Discussion
Clearly, definition (4) implies definition (3). Let us show the converse, assuming local compactness. So let G be a locally compact group satisfying (3). By Theorem 1.3.1 of Bekka et al., G is compactly generated. Therefore, Remark 1.1.2(v) of Bekka et al. tells us the following. If we take K to be a compact generating set of G, and let ε be any positive real number, then a unitary representation of G having an (ε, K)-invariant unit vector has (ε', K ')-invariant unit vectors for every ε' > 0 and K ' compact. Therefore, by (3), such a representation of G will have a nonzero invariant vector, establishing (4).
The equivalence of (4) and (5) (Property (FH)) is the Delorme-Guichardet Theorem. The fact that (5) implies (4) requires us to assume that G is σ-compact (and locally compact) (Bekka et al., Theorem 2.12.4).
Read more about this topic: Kazhdan's Property (T)
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