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On irreducible decompositions of the regular representation of free groups. (English) Zbl 0586.22004
Let 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. Figà- 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.
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