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

Kajiwara, Tsuyoshi

Boll. Unione Mat. Ital., VI. Ser., A 4, 425-431 (1985).

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

Keywords:

unitary representations; unitary equivalence; irreducible representations; direct integral decomposition; completely different decompositions; free group; finite number of generators; regular representation; induction process; \(C^*\)-algebras