Harmonic analysis on the free product of two cyclic groups.

*(English)*Zbl 0619.43003Radial 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.

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 |

##### Keywords:

radial functions; semi-radial functions; spectrum of convolution operator; commutative \(C^ *\)-algebra; finitely generated free groups; free products; orthogonal polynomials; Plancherel measure; spherical functions
PDF
BibTeX
XML
Cite

\textit{D. I. Cartwright} and \textit{P. M. Soardi}, J. Funct. Anal. 65, 147--171 (1986; Zbl 0619.43003)

Full Text:
DOI

##### References:

[1] | Akhiezer, N.I, The classical moment problem, (1965), Oliver & Boyd Edinburgh/London · Zbl 0135.33803 |

[2] | Betori, W; Pagliacci, M, Harmonic analysis for groups acting on trees, Boll. un. mat. ital., 3-B, 333-350, (1984), (6) · Zbl 0579.43014 |

[3] | Cartier, P, Harmonic analysis on trees, (), 419-424 · Zbl 0309.22009 |

[4] | Cohen, J.M; Trenholme, A.R, Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis, J. funct. anal., 59, 175-184, (1984) · Zbl 0549.43002 |

[5] | \scJ. Faraut and M. A. Picardello, The Plancherel measure for symmetric graphs, preprint. · Zbl 0565.43005 |

[6] | Figà-Talamanca, A; Picardello, M.A, Spherical functions and harmonic analysis on free groups, J. funct. anal., 47, 281-304, (1982) · Zbl 0489.43008 |

[7] | Figà-Talamanca, A; Picardello, M.A, Harmonic analysis on free groups, () · Zbl 0536.43001 |

[8] | Gerl, P, Continued fractions methods for random walks on \(N\) and on trees, (), in press |

[9] | \scP. Gerl and W. Woess, Simple random walks on trees, preprint. · Zbl 0606.05021 |

[10] | Iozzi, A, Harmonic analysis on the free product of two cyclic groups, Boll. un. mat. ital., 4-B, 167-177, (1985), (6) |

[11] | Iozzi, A; Picardello, M.A, Spherical functions on symmetric graphs, (), 344-387 |

[12] | \scA. Iozzi and M. A. Picardello, Graphs and convolution operators, in “Topics in Harmonic Analysis,” Istituto Nazionale di Alta Matematica, in press. |

[13] | Karlin, S; McGregor, J, Random walks, Illinois J. math., 3, 66-81, (1959) · Zbl 0104.11804 |

[14] | Kuhn, G; Soardi, P.M, The Plancherel measure for polygonal graphs, Ann. mat. pura appl., 144, 393-401, (1983) · Zbl 0536.43021 |

[15] | Magnus, W; Karrass, A; Solitar, D, Combinatorial grup theory, (1976), Dover New York |

[16] | Pedersen, G.K, \(C\^{}\{∗\}\)-algebras and their automorphism groups, (1979), Academic Press London/New York/San Francisco |

[17] | Picardello, M.A, Spherical functions and local limit theorems on free groups, Ann. mat. pura appl., 143, 177-191, (1983) · Zbl 0527.60011 |

[18] | Pytlik, T, Radial functions on free groups and a decomposition of the regular representation into irreducible components, J. reine angew. math., 326, 124-135, (1981) · Zbl 0464.22004 |

[19] | Sawyer, S, Isotropic random walks in a tree, Z. wahrsch., 12, 279-292, (1978) · Zbl 0362.60075 |

[20] | Szegö, G, Orthogonal polynomials, () · JFM 65.0278.03 |

[21] | \scA. R. Trenholme, Maximal abelian subalgebras of function algebras associated with free products, preprint. · Zbl 0665.46049 |

[22] | Woess, W, Random walks and periodic continued fractions, Adv. appl. probab., 17, 67-84, (1985) · Zbl 0554.60069 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.