Hyperfinite approximations of compact groups and their representations (Q1921779)

The concept of approximating topological abelian groups by finite groups that has been introduced earlier [see \textit{D. Handelman}, Ergodic Theory Dyn. Syst. 6, 57-79 (1986; Zbl 0748.28009); \textit{E. I. Gordon}, Acta Appl. Math. 25, 221-239 (1991; Zbl 0769.26009), Hyperfinite approximations of commutative topological groups, in: S. A. Albeverio (ed.), Advances in analysis, probability and mathematical physics: Contributions of nonstandard analysis. Math. Appl., Dordrecht 314, 37-45 (1995; Zbl 0830.43019)]\ is generalized to arbitrary topological groups. It is then proved that irreducible unitary representations of compact groups are approximated in some sense by those of the corresponding finite groups. This result is a rather new procedure for constructing irreducible representations of groups that admit such finite-group approximation. Locally compact abelian groups, profinite and inductive limits of finite groups are particularly referred to this class. The procedure given is illustrated by the example of the semidirect product of the additive group of the ring of \(p\)-adic numbers and the multiplicative group of the invertible elements of this ring.