u.} -n~ of the given f by functions in W~,{~.) yields

lim [I f f(t)T(t)x d#n(t) ll = 0.

R e m a r k s . 1. In the proof of (iii) implies (ii), condition (iii) is used only for proving convergence on Xp, while the assumption of approximation of f by

functions in LJ~>I W~,{u,} is needed only for the convergence to 0 on Xo.

2. If {#,~} is only weakly WAP-averaging (hence weakly WAPR-averaging by

Theorem 1.2), we can obtain the equivalence of conditions (i),(iii), and (iv). The proof of (iii)~(i) is then a great simplification of the previous proof of (iii)~(ii):

the construction of the representation R(t) is not needed, since (1.2) and Proposition

1.1 immediately yield f ]{z*, T(t)x} ]d#~ --* 0 for x E Xo and z* E X*.

3. If {#,~} is WAP-averaging and satisfies the previous theorem, then it is in

fact WAPR-averaging (use f = 1).

Since for G nilpotent non-compact WAP(G)/B~(G) contains a subspace

isomorphic to g~ [C], we expect that unitary averaging does not imply WAPR-

averaging. If this is so, our next result does not follow from Theorem 2.2.

T h e o r e m 2.3. Let {#~} be a unitarily averaging sequence of probabilities on G, and let f be in the ]r " Ih,(..}-closure (in wl,{u.}) ofU~>~ w~,(u.}. Then the following are equivalent:

(i) For every unitary representation U(t) in a Hilbert space, f f(t)U(t)x d#~ (t) converges weakly.

(ii) For every unitary representation U(t) in a Hilbert space, f f(t)U (t)x d ~ (t) converges strongly.


(iii) For every irreducible unitary representation U(t) in a finite-dimensional Hilbert space, f f(t)U(t)x d#~(t) converges strongly.

(iv) f f(t)9(t) d#~(t) converges for every g 6 AP(G),

Proof . (i) implies that for every g E B~(G) the sequence f f(t)g(t)d#~(t) converges. Since AP(G) C B~(G), (iv) holds. (iv)~(iii) is in Proposition 2.1.

It remains only to prove (iii)=~(ii). Fix a unitary representation U(t), and let

X = Xp | X0 be its Jacobs-DeLeeuw--Glicksberg decomposition. I f X ~ is a

finke-dimensional U-invariant subspace, then (iii) implies that f f(t)U(t)x d~,(t) converges strongly for x e X'. Since Xp is the closed linear manifold gen-

erated by the finite-dimensional U-invafiant subspaces, we have the required

convergence on a linear manifold dense in Xp. Since sup,~ H f f(t)U(t)d#,~(t)]] < sup~ f [f(t)l d#~(t) < oo, we have the convergence on all of Xp.

Fix x 6 32o. Then the weakly almost periodic function h(t) = ](u(t)x,x)J satisfies M(h) = 0 and M(h 2) = 0, by (1.2). The function hi(t) = (U(t)x,x) is in

B~(G). so also hi and [hl[ 2 = h 2 are in B.(G) (which is a self-adjoint a lgebra- see the previous section). Since {~n} is unitarily averaging, Theorem 1.4 yields

#~, h 2 -~ M(h 2) = 0 uniformly on G and thus, by the Cauchy-Schwartz inequality,

~ �9 h ~ 0 uniformly on G. Hence

sup f I(U(s)x, U(t)x> Id#n (s) : sup f I<U(s)x, U* (t-1)x}ld#. (~) t6G J tEG d

f I<U(t-ls)x,x)id#~(s) = s u p f ](U(ts)x,x}id#n(s ) = IJ#~ * hl[oo --* O. s u p tEG J tEG J

Let 1 < r < c~, let q = r/(r - 1) be the dual index, and let g e W,r,{u, }. Then

sup f Ig(s)] I<g(s)x, g(t)x>ld.~(s) <_ sup ]lgllL=(~)[f I<g(s)x, U( t )x ) lqd~ tn (8 ) ] 1/q t6G J t6G J

I1~11 ~("-l)/q[[ I(U(~)~, u(t)~>ldlz~ (s)] 1/q --+ O. < Ilglls,.(.~) sup tGG J

Fix e > 0, and let no such that suPtea f Ig(s)l [(u(s)x, u(t)x)Id#.~(s) < e for n > no.

Then for such n we have

l[ f g(t)U(t)x@,~(t)[' 2= f f g(s)o(t)<U(s)x, u(t)x} dl~n(s)d#,~(t)

_< f Ig(t)l[f 19(s)[ I(U(s)x,U(t)x>ld#~(s)]d#~(t)

/ Ig ( t ) l dun(t) <_ esu.p /19(t) l duj(t). < e J 3 J

Hence lim,~ IIf g(t)U(t)x dm~(t)ll = 0 for 9 6 W~,{u, }. Approximating the given f

by such g, we obtain lim~ [J f f(t)U(t)xdu~(t)j I = o.


C o r o l l a r y 2.4. Let f e WAP(G). Then

(i) For every WAPR-averaging sequence {#~ } and every weakly almostperiodic representation T(t) in a Banach space, f f(t)T(t)x d#~(t) converges strongly.

(ii) For every WAP-averaging sequence {#~ } and every unitary representation U(t) in a Hilbert space, f f(t)U(t)x d#~(t) converges strongly.

(iii) I f f E B~(G), then for every unitarily averaging sequence {#~} and every unitary representation U(t) in a Hilbert space, f f(t)U(t)x d#,(t) converges strongly,

Furthermore, the limits depend only on the representation, but not on the averaging sequence, and define a bounded linear operator

Ex = l ira / f(t)T(t)x d#n(t).

Proof . Let f E WAP(G). Since WAP(G) is an algebra, f satisfies the

hypothesis of Proposition 2.1 for any weakly WAP-averaging sequence {#,~}. Hence condition (iv) of Theorems 2.2 and 2.3 is satisfied, so (i) and (ii) hold. Since

B~, (G) is a sub-algebra of WAP(G), the hypothesis of Proposition 2.1 is satisfied by f c B~,(G) for every {#~} unitarily ergodic, so (iii) holds by Theorem 2.3. The

proofs of these theorems show that for a given representation, the limit is zero on Xo. On Xp the limit is determined by limits for finite-dimensional representations,

for which it does not depend on the averaging sequence, by remark 1 to Proposition

2.1.

Proposition 2.5. Let {#~} be an AP-averaging sequence, and let T(t) be a bounded continuous representation of G in a Banach space X, and x c X. Then the set o f f in Wl,{u.}, such that f f(t)T(t)xd#n(t) converges strongly, is closed

in Wl,i/zn }.

Proof . Let [if - fj[Jl,{u~} -+ 0, with f fj(t)T(t)xd#n(t) strongly convergent

for each j. Then { f f(t)T(t)x d#n (t)} is a Cauchy sequence, by the inequalities

[] / f ( t )T( t )xd#~( t ) - f f(t)T(t)xd#k(t)l[


_<11 f I(t)T(t)x d,n(t)- f fs(t)T(t)x + I[/fj(t)T(t)xd#~(t)- f fj(t)T(t)xd#k(t)l I

+ I I / f(t)T(t)xd#k(t)- f fj(t)T(t)xd#k(t)l I

_<[[x[] sup IIT(t)]l [ If(t) - fj(t) Id#u(t) tEG J

+ I] f fj(t)T(t)xd#~(t)- f fj(t)T(t)xd#k(t)l [

+ ][x[[ sup ] ]T(t )H/If( t ) - fj(t)[d#k(t). tEG J

Extending the "generalized almost periodic functions" on R introduced by

Besicovitch [B], we will say that f is r-Besicovitch for {#,~} if it is in the "r-

seminorm closure (in Wr,{~,~}) of AP(G) (i.e., for e > 0 there is a g E AP(G) with

Ill - gllr,{~} < e), and denote this class by B~,{u~}. We denote

o o 1

L~,{u~} is the space of all #-essentially bounded functions on G with the

ess sup-seminorm; B~o,{..d = BI,{u~}ML~,{., d. If1 < r < p _< 0% the well-known

integral inequality implies [[fl[~,{u~} _< llf[ip,{~} and hence Bp,{u~} c B~,{,~}.

T h e o r e m 2.6. Let {#,~} be an AP-averaging sequence of probabilities on a locally compact a-compact group, and let f E BI,{~,}. Then

(i) M(fg) := limoo ] f9 dl~ exists for any 9 E AP(G).

(ii) I f {#n} is unitarily averaging, then for every unitary representation U(t) in a Hilbert space, f f(t)U(t)x d#~ (t) converges strongly.

(iii) I f {/~,~} is WAPR-averaging, then for every weakly almost periodic representation T(t) in a Banach space, f f(t)T(t)x d#~(t) converges strongly.

Proof . (i) Apply Proposition 2.5 to the representation by translations in AP(G) (using remark 1 following Proposition 2.1).

(ii) and (iii) follow from combining Corollary 2.4 (for f E AP(G) ) with

Proposition 2.5.

R e m a r k s . 1. Theorem 2.6(i) yields that for {#~} AP-averaging, lira f f d/~

exists for any f E BI,{u,}, which is a generalization of the classical mean value the-

orem for Besicovitch functions on R (see, e.g., [Le]; see also [T-3] for commutative

groups).


2. Theorems 2.2 and 2.3 are more general than Theorem 2.6 - an example

in [Be-Lo] (on Z, with respect to Cesaro averages) shows a bounded function

(sequence) satisfying the hypothesis o f Proposition 2.1 (hence the consequences

of Corollary 2.4), which is not a Besicovitch sequence.

Let f E WAP(G) . By Eberlein's theorem [El (see also [Bu, pp. 29--30]) we

have f = f l + fo where f l E AP(G) and M(lf0l) = 0. Hence also M(lfol ~) = 0

for 1 _< r < co. Thus, for a weakly WAP-averaging sequence {~z,~} we have

f0 ;{,~t" = l i m ~ f [fol~d#n(t) = M(lf0[ ~) = 0, so WAP(G) C B~,(,~} for every I _< r < oo (see also [L-O, Proposition 2.3]). A similar result holds for

f E B~(G) and {#n} weakly unitarily averaging.

3 M o d u l a t e d ergodie theorems for group act ions

In this section we look at measure preserving actions of G, and look for point-

wise modulated ergodic theorems. The Abelian case was treated in [T-3], and

extended in [L-O].

D e f i n i t i o n . Let (f~, Jr, m) be a o--finite measure space. We shall say that a

system {Or : t E G} is an action of G in (f~, Jr, m) i f

(1) every Ot is an invertible measure preserving transformation in f~;

(2) OtlOt2 = O~lt 2, t l , t z E G; (3) for any function ( E Ll(m) the function (w, t) ~ ((0~co) is Y" x 13 measurable.

Condition (3) is clearly implied by

(3') (w, t) H Otw is j r x B measurable.

Conversely, i f A is a measurable subset of f~ with finite measure, then 1A E

L l ( a , m ) , and by (3), {(~o,t) : Otto E A} is measurable. Since m is ~-finite, (3')

holds. Thus (3) and (3') are equivalent.

An action is called ergodic i f all invariant functions with respect to {0~ : t E G}

are constant a.e.

For an action {Or} define the induced representation T( t)~(w) = ~(0~ lw), which

is clearly a weakly measurable representation o f G by invertible isometries o f

Lp(f~,m), I < p < ~ . Moreover, for 1 < p < co the induced representation in Lp

is continuous IT-4, p. 347].

L e m m a 3.1. Let {Or : t E G} be an action o f G in (f~,Yr m), let # be a probability measure on G, and let f E LI(G,t~). Fix 1 <_ p < c~, and let ~ E

Lp(a, m). Then the integral fG f(t)~(Otw)dlz(t) is well-defined for m-a.e, w E ~2,

and defines a function ~ E Lp(a,m), with 11r <-II~llLp(,~)f [f(t)[d#.


P r o o f . By definition, the function f(t)~(Stw) is 5 r x B-measurable when ~ E

Ll(m) ;q Lp(ra), so it is measurable for any ~ E Lp(m). The ca sep = c~ is obvious,

so l e t p < c~. Let q = p/(p - 1) be the dual index, and take r e Lq(m) with r > 0

a.e. Let ~ 6 Lp(f~, m). By H61der's inequality

f lr162 din(w) < HT(t-~)(HLp(,~)IIr -- ]Ir162

B y the Fubini-Tonell i theorem

~ • If(t)'(O~w)[r • m ) = ~ ,f(t)[ ~ ,,(Otw)]r d#(t)

-< IIr162 If] d#.

Hence the function ~(w) = fa f(t)~(Otw)d#(t) is well-defined for a .e .w. F o r p = 1,

r _-- 1 yields the integrability o f C. Let 1 < p < oo. Replacing r in the above

argument by an arbitrary r c Lq(ft, re) we obtain that f ~r dm is finite for any

r 6 Lq. The Banactr--Steinhaus theorem now implies that ~ E Lp(fL m), and the

estimate on the norm follows from the above computations.

N e m a r k s . 1. For 1 < p < oo, f f( t)T(t-~)~d#(t) is defined in the weak

topology o f Lv(f~,m), and the previous lemma identifies this Lp element as a

pointwise integral.

2. The p r o o f o f L e m m a 3.1 did not use the fact that T is a representation. Hence

atso fc f(t)~(Ot-~w)d#(t) is well-defined, and defines a function in Lp(ft, m).

T h e o r e m 3.2. Let {#~ } be a unitarily averaging sequence o f probabilities on G, and let f be in the ]l" ]]l,{,~}-closure (in WI,{~,~}) ofU~> 1 w~,{,~}. Assume that f f(t)g(t) d#,~(t) converges for every 9 ~ AP(G). Then for every group action {at} in (f~, U, m), f f(t) ~ o O~ 1 d~,(t) converges in L~-norm for every ~ ~ Lp(f~, m), 1 < p < c~, and when m isfinite also for p = 1.

P r o o f . Let T(t)~ = ~ o a~ -1 be the continuous representat ion induced in

Lp(f~, m) by the action {at}. The result for p = 2 follows f rom Theorem 2.3.

We denote A,~ = f f(t)T(t)~ d#~(t). For 1 < p < ~ fixed, we have sup~ HAn llz, p < sup s f If(t)ld#~(t). When m(f~) = 1, we have I]A,~ - Ak~][x < [ IA~ - Ak~ll2

for ~ C L2(m). Hence A,~ converges in Ll(m) for ~ in a dense subspace, and thus

on all o f Ll(m) by uniform boundedness in L1 o f {A,~}. For 1 < p < ~ , take

bounded. Then

f IAor A r J rAor A eldm _< IIA, r - Ak lh[2l[r sup IIA .[I ] p - 1

J


yields convergence in Lp-norm on a dense subspace, hence on all of Lp(m). It remains to prove the convergence for I < p < c~ in the general case. We

prove by induction on j that the theorem is true for p >_ (j + 1)/j. For j = 2, fix

p _> 1.5, and let ~ E Lp(m) be bounded. Since 2(p - 1) _> 1, we can use HOlder's

inequality to obtain

i IAn~ -- Ak~lPdm = l IAn~ - Ak~liAn~- Ak(IP-l dm

[ l I A ~ - Ak~12(P-1) dm] �89 = H A ~ - Ak~ll2 [IIA,~- Ak~N2(p-~)] p-~ <_ NA~ Ak~H2

< IIAn~ - A~-[I2 [211~112r sup IlA~ll2(p_~)] p-t.

Now the convergence in L2 implies that {An~} is Cauchy in Lp. Thus we have

convergence of {A~(} on a dense subspace, so we have convergence on all of Lp,

and the theorem is true for p > 1.5. Now we assume that the theorem is true for

p >_ ( j + l ) / j , andfixp >_ ( j+2) / ( j+ l ) . For~ E Lp(m) bounded, ( j + 1)(p- 1) _> 1

and HOlder's inequality yield

./" ]An~- Ak~iPdT~ -~- f [An~- Ak~iP-llgn~- Ak~id?7~

= I IA ,~ - A k ~ I I ( s + ~ ) i S [ I I A n ~ - A~<~:II(i>-~)0"+~)F -~

<- I I & ~ - A,~<011(S+~)/S [211~7110+~)(.-~) sTP I IAd l (s+~)( . -~) ] " - 1 .

Now the convergence in L(j+I)Ij implies that {A,~} is Cauchy in Lp. Thus we

have convergence on a dense subspace, so we have convergence on all of Lp. This

proves our theorem.

C o r o l l a r y 3.3. Let {#,~} be a unitarily averaging sequence of probabilities on G, and let f E BI,{,:). Then for every group action {0t} in (f~, Y:, m), f f(t) ~ o O~ -1 dpn(t) converges in Lp-norm for every ~ E Lp(f~, m), 1 < p < co. When m is finite, the result holds also for p = 1.

Proof . f satisfies the hypotheses of the previous theorem, by Theorem 2.6(i).

R e m a r k . If f E B,, (G), then f E BI,{~. ) for any unitarily averaging sequence.

In that case, the limit in the previous corollary does not depend on the particular

sequence {#,~}.

For any measurable function f on G, let f( t) = f( t-1) . If f c (W)AP(G), so is f , since (weak) almost periodicity for left and right translations is the same


[Bu, p. 51. If f E B~(G) is a coefficient function f(t) = <U(t)x, y), then ](t) = (x, U(t)y> = (U(t)y,x} shows that f E B~(G) since B~(G) is self-adjoint. Hence

f E B~(G) for f E B~(G). Now, for a probability # on G define the measure

/2 by #(A) = #(A-l) . The equality f f d# = f ] d # shows that a sequence {#,~} is weakly (W)AP-averaging (weakly unitarily avearging) if and only if {/2n} is

weakly (W)AP-averaging (weakly unitarily averaging). Hence, by Theorem 1.6,

{#n} is AP-averaging if and only if {#n} is AP-averaging. For an AP-averaging

sequence {#n} we have f E W~,{,~} if and only i f f E W~,{~}, and f 6 B~,{~,~} if and only i f ] 6 B~,{~}.

The following statement generalizes Theorem C in [%3] and the statement

a ) ~ d~,s for a _> 1 of Lemma 4.1.8 in [%4].

C o r o l l a r y 3.4. Let {#n} be a unitarily averaging sequence of probabilities on G, and let f E Bl,{u~}. Then for every group action {Or} in (f~,~',m),

f f(t) ~(Otw)d#~(t) converges in Lp-norm for every ~ E Lp(ft, m), 1 < p < oo. When m is finite, the result holds also for p = 1.

Proof . We have f f(t)~(Otw) d#n(t) = f f(t)T(t){d#n(t). We can now apply the previous corollary.

h order to deal with pointwise convergence in the previous corollary, it is necessary to have a.e. convergence without modulation.

Defini t ion. A sequence {#,~} of regular probability measures on G is called

pointwise averaging if for any action {Or : t E G} of G in a cr-finite measure space (~, 5 r, m) and for every ~ E Ll(m) the limit

(3.1) lina f ~(Otw) d#~ (t) dej

G

exists for m-almost all w e ~, and ~ is invariant wkh respect to {Or}. If {#,~} is pointwise averaging, then for actions in finite measure spaces the

convergence in (3.1) is also in Ll-norm.

R e m a r k . Many proofs of pointwise convergence show first that {#,~} is

maximal ergodic, i.e., for every action {Or} there is a constant C such that

sup f l~(O~w)ld#~(t) >_ e} < (c/~)ll~]ll (~ e Ll(m), e > 0). r e { t o : n d

If G is Abelian and {/zn} is pointwise averaging, then for every ergodic action in a

finite measure space the above maximal inequality holds for some C, by Sawyer's Theorem [Ga, p. 7].


Def in i t ion . A sequence {A,~} of measurable sets in G is called a pointwise

averaging sequence o f sets i f0 < ),(An) < oo for every n, and the sequence {#n} with

1 d#n - A(An) IA~d)~

is pointwise averaging. Such sequences exist on every connected locally compact

~r-compact group (see [T-4]).

If G is amenable, and {A~} is a sequence of Borel sets (of finite positive

measure) with the following properties: A(tAn/kA~)

l) l i m )~(An) - 0, t C G (Folner's condition);

2) A 1 c A 2 C - . - ;

3) sup A*(AnlA~) < co ()~* is the outer measure induced by )0; )~(A~)

then {An } is a pointwise averaging sequence of sets, which is also maximal ergodic

[T-l], IT-2] (see also [T-4], Ch. 6, w For example, in R ~, any increasing sequence

of bounded convex sets (A,~) with intrinsic diameters d(A~) --* oo satisfies the above

three conditions (so is pointwise averaging and maximal ergodic).

The following result is a special case of Corollary 6.6.2 of [T-4]. We give a

straightforward proof.

Proposition 3.5. Let # be an adapted probability on G, and define #~ = J. n "~ ~ j = l #J" Then the sequence {#,~} is pointwise averaging and maximal ergodic.

Proof . Let {0t : t E G} be an action of G in a or-finite measure space

(f~, ~-, m) and let P be the Markov operator defined by P~(w) = f ((Otw)d#(t) for

E Lp(f~, m), which preserves m by Fubini's theorem. We then have

n j=l

which satisfies the maximal inequali W (with C = 1), and converges a.e. for ~ Lp(~t, m), i < p < o~, by the pointwise ergodic theorem (e.g., [K]). For

1 < p < c~ convergence holds also in Lp-norm, and the limit is the projection on

the fixed points of P, which are the common fixed points, by uniform convexity

and Lp-continuity of the representation, since # is adapted. Hence the limit is

{Ot}-invariant. The same applies for p = I when m is finite.

Assume now m infinite. Then ~t is decomposed into two P-invariant sets; on

the first P has a finite invariant measure equivalent to m, and on the other there

is no finite invariant measure < < m. By using the continuity in L2, it is not hard

to prove that these invariant sets are {Or }-invariant. For functions supported in the

258 M. L1N AND A. TEMPELMAN

first set we apply our argument for p = I in the fmite case, while on the second set

the pointwise limit is zero for all L~ functions.

T h e o r e m 3.6. Every pointwise averaging sequence is AP-averaging.

Proof . Let {#n} be a pointwise averaging sequence. Let U(t) be a continuous

unitary representation in the Hilbert space Hk of dimension k. Let/d(k) be the

group of k x k unitary matrices, which is compact [H-Ro, (4.26)], and let b/be the

closure of {U(t) : t c G}, which is a compact subgroup of/d(k), with normalized

Haar measure ),u- The mapping atA = U(t)A for A ~/d is continuous on/.4 x G~

so it is bimeasurable, and thus is an action of G in (b/, ;~u). Let {xl, xz, ..., xk} be

an orthonormal basis of ilk. Each function r = (Axi, xj) is continuous on/d,

s o m ' * P

/ ( U ( t ) A x i , x j } d#~(t) = lim / r lim Ja n J G

exists for a.e. A C L/, and is {at}-invariant--hence a constant on/d. The value of

the constant is

n J G J M n J G J~4

= .~.~ (Axg, xj }dAu (A)

by Lebesgue's theorem and Fubini's theorem, since :Xu is shift-invariant.

By excluding a finite union of null sets, we have that for a.e. A C Z~,

limo s (V(t)Ax, y),t.o(t) = y/d (m

for every x, y C Hk. Fixing A in the convergence set and substituting A - I x for x,

we have that

for every x, y E Hk. Fix x, y E Hk, and let f(t) = {U(t)x, y). For s E G substitute

in (3.2) U*(s)y = U(s-1)y for y, to obtain

Hence the limit does not depend on s E G, so it is M(]). Taking all possible finite-

dimensional representations and using the Approximation Theorem, we conclude

that {#~} is pointwise AP-averaging, so it is AP-averaging by Theorem 1.6.


Proposition 3.7. Let {#~) be a po in~ i se averaging sequence. For {Or : t E G} a group action o f G in (ft, Sr, m) and 1 <_ p < ~ , the limit (3.1)

exists a.e. f o r any ~ E Lp(f~, rn).

P r o o f . For p = 1 the convergence holds by definition. The case that m is finite

is trivial since then Lp(m) C La(m). Assume now m infinite. Fix 1 < p < ~ . For

�9 Lp, we obtain for a.e. w

.-f f l~(0,..)i ...(t) _< .u.[i i~(0,~.)l"..(,)]'/" < oo,

since [~1 p �9 L1 (m), and thus f I~(O~w)[P d#, (t) converges a.e. Since f ~(O~w) d#, (t) converges a.e. for ~ �9 Lp n L1, the Banach principle implies convergence for every

~ � 9

L e m m a 3.8. Let {Or } be an action o f G in (0, jz m) and let { #n } be a sequence

ofprobability measures on G. Let f �9 Wq,{u.} and ~ �9 Lp(f~, m), with 1 <_ p < c~ and l i p + 1/q = 1, and assume that ~ = l i m n - ~ f f(t)~(Otw) d#,(t) exists a.e.

Then ~ �9 Lp(m) and II~IIL.<.~) -< Ilfllq,<..}ll~llL.(.~)

P r o o f . The case p = ~ is easy. Let 1 < p < c~. Employing Fatou~s lemma,

the Fubini-Tonel l i theorem, and H61der's inequality (when p > 1), we obtain

f ,~,P d r a = / ] l i r a f f(t)~(Otw)d#~(t),Pdm(w) f~ G

< l i r a / [ / I f ( t ) ~ ( o , ~ ) l d,.(~)Fdm(~)

f~ G

o G G

f/ fp -< II [[q,(.,0 lira j[j G f~

- f P /'l~]Pdm < o o .

f~

T h e o r e m 3.9. Let {/z~} be a pointwise averaging sequence. Then for any

group action {Or : t �9 G} in a or-finite measure space (f~,~-,ra), any ~ �9 Lp(~2,m), I _< p < c~ andany f �9 Bq,{u.} with l i p + 1/q = 1, the limit

f f(t)~(o,~) du.(t) ~'J ~(~) (3.3) , o r

G

exists m-a.e., and the limit function ~(.) is in Lp.


P r o o f . Let {Or} be a group action in the a-finite measure space (f~, U, m). Fix

E Lp(f~,m), 1 _< p < ec and any S E Bq,{u,~ } . The integrals on the left-hand side

o f (3.3) are well-defined, by Lemma 3.1.

We first prove the Theorem for coefficient functions o f finite-dimensional uni-

tary representations: let U(t) be a continuous unitary representation in the Hilbert

space Hk with basis {x3 }l_<j<k. We use the notation of the proof of Theorem 3.6.

Let Pt = at x 0, be defined on (5/x f~, Au x m). It is clearly bimeasurable, so

defines an action in (U x f~, )~u x m).

The function ~bi,j(A, co) = (Axi, xj}~(co) is in Lp(~U x m), so

(3.4) lira s (U(t)Axi,xj)~(Otw) d#~(t) = lira s r d#,~(t)

exists )~u x m a.e., and is {flt}-invariant. Hence for m-a.e, co, the limit in (3.4)

exists for )~u-a.e. A E gt. After excluding finitely many null-sets in ft, and then

removing finitely many null-sets in /g , we have that for a.e. co there is a set 5t'

(depending on co) with Au(b/') = 1, such that (3.4) holds for every i,j and A E/4' .

Hence

lim ~ (U(t)Ax, y)~(Otco) d#~(t)

exists for every x, y E Hk and every A E/,/ '. Now fix such A, and put A-ix for x

to conclude that for a.e. co,

fG (v(t)x, co) (t)

exists for every x, y E H~. Hence the Theorem holds for coefficient functions o f

finite-dimensional unitary representations.

For p > 1 we have q < co, and by the Approximation Theorem, the closure in

Wq,{o,} of the linear span of the coefficient functions is Bq,{,,~}. We show that

the set o f functions f E Wq,(,,} for which (3.3) holds (for a given ( E Lp(~2, m))

is closed, and this implies the Theorem for p > 1. So let Ilfj - fllq,(~,~} --* c~,

with fj E Wq,(u,} satisfying (3.3). Since {#~} is pointwise averaging, ~(w) =

lim~ fc I((Otw)l pd~'~(t) exists m-a.e. Hence for a.e. w we have

G G

AVERAGING SEQUENCES A N D M O D U L A T E D E R G O D I C T H E O R E M S 261

G G

+If / G G

-4-1 r

G G

- I ' i " ' f " I ' < I f - D l "dw ]~ [ I~(e,<o)l dv,~(t)], + IS - Dl~dvk]~[ I((O,~o)l~d#k(t)] ~ G (7 G G

+1 f fy(t)((Otw)d#~(t)- f fj(t)((Otw)dl~k(t)l. G G

Hence {fc S(t)((Otw) d#~(t)} is a Cauchy sequence for m-a.e, w, since

sup I/f(t)((Otw) d#n(t) - i f(t)((Otw) dpk(t)l < 2r IIS - DIIq,(.o)- lim J J G G

Now letp = 1. Fix f E Boo,{u~}. By [L-O], f E B2,{u~)- Since f is #~-essentially

bounded and {#~} is point-wise averaging, for any ( E L1 (m) we have m-a.e.

esssup I. f f(t)((Otm)d#,.(t)U _< lifll~,{..:} sup f I((Otw)] d#n(t) < oo. We have proved that (3.3) holds m-a.e, i f ( belongs to the set L1 (ra) n L2(m) which

is dense in Ll(m); a.e. convergence for any ( E Ll(m) and f E Boo,{u,} follows from the Banach principle.

Finally, the limit function is in Lp(m) by the previous Lemma.

R e m a r k s . 1. Instead of using the result of [L-O] for p = 1, we can show

by computations similar to those forp > I that {fc f(t)((Otw) d#~(t)} is a Cauchy

sequence for ( E Ll(m) N Lo~(ra), since f E Boo,{u,} is the limit in W~,{u, d

semi-norm of {fj} bounded functions for which (3.3) holds.

2. The result for G Abelian (with some restrictions on {#n}) was proved in

[T-3] for p > 1, and in [L-O] for p = 1.

3. For pointwise modulated ergodic theorems for Dunford-Schwartz operators

see [Ba-O] for a single operator, and [J-O] for finitely many commuting operators.

Since these operators are not obtained from group (or semi-group) actions, the

results do not follow immediately from group action results. A direct proof for

Besicovitch modulation, based on the Dunford-Schwartz theorem, is given in

[J-O].

Although we do not know if the reflected sequence of a pointwise averaging

sequence is unitarily averaging (so Corollary 3.4 cannot be used), we can still show


that under the conditions of Theorem 3.9 we have also Lp-norm convergence when

m is finite.

T h e o r e m 3.10. Let {#,~} be a pointwise averaging sequence o f probabilities on G, and let f E Bl,{t~,}. Then for every group action {0t} in a finite measure space (a, .~, m), f f ( t ) ~(Otw) d#n(t) converges in Lp-norm for every ~ E Lp(12, m),

l _ < p < c~.

P roof , Let {Or} be a group action in a finite measure space (12,~',m).

For ~ E L~(f~,m) and f E AP(G) the a.e. convergence of (3.3) proved in

Theorem 3.9 implies Lp-norm convergence by Lebesgue's Theorem. The proof of

Proposition 2.5 does not require that T be a representation, and yields, for fixed

�9 Lo~, that f f ( t ) ~(Otw)d#~(t) converges in Lp-norm for every f �9 Bi,{;,n}.

Now fix p and f �9 BI,{u~}, and define A,~(w) = f f ( t ) ~(Otw)d#,~(t). Then

supn IIA~llL, _< supn f If(t)[ d#~(t) < oo, so norm convergence on a dense sub-

space implies convergence on all of Lp(~2, m).

The next result, which is a generalization of the Wiener-Winmer Theorem,

yields the generation of functions which satisfy the conditions of Proposition 2.1,

without necessarily being Besicovictch functions. Moreover, these functions will

satisfy all the conditions of Theorem 2.2.

T h e o r e m 3.11, Let {#~ } be an ergodic pointwise averaging sequence o f probabilities on a separable locally compact amenable group G, and let {0, : t E G} be an ergodic action o f G in a probability space (12, 5 c, m) with (f~, F) a standard Borel space [Va]. Then f o r ~ integrable there exists I2o E Jr

with re(a0) = 0, such that for any w q~ f~o the function f ( t ) = ~(Otw) is in WI,{u~},

and f f(t)g(t) d#,~(t) converges for every g �9 AP(G). Moreover, f o r every weakly almost periodic representation T( t ) in a Banach space X and x �9 X, the sequence

f f ( t )T( t )x d#n(t) converges strongly.

PrOOfo Taking 9 = 1, we see that {#,,} must be pointwise averaging (at least

for actions in finite measure spaces). Pointwise averaging implies that f �9 WI,{;,,}

for a.e. aJ �9 f~. The proof of Ornstein and Weiss [Or-We], given for G discrete and {#~}

defined by a pointwise averaging Folner sequence, can be adapted to prove the

first part o f the theorem for ~ bounded: separability of G and the standard Borel

space are needed for the construction of a compact metric space model [Va], and

the ergodicity assumption (which implies amenability of G [L-W]) is used for

obtaining certain invariant probabilities in the topological model. The second part

of the theorem now follows from Theorem 2.2.


For ( integrable, let (j be bounded functions converging to ( pointwise and

in Ll(rn). By ergodicity of the action and pointwise ergodicity of {#~}, for each

j and a.e. co we have l im~_~ f I(~(Otco) - ((Otw)[ d ~ ( t ) = IIs - CI]I. Removing

the null sets where the convergence does not hold, and those obtained for each (~

from the theorem, we obtain for co in the remaining set that the bounded functions

f j(t) = ~j(0tw) converge to f ( t ) = ~(Otco) in Wl,{u.}-seminorm. By Proposition

2.5, f f(t)g(t)d#~(t) converges for every g E AP(G). Since {Iz~} is ergodic, it

is WAPR-averaging, and Theorem 2.2 yields the modulated ergodic theorem for

every weakly almost periodic representation.

R e m a r k s . 1. For {tz,~} and G as in Theorem 3.11, the functions f ( t ) obtained

in the theorem satisfy the convergence assertion of Theorem 3.2, so our result

extends Theorem 2 of [Or-We].

2. Let G be amenable. Then there exists an adapted probability such that Izn = 1 n ]~k ~ k = l is ergodic [R]. By Proposition 3.5, {#,~} is also pointwise averaging.

3. I r a is a group with AP(G) consisting only of constants (e.g., [V]), and {#n}

is pointwise averaging, then f ( t ) defmed from an action as in Theorem 3.11 clearly

satisfies the properties of Proposition 2.1. Thus amenability of G is not necessary.

4 I d e n t i f i c a t i o n o f t h e l i m i t i n H i l b e r t s p a c e s

In this section we study the limit operator obtained in the modulated ergodic

theorems: we identify it for unitary representations, and obtain some general

results. For G commutative and f 2-Besicovitch, the limit was identified in [T-3] for

unitary representations, and in [L-O] for weakly almost periodic representations.

In [O] the limit was identified for Besicovitch-weighted averages of the powers of

a contraction in a Hilbert space.

Let {/~} be an appropriately averaging sequence (unitary or WAPR).

Theorems 2.2 and 2.3 show that, for f E W~,{u, t (r > 1), anecessary and sufficient

condition for the modulated ergodic theorem is the condition of Proposition 2.1.

For convenience, for an AP-averaging sequence {/~n} we denote

{f e Wa,{~} : M( fg ) := npfy(t)g(t)d .(t) exists for every g E AP(G)}.

Theorem 2.6 shows that BI,{,~} C F{~}. Clearly M(fg) = M(fg) for f E AP(G). Theorem 6 in [Ka, p. 73] shows that there may be many bounded non-zero functions

f in F(~,~} with M(fg ) = 0 for every g E AP(G) (and thus are not Besicovitch).


The proofs of Theorems 2.2 and 2.3 show that the limit operator in the modulated

ergodic theorem is zero on X0 (this convergence was the difficult part in these

theorems). On Xp the modulated ergodic theorem holds for any f E F{~},

without additional conditions on f , and the limit is obtained from the limits on the

finite-dimensional invariant subspaces, which we now study.

Let U(t) be a continuous unitary representation of G in a finite-dimensional

Hilbert space H, and let f E F{u~}. By Proposition 2.1, the operator

is well-defined, and is denoted by M(fU*) , or M[f(t)U*(t)]. Let Uij(t) be the

matrix of U(t) with respect to a given orthonormal base {ei} in H. Since

the representation is AP, each function Uij(t) = (U(t)ej, e~} is AP [Bu, p. 36].

Hence Mi~ = M(fUji) is well-defined. The matrix Mij represents the operator

M[f(t)U* (t)] in the given base.

Let I" be the set of equivalence classes of irreducible finite-dimensional unitary

continuous representations of G. Denote by U-~ a representative of the class -y a F,

acting in the finite-dimensional Hilbert space H.y of dimension d.y; For a fixed

orthonormal base in/ /7 , denote by U~J (t) and i3 A. r (f) the matrices of the operators

U~ and &(f) = d.tM[f(t)U~(t)]. The family {A-~(f) : 7 e r} will be called

the "Fourier operators" of f E F{~,,}. The set c~f := {7 e I" : A~(f) r 0}

is called the spectrum of f . For G Abelian (where I" is the set of continuous

characters), the Fourier operators reduce to the Fourier coefficients o f f , defined

by a~(f) --- M( f9 ) = limn f f(t)~(t) d..(t) for any continuous character 7-

The following temma follows from the definitions.

L e m m a 4.1. Let {~,} be an AP-averaging sequence and T a weakly almost periodic representation. Let Y be a finite-dimensional minimal T-invariant subspace such that Tiy is equivalent to U~. Then for y E Y and f e F{u,~ } we

have

f f ( t )T(t)y d ~ ( t ) = S-1M[f(t)U.y(t)]Sy = d~IS-1A~(])*Sv, jim

where S : Y --* H. l is an isomorphism such tha tT = S-1U.rS.

The identification of the limit in the modulated ergodic theorems thus depends

on the Fourier operators o f f , and on an analysis of the finite-dimensional minimal

invariant subspaces ofXp. Thespectrumofaweaklyalmostperiodicrepresentation

T, denoted by a~,, is defined as the set of-'/E 1" such that T has a minimal invariant

subspace on which its restriction is equivalent to U~. Thus, T has an empty

spectrum if and only if X = )Co.


Proposition 4.2. Let U be a continuous unitary representation o f G in a

Hilbert space H. Then there exists an orthogonal family o f minimal U-invariant

subspaces {H~,~ : "y G av, a E Z.r} (g~ an index seO, such that each H~,= is

o f finite dimension d r, lip = ~ e ~ v ~ e z , | and U restricted to Hz,~ is (isometrically equivalent to) U~.

The proof follows from the following facts: (i) Two different minimal invariant

subspaces have zero intersection. (ii) The orthogonal complement of an invar/ant

subspace is invariant. (iii) Every invariant subspace of Hp contains a minimal

invariant subspace (since the restriction of U to an invariant subspace of Hp is

almost periodic).

R e m a r k s . 1. Jacobs [Ja, p. 105] has H , as the closed linear manifold generated

by the H-~,~, but this easily yields the direct sum representation.

2. The proposition is false in general Banach spaces - take the identity repre-

sentation in a separable Banach space without a Schauder basis.

Theorem 4.3, Let {#n} be a unitarily averaging sequence in G, and let U be a continuous unitary representation in H. Let Ex,~ be the orthogonal projection

from H onto H~,~. I f f E F0,~} is in the Wl,{u~}-elosure o f [.J~>l W~,{u,}. then for

every x E H we have

f f ( t )U(t )xd#, ( t ) = ~ ~ M[f(t)UT(t)]E~,~x lim 7Eau aEZ. r

E E - 1 - . = d2~ AT(f ) E.y,~x, ~E~ u aEZ,~

with strong convergence on both sides (and countably many non-zero terms in the

s u m ) .

Proof . The convergence of the left hand side follows from Theorem 2.3. I f

U has empty spectram, H = H0 and the limit is 0, and also the sum is O (empty

sum). Otherwise, we use the previous proposition (remembering that Ho and H v

are orthogonal) to obtain by the orthogonality that for each x the sum has only

countably many non-zero terms. On HT,~ the limit is M[f(t)UT(t)] = d~lA~(f) *. The convergence of the sum follows from the previous proposition.

R e m a r k s . 1. If H = lip, then the above result holds for {#n} AP-averaging

and any f E F{u~}. 2. If G is amenable, a similar result holds for any bounded continuous repre-

sentation in a Hilbert space, since such a representation is equivalent to a unitary

one [Ly, p. 83].


When the group G is commutative, the irreducible representations are l-

dimensional (characters), and P can be identified with G, the group of characters.

We shall denote the characters by ~(-). The Fourier operators o f f E F{un} become

the Fourier coefficients a n = lim~ f f ( t )2( t ) dl~(t).

Let X x = {x : T( t )x = x( t )x Vt E G} be the eigenspace of T corresponding

to the character X. The spectrum rrT is now the set of all characters for which

r {0}.

T h e o r e m 4.4. Let {#,~} be a unitarily averaging sequence in a cr-compact

LCA group G. Let T be a continuous bounded (unitary) representation in a Hilbert

space H, with P• the (orthogonal) projection from H onto the eigenspace of

E G. I f f E F{u,d has Fourier coefficients {ax}, and is in the Wl,{u,}-closure of

U~>I wT,{un}, then for every x E H we have

(4.1) lira / f(t)T(t)x d#,~(t) = ~ a2Pxx, x

with strong convergence on both sides (and countably many non-zero terms in the

sum).

Proof . The unitary case follows from Theorem 4.3. If T is bounded, then

T(t) = SU(t )S -1 for some unitary representation U, since G is amenable. The

result now follows from the unitary case.

R e m a r k . The conditions are satisfied by any bounded f E F{~,.), and atso

by f E BI,{~,.}. The identification in IT-3] (and in [L-O]) was obtained onty for

f E B2,{~}.

Note added in proof. An example of a unitarily averaging sequence which is

not even weakly WAP-averaging will be included in a joint work with J. Rosenblatt

(in preparation).

Acknowledgment The first author wishes to thank Penn State University, where part of this

research was carried out, for its warm hospitality.

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[T-3] A. Tempelman, Ergodic theorems for amplitude modulated homogeneous random fields, Lithuanian Math. J. 14 (1974), 221-229 (in Russian). English transl, in Lithuanian Math. Trans. 14 (1974), 698-704.

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[T-4]

[Va]

[V]

Michael Lin DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

BEN-GuRION UNIVERSITY OF THE NEGEV BEER-SHEVA 84105, ISRAEL

emalh I~n(~math.bgu.ac.il

Arkady Templeman DEPARTMENT OF MATHEMATICS

THE PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PA 16802, USA

email: arkady~stat.psu.edu

(Received May 11, 1998 and in revised form November 15, 1998)

averaging sequences and modulated ergodic theorems for weakly almost periodic group representations

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