## 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.

Zbl 0782.30035
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