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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)
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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