Well, in the case X is also a group, your question somehow relates with cellular automata, i.e your map \phi $\phi :G^X\rightarrow G^XG^X$ is continuous and X-equivariant. When X is amenable,\phi$\phi$ is pre-injective if and only if \phi$\phi$ is surjective. Thus, your dual \widehat{\phi}$\widehat{\phi}$ will be injective in this case. It is not true anymore when X=F_2$X=F_2$, the free group with 2 generators, see section 5.11 of Ceccherini-Silberstein, Coornaert's book "Cellular Automata and Groups" for an example of pre-injective but not surjective cellular automaton over F_2$F_2$.
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Well, in the case X is also a group, your question somehow relates with cellular automata, i.e your map |
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