Random walks on free products

M. Gabriella Kuhn

Annales de l'institut Fourier (1991)

  • Volume: 41, Issue: 2, page 467-491
  • ISSN: 0373-0956

Abstract

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Let G = * j = 1 q + 1 G n j + 1 be the product of q + 1 finite groups each having order n j + 1 and let μ be the probability measure which takes the value p j / n j on each element of G n j + 1 ∖ { e } . In this paper we shall describe the point spectrum of μ 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 μ in C reg * ( G ) in several cases. A family of irreducible representations of G , parametrized on the continuous spectrum of μ , is here presented. Finally, we shall get a decomposition of the regular representation of G by means of the Green function of μ and the decomposition is into irreducibles if and only if there are no true eigenspaces for μ .

How to cite

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Kuhn, M. Gabriella. "Random walks on free products." Annales de l'institut Fourier 41.2 (1991): 467-491. <http://eudml.org/doc/74925>.

@article{Kuhn1991,
abstract = {Let $G=*^\{q+1\}_\{j=1\}G_\{\{n_j\}+1\}$ be the product of $q\{+\}1$ finite groups each having order $n_ j\{+\}1$ and let $\mu $ be the probability measure which takes the value $p_ j/n_ j$ on each element of $G_\{\{n_j\}+1\}\setminus \lbrace e\rbrace $. In this paper we shall describe the point spectrum of $\mu $ in $C^*_\{\rm reg\}(G)$ 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 $C^*_\{\rm reg\}(G)$ 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 $.},
author = {Kuhn, M. Gabriella},
journal = {Annales de l'institut Fourier},
keywords = {free products; point spectrum; irreducible representations; decomposition of the regular representation},
language = {eng},
number = {2},
pages = {467-491},
publisher = {Association des Annales de l'Institut Fourier},
title = {Random walks on free products},
url = {http://eudml.org/doc/74925},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Kuhn, M. Gabriella
TI - Random walks on free products
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 2
SP - 467
EP - 491
AB - Let $G=*^{q+1}_{j=1}G_{{n_j}+1}$ be the product of $q{+}1$ finite groups each having order $n_ j{+}1$ and let $\mu $ be the probability measure which takes the value $p_ j/n_ j$ on each element of $G_{{n_j}+1}\setminus \lbrace e\rbrace $. In this paper we shall describe the point spectrum of $\mu $ in $C^*_{\rm reg}(G)$ 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 $C^*_{\rm reg}(G)$ 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 $.
LA - eng
KW - free products; point spectrum; irreducible representations; decomposition of the regular representation
UR - http://eudml.org/doc/74925
ER -

References

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  14. [S] T. STEGER, Harmonic analysis for anisotropic random walks on homogeneous trees, Doctoral Dissertation, Washington University, St. Louis. Zbl0836.43019
  15. [T1] A. R. TRENHOLME, Maximal abelian subalgebras of function algebras associated with free products, J. Funct. Anal., 79 (1988), 342-350. Zbl0665.46049MR90c:46073
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