Kazhdan's property \((T)\) and amenable representations (Q1319292)

Let \(G\) be a separable locally compact group. It is shown that a relative version of property \((T)\) involving tensor products of representations of \(G\) is sufficient to imply Kazhdan's property \((T)\) of \(G\). As a consequence, groups with property \((T)\) are characterized by the fact that their amenable representations (in the sense of \textit{M. E. B. Bekka} [Invent. Math. 100, 383-401 (1990; Zbl 0702.22010)]) contain finite dimensional subrepresentations. New and simple proofs of some previously known results about property \((T)\) are deduced. Moreover, it is shown that simple noncompact Lie groups have no nontrivial amenable irreducible representations.