Positive definite radial functions on free product of groups. (English) Zbl 0658.43004

Consider the free product \(G=G_ 1*G_ 2*...*G_ n\) of n discrete groups \(G_ j\). Every \(x\neq e\) in G can be uniquely written as \(x=g_ 1g_ 2...g_ n\) with \(g_ i\in G_{j_ i}\), \(g_ i\neq e:\) \(n\equiv \| x\|\) is called the length of x. By a result of M. Bożejko [Boll. Unione Mat. Ital., VI. Ser. A 5, 13-21 (1986; Zbl 0591.43009)], the exponential \(P_ r(x)=r^{\| x\|}\) is positive definite if \(0<r\leq 1\). The present paper proves that \(P_ r\) belongs to the Fourier-Stieltjes algebra B(G) if \(-1/(n-1)<r\leq 1\), but is not irreducible if \(0<r<1\). The case of the free product \(G_{k,n}\) of n copies of the cyclic group \({\mathbb{Z}}_ k\) has been previously studied by A. Iozzi and the reviewer [Lect. Notes Math. 992, 344-386 (1983; Zbl 0535.43005)], who introduced and studied the so-called spherical functions. The author shows that a spherical function on \(G_{k,n}\) is positive definite if and only if its value \(\gamma\) on words of length 1 satisfies the inequalities \(-1/(n-1)\leq \gamma \leq 1:\) this corrects a wrong statement in the last reference quoted above. An important tool is the fact that the radial expectation maps positive definite functions onto positive definite functions: this fact, observed in the last reference, is proved here in full generality.
Reviewer: M.Picardello


43A35 Positive definite functions on groups, semigroups, etc.
43A85 Harmonic analysis on homogeneous spaces
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations