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Let G = S∞(T ) be the wreath product of a compact group T with the infinite symmetric group S∞. We study the characters of factor representations of finite type of G, and give a formula which expresses all the characters explicitly. 1. Characters of factor representations of finite type 1.1. Characters. Let G be a Hausdorff topological group, P(G) the set of continuous positive definite functions on G, K(G) the set of f ∈ P(G) which are invariant under inner automorphisms, K1(G) the set of f ∈ K(G) normalized as f(e) = 1 at the identity element e ∈ G, and E(G) the set of extremal points in the convex set K1(G). Let π1 and π2 be two continuous unitary representations (= URs) of G, and Ui = πi(G) ′′ (i = 1, 2) the von Neumann algebra generated by πi(G). We say that π1 and π2 are quasi-equivalent if there exists an isomorphism Φ from U1 onto U2 as ∗-algebras such that Φ(π1(g)) = π2(g) for g ∈ G. Theorem 1.1 ([HH3, Theorems 1.5.4 and 1.6.1]). Let π be a continuous unitary representation (= UR) of G such that the von Neumann algebra U = π(G) has a faithful normal finite trace t on the set U of positive elements in U. Normalize t as t(I) = 1 2000 Mathematics Subject Classification: Primary 22A25; Secondary 20C32, 22D25, 46L10.

Character formula for wreath products of compact groups with the infinite symmetric group