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Unitary representations of the group \(SO_ 0(\infty,\infty)\) as limits of unitary representations of the groups \(SO_ 0(n,\infty)\) as n\(\to \infty\). (English. Russian original) Zbl 0628.22009
Funct. Anal. Appl. 20, 292-301 (1986); translation from Funkts. Anal. Prilozh. 20, No. 4, 46-57 (1986).
The author introduced the (G,K)-pair for infinite-dimensional groups in 1983. The (G,K)-pair is said to belong to type I when the von-Neumann algebra generated by its admissible representations is of type I.
In the present paper, let \(G=SO(\infty,\infty)\) and \(K=SO(\infty)\times SO(\infty)\). The following results are proved:
1) The pair (G,K) belongs to type I and some completion of G is a type I topological group.
2) In the decomposition of irreducible admissible representations of G with respect to K, multiplicities are finite.
3) Approximation of irreducible admissible representations of G by those of SO(n,\(\infty).\)
Also a formula for spherical functions is given for the representations with an SO(\(\infty,\infty)\)-invariant vector.
Reviewer: Y.Asovo

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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