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It might be better to rewrite this question as suggested below: Let $g$ be a non-identity element in a torsion-free amenable group, does there exist an unitary representation $\pi$ with $\pi(g)\neq 1$?

Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?

(The word "finite-dimensional" was initially omitted: as mentioned in the comments the answer is a trivial "yes" then, by considering the left regular representation.)

It might be better to rewrite this question as suggested below: Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?

It might be better to rewrite this question as suggested below: Let $g$ be a non-identity element in a torsion-free amenable group, does there exist an unitary representation $\pi$ with $\pi(g)\neq 1$?

It might be better to rewrite this question as suggested below: Let $g$ be a non-identity element in a torsion-free amenable group, does there exist an unitary representation $\pi$ with $\pi(g)\neq 1$?

It might be better to rewrite this question as suggested below: Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?