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Random walks on free products. (English) Zbl 0725.60009
Let $$G=*^{q+1}_{j=1}G_{n_ j+1}$$ be the free product of $$q+1$$ finite groups each having order $$n_ j+1$$. Choose $$q+1$$ positive numbers $$p_ j$$ with $$\sum^{q+1}_{j=1}p_ j=1$$. Consider the probability measure $$\mu$$ which takes the value $$p_ j/n_ j$$ on each element of $$G_{n_ j+1}\setminus e$$. In this paper we shall describe the point spectrum of $$\mu$$ in $$C^*_{reg}(G)$$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $$n_ j$$. We also compute the continuous spectrum of $$\mu$$ in $$C^*_{reg}(G)$$ in several cases. A family of irreducible representations of G, parametrized on the continuous spectrum of $$\mu$$, is here presented. Finally, we shall get a decomposition of the regular representation of G by means of the Green function of $$\mu$$ and the decomposition is into irreducibles if and only if there are no true eigenspaces for $$\mu$$.
Reviewer: G.Kuhn (Milano)
##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G50 Sums of independent random variables; random walks 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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