Sequential convergence to invariance in 𝐵𝐶(𝐺)

Abstract

In this note it is shown that weak and strong convergence to invariance are equivalent for a sequence of probabilities acting on B C ( G ) BC(G) of a noncompact locally compact group. This result was known for G = Z G = Z . For other generalizations of the bounded sequences on Z Z , say B U C ( G ) BUC(G) , the result does not hold.

Weak Convergence Is Not Strong Convergence For Amenable Groups

“…Despite so much having been written on amenable groups, there does not seem to have been a construction of nets which are weak* converging to invariance but not converging strongly to invariance. There is something to prove here since Sine [7] showed that for sequences, weak* convergence to invariance actually does imply strong convergence to invariance. This article will construct nets which are weak* converging to invariance but not converging strongly to invariance, by considering some aspects of Cayley graphs on the group which are equivalent to amenability of the group.…”

“…There are seven configurations corresponding to the partition, namely: C (1) = (1, 1, 2), C (2) = (2, 1, 2), C (3) = (3, 1, 2), C (4) = (3, 3, 2), C (5) = (3, 2, 2), C (6) = (3, 1, 3) and C (7) = (3, 1, 1). These configurations are shown in Figure 3.…”

Modulated and subsequential ergodic theorems in Hilbert and Banach spaces

“…In fact, the same is true for the convergence of nets {µ α } (by combining Theorem 1.2.5, Corollary 4.8.11 and Lemma 4.8.11 of [28]). However, we have the result from [22], that weak convergence to invariance of nets does not imply strong convergence (in any amenable locally compact group), although for sequences it does [26]. This of course raises the question for weakly almost periodic functions: If {µ n } is an averaging sequence of probabilities, and µ n * f → M(f ) pointwise for every f ∈ WAP(Z), is the convergence uniform for every f ∈ WAP(Z)?…”