Let G = Zr * Zx = 〈 a, b | ar = bs = e 〉, where r > s ≥ 2. A natural length is defined for the elements of G and χ1 denotes the characteristic function of the set of elements of length 1. We study the convolution C*-algebra A(χ1) generated by χ1. We obtain explicitly the associated Plancherel measure, orthogonal polynomials, and spherical functions, and show that A(χ1) is not a maximal abelian subalgebra of the reduced group C*-algebra of G. Closely related to work by Figà-Talamanca, Picardello, Pytlik, and others on free groups, this study is complicated by the fact that A(χ1) does not consist of radial functions. © 1986.
Cartwright, D., Soardi, P. (1986). Harmonic analysis on the free product of two cyclic groups. JOURNAL OF FUNCTIONAL ANALYSIS, 65(2), 147-171.
Harmonic analysis on the free product of two cyclic groups
SOARDI, PAOLO MAURIZIO
1986
Abstract
Let G = Zr * Zx = 〈 a, b | ar = bs = e 〉, where r > s ≥ 2. A natural length is defined for the elements of G and χ1 denotes the characteristic function of the set of elements of length 1. We study the convolution C*-algebra A(χ1) generated by χ1. We obtain explicitly the associated Plancherel measure, orthogonal polynomials, and spherical functions, and show that A(χ1) is not a maximal abelian subalgebra of the reduced group C*-algebra of G. Closely related to work by Figà-Talamanca, Picardello, Pytlik, and others on free groups, this study is complicated by the fact that A(χ1) does not consist of radial functions. © 1986.
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