Weak convergence is not strong convergence for amenable groups (Q2732643)

Let \(G\) be an infinite discrete amenable group or a non-discrete locally compact amenable group. It is well known that there exists a net \(\{f_\alpha\}\) of positive, normalized functions in \(L_1(G)\) such that the net is weak* convergent to invariance but does not converge strongly to invariance. In this paper it is shown how to construct actually such nets by considering some aspects of Cayley directed graphs on the group which are equivalent to amenability of the group. By the method of the construction of such nets the following theorem is proved. Theorem. There exists a net \(\{f_\alpha\}\) in \(P(G)\) which is weak* convergent to invariance, such that for every \(x\in G\), \(x\neq e\), \(\|f_\alpha-_xf_\alpha\|_1=2\).