A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we characterise up to isomorphism the countable subgroups Δ of the unitary group of a separable infinite dimensional Hilbert space ℋ of which the bicommutants Δ'' (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on ℋ.
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Bachir Bekka, Pierre de la Harpe, Irreducibly represented groups. Comment. Math. Helv. 83 (2008), no. 4, pp. 847–868DOI 10.4171/CMH/145