On irreducible decompositions of the regular representation of free groups.

*(English)*Zbl 0586.22004Let G be a free group with a finite number of generators. Irreducible decompositions for the regular representation of G were obtained by H. Yoshizawa [Osaka Math. J. 3, 55-63 (1951; Zbl 0045.301)] and, recently, by S. Kawakami [Math. Jap. 28, 337-340 (1983; Zbl 0531.22007)]. Another decomposition of the regular representation into irreducible ones has been obtained by T. Pytlik [J. Reine Angew. Math. 326, 124-135 (1981; Zbl 0464.22004)] and by A. FigĂ - Talamanca and the reviewer [J. Funct. Anal. 47, 281-304 (1982; Zbl 0489.43008)].

The aim of the present paper is to construct a family of irreducible decompositions of the regular representation of G which interpolate between Yoshizawa and Pytlik’s decompositions. It turns out that every two decompositions of the family constructed here are disjoint. The construction is based upon the induction process for \(C^*\)-algebras, introduced by M. A. Rieffel [Adv. Math. 13, 176-257 (1974; Zbl 0284.46040)]. Free groups over an infinite set of generators are also considered.

The aim of the present paper is to construct a family of irreducible decompositions of the regular representation of G which interpolate between Yoshizawa and Pytlik’s decompositions. It turns out that every two decompositions of the family constructed here are disjoint. The construction is based upon the induction process for \(C^*\)-algebras, introduced by M. A. Rieffel [Adv. Math. 13, 176-257 (1974; Zbl 0284.46040)]. Free groups over an infinite set of generators are also considered.

Reviewer: M.Picardello

##### MSC:

22D10 | Unitary representations of locally compact groups |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

22D30 | Induced representations for locally compact groups |

46L05 | General theory of \(C^*\)-algebras |

20E05 | Free nonabelian groups |