## 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

### MSC:

 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

### Citations:

Zbl 0591.43009; Zbl 0535.43005