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Let $\tau_1,\tau_2 $ be topologies on group $G$ such that $(G,\tau_1),(G,\tau_2)$ be a locally compact group. Let $\tau_1\subseteq\tau_2$ and $(G,\tau_2)$ be an amenable group, when $(G,\tau_1)$ is amenable?

(Particularly, let $\tau_2 $ be the discrete topology on $G$, then $l^1(G)$ be amenable. When $L^1(G,\tau_1)$ is amenable?)

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    $\begingroup$ The identity map $(G,T_2)\to (G,T_1)$ is continuous. Amenability passes to targets of homomorphisms with dense image (e.g., by the fixed point criterion). Hence amenability of $(G,T_2)$ implies that of $(G,T_1)$. $\endgroup$
    – YCor
    Feb 2, 2017 at 17:52
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    $\begingroup$ @YCor has answered your question. If you are more used to invariant means than fixed point arguments, then note that we have $C_b(G,\tau_2) \to C_b(G,\tau_1)$ and so a left-invariant mean on $C_b(G,\tau_1)$ gives a left-invariant mean on $C_b(G,\tau_2)$ $\endgroup$
    – Yemon Choi
    Feb 12, 2017 at 20:37

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