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Irreducible representations of free products of infinite groups. (English) Zbl 0841.43015

Let \(G\) be a free product \(*_{i \in I} G_i\) of discrete groups. For \(x = g_1 \dots g_n \in G\), such that \(n \geq 0\), \(g_k \in G_{i_k} \backslash \{e\}\), \(i_k \neq i_{k + 1}\), we define its type \(t(x) = i_1 \dots i_n\). A function \(f\) on \(G\) is said to be type-dependent if \(f(x) = f(y)\) whenever \(t(x) = t(y)\). Denote by \(S\) the set of all types. We regard \(S\) as a unital *-semigroup generated by \(I\) and defined by the relations \(ii = i^* = i\) for \(i \in I\). For every *-representation \(\sigma\) of \(S\), acting on a Hilbert space \(H_0\), we construct a unitary representation \(\pi\) of \(G\), acting on \(H \supset H_0\), such that for every \(x \in G\), \(\xi \in H_0\) we have \(P_0 \pi (x) \xi = \sigma (t(x)) \xi\) \((P_0\) denotes the orthogonal projection of \(H\) onto \(H_0)\). Moreover, if \(\sigma\) is irreducible and nontrivial (i.e. \(\sigma (u) \neq 0\) for some \(u \in S \backslash \{e\})\) and all \(G_i\)’s are infinite then \(\pi\) is irreducible too. This means, in terms of positive definite functions, that if all \(G_i\)’s are infinite then a type-dependent function \(\varphi \circ t\) is (extreme) positive definite on \(G\) if and only if \(\varphi\) is (extreme, not equal to \(c \delta_e)\) positive definite on \(S\).
In the last part we find, for finite \(I\), an analytic family \(\{\sigma_z\}_{|z |< 1}\), of uniformly bounded representations of \(S\), which leads to such a family \(\{\pi_z\}_{|z |< 1}\) of representations of \(G\). We show that the representations \(\pi_z\) are equivalent to those studied by J. Wysoczánski [Pac. J. Math. 157, 373-387 (1993; Zbl 0782.30035)].

MSC:

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A35 Positive definite functions on groups, semigroups, etc.

Citations:

Zbl 0782.30035
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