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# Weak Convergence Is Not Strong Convergence For Amenable Groups

Published:2001-06-01
Printed: Jun 2001
• Joseph M. Rosenblatt
• George A. Willis
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## Abstract

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 - Means on groups, semigroups, etc.; amenable groups