Generalized Summing Sequences and the Mean Ergodic Theorem

Conditions are found on a sequence of probability measures pn on a locally compact abelian group G so that, for any strongly continuous unitary representation of G, J" Utfd/x„ will converge to a [/-invariant function. These conditions are applied in the case where the group is the integers. Introduction. In a recent paper [3] Greenleaf discusses summing sequences in amenable locally compact groups. These are defined as sequences of compact sets {Un} in G such that the following hold: (1) int(C/n+1) 2 Un and G = G Um; 71=1 (2) \kU„& Un\l\Un\-*0 as n -* co, for every k e G, where |£| is the left Haar measure of E. These sequences have the property that if G x S—>S is a measurable action of G on a measure space (S, 38, p) such that p is G-invariant, then, for/in Lp, Wn\~l fVn figs)dg converges in Lp to a G-invariant/*. Such results are proved using convexity and fixed point arguments. In this paper attention is restricted to locally compact abelian groups thus allowing the full power of harmonic analysis to come into play. In §1 more general summing sequences for the purpose of ergodic theorems as above are found and shown to include ordinary summing sequences. In §2 more specific results are obtained on summing sequences when the group G is the integers. 1. Generalized summing sequences. Let U be a strongly continuous unitary representation of a locally compact abelian group G on a Hubert space H. If ¿a is a probability measure on G, ¡ Usfdp is defined weakly for each fin H so that, for all h in H, (3) (jujdp, hj « j(UBf, h) dp(g). Received by the editors May 15, 1973. AMS (MOS) subject classifications (1970). Primary 28A65, 43A65, 43A25. 1 Supported by NSF grant GP-25736. 2 Supported by NSF grant GP-37771. © American Mathematical Society 1974 423 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 424 JULIUS BLUM AND BENNETT EISENBERG [February A sequence of probability measures pn is called a generalized summing sequence if, for every unitary representation U on H and every/in H, J" Ugfdp.n converges in mean to an invariant element of H. Theorem 1. The following are equivalent: (i) pn is a generalized summing sequence. (ii) For every character x on G not equal to the identity the Fourier transforms fin(x) converge to 0. (iii) pn considered as restrictions of measures on the Bohr compactification G of G converge weakly to Haar measure on G. (iv) For every character x of infinite order, the measures pxn induced by x on the unit circle in the complex plane converge weakly to normalized Lebesgue measure on the circle, and, for every character x of order m, m=0, 1, 2, • • • , the measures pxn converge weakly to Haar measure on the mth roots of unity. Proof, (ii) implies (i). Assume that fin(x)-*0 for every character x not the identity. Let U be any unitary representation of G and let Pf be the projection of / on the subspace of H invariant under U. Then J UgPfdpn(g) equals Pf for all n and hence converges to Pf. \ju,(f Pf) dpn 2= (J"i/„(/ pf) dpn, juh(f pf) 4u„) \(Ug(f Pf), Uh(f Pf)) dpn(g) dpn(h), -if< by (2). Denote (Ug(f-Pf),f-Pf) by R(g). Then by Stone's theorem, R(g)=)(g,x)dF(x), where dF(A)=(EA(f-Pf),f-Pf) and EA is the spectral resolution of U over the dual group G. Hence |/wrn<ip,T -Ii Ja Jc R(gh-1) dpn(g) dpn(h) o = f f ¡/fgh'\x)dF(x)dMn(g)dpn(h) JgJgJu -f f \(g,x)(h-\x)dpn(g)dpn(h)dF(x) + [ dF(x) -i 1/ Jx*l \J( (g, x) dfin(g) dF(x) + dF(\). By dominated convergence and assumption (ii) this converges to dF(l). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1974] SUMMING SEQUENCES AND THE MEAN ERGODIC THEOREM 425 But Ex is the projection on the subspace of H invariant under U, and since fi-Pf is orthogonal to this subspace, dF(l)=Q. Thus J Ug(f-Pf) dpn converges to 0 in mean. Putting the pieces together f UJdpn=j UgPfdpn+} Us(fi-Pfi) dpn converges to 7/in mean. (i) implies (ii). Assume ßn(x) does not converge to 0, where x is not identically one on G. Consider the unitary representation, where Ug is multiplication by (g, x) on some Hubert space H. Since (g, x) is not identically one, Pf=0 for all/in //. But jujdpn(g) 2=jy<g, x)(h-\ x) ii/ii2 dpn(g) dpn(h)