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Weak Convergence Is Not Strong Convergence For Amenable Groups

Open Access article
 Printed: Jun 2001
  • Joseph M. Rosenblatt
  • George A. Willis
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Let $G$ be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net $(f_\alpha)$ of positive, normalized functions in $L_1(G)$ such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.
MSC Classifications: 43A07 show english descriptions Means on groups, semigroups, etc.; amenable groups 43A07 - Means on groups, semigroups, etc.; amenable groups

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