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Harmonic analysis on the free product of two cyclic groups. (English) Zbl 0619.43003
Radial harmonic analysis on finitely generated free groups and on free products of a finite number of finite groups of the same order has been studied in several recent papers [see appropriate references in A. Figà-Talamanca and the reviewer, Harmonic analysis on free groups (Lect. Notes Pure Appl. Math. 87) (1983; Zbl 0536.43001); A. Iozzi and the reviewer, Lect. Notes Math. 992, 342-386 (1983; Zbl 0535.43005); J. M. Cohen and A. R. Trenholme, J. Funct. Anal. 59, 175-184 (1984; Zbl 0549.43002); G. Kuhn and P. M. Soardi, Ann. Mat. Pura Appl., IV. Ser. 134, 399-401 (1983; Zbl 0536.43021); A. Trenholme, ”Maximal abelian subalgebras of function algebras associated with free products”, in print in J. Funct. Anal.; J. Faraut and the reviewer, Ann. Mat. Pura Appl., IV. Ser. 138, 151-155 (1984; Zbl 0565.43005)].
More recently, similar problems have been studied in the non-radial setting [K. Aomoto, J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 297- 318 (1984; Zbl 0583.60068); ”Algebraic equations for Green kernels on a free group”, to appear; D. I. Cartwright and P. M. Soardi, Nagoya Math. J. 102, 163-180 (1986; Zbl 0581.60055); A. Figà- Talamanca and T. Steger, ”Harmonic analysis for anisotropic random walks on homogeneous trees”, in print in Mem. Am. Math. Soc.; W. Woess, Boll. Unione Mat. Ital., VI. Ser., B 5, 961-982 (1986)].
Radial harmonic analysis is primarily concerned with the spectral resolution of the convolution operator by the characteristic function \(\chi_ 1\) of the set of words of length one (that is, the generators and their inverses in a free group, or the nontrivial elements of the factor subgroups in a free product). The present paper studies the free product of two finite groups of different orders (for simplicity, the results are stated for the free product of two cyclic groups, \(G=({\mathbb{Z}}/k)*({\mathbb{Z}}/r))\). In this context, the convolution algebra \({\mathcal A}_ 1\) generated by \(\chi_ 1\) is not radial (that is, its elements are functions on G which can attain different values on words of the same length).
\({\mathcal A}_ 1\) is a subalgebra of the noncommutative finitely generated algebra \({\mathcal A}\) of ”semiradial” functions: for each \(n>0\), these functions are constant on the sets of all words of length n whose first ”letter” belongs to the first (respectively, the second) factor. It turns out that the closure \({\mathcal A}^*_ 1\) of \({\mathcal A}_ 1\) in the regular \(C^*\)-algebra of G is not maximal abelian; this should be compared with the results of the paper of A. Trenholme quoted above.
Another significant result is the computation of the Plancherel measure of \({\mathcal A}^*_ 1\). In particular, the \(\ell^ 2\)-spectrum of \(\chi_ 1\) is shown to consist of two intervals and two discrete points. The orthogonal polynomials associated with the Plancherel measure are computed. Unlike the previously known cases, here they satisfy a recurrence relation with non-constant coefficients; this fact makes spectral theory considerably harder. Nevertheless, the authors are able to compute the spherical functions (that is, the multiplicative functionals of the commutative algebra \({\mathcal A}_ 1).\)
The fact that this pretty and complete analysis can be carried out for such an intractable convolution algebra is surprising. Conceivably, there should be a deep underlying reason to explain why. However, if an explanation exists, it is certainly difficult to grasp, and the arguments in this paper, based upon long and patient (though smart) computations, do not always offer a good insight.
Reviewer: M.Picardello

MSC:
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A90 Harmonic analysis and spherical functions
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L05 General theory of \(C^*\)-algebras
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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