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Partially defined semigroups relative to multiplicative free convolution. (English) Zbl 1092.46046
The purpose of the paper under review is to prove existence of continuous convolution powers of a Borel probability measure (on $$\mathbb R_+$$ or the unit circle $${\mathbb T}$$) with respect to free multiplicative convolution. Such a result for free additive convolution was already proven by the author and D. Voiculescu [Probab. Theory Relat. Fields 103, No. 2, 215–222 (1995; Zbl 0831.60036)], and also by A. Nica, R. Speicher [Am. J. Math. 118, No. 4, 799–837 (1996; Zbl 0856.46035)].
For two such measures $$\mu$$ and $$\nu$$, let us denote by $$\mu \boxplus \nu$$ and $$\mu \boxtimes \nu$$ their free additive and free multiplicative convolution respectively. For a Borel probability measure $$\mu$$ on $$\mathbb R_+$$ which is not $$\delta_0$$, an analytic function $$\Sigma_\mu$$ is defined on the domain $$\Omega={\mathbb C}- \mathbb R_+$$ which satisfies $$z \Sigma_\mu(z)=\eta_\mu^{-1}(z)$$, where $$\eta_\mu(z):=\frac{\Psi_\mu(z)}{1+\Psi_\mu(z)},$$ with $$\Psi_\mu(z):=\int \frac{tz}{1-tz} \,d \mu(t)$$ (the integral is over all $$t$$ for which the integrand makes sense). It can be shown that there is a unique analytic function $$S_\mu$$ (with suitable domain) which satisfies $$\Psi_\mu(\frac{z}{z+1}S_\mu(z))=z$$ and this $$S_\mu$$ plays an important role in the context of free multiplicative convolution, since it satisfies $$S_{\mu \boxtimes \nu}=S_\mu S_\nu$$.
The fundamental result proved in the present paper is the existence of a family of Borel probability measures $$\mu_t$$ (also denoted by $$\mu^{\boxtimes^t}$$), for $$t \geq 1$$, satisfying $$S_{\mu_t}=S_\mu^t$$. In terms of the function $$\Sigma$$, this translates into $$\Sigma_{\mu_t}(z)={\Sigma_\mu(z)}^t$$ for $$t \geq 1$$ and for all negative $$z$$ sufficiently close to $$0$$. Moreover, it is shown that there exist analytic functions $$\omega_t$$ from $$\Omega$$ to $$\Omega$$ such that $$\eta_{\mu_t}(z)=\eta_\mu(\omega_t(z))$$ for all $$z \in \Omega$$.
A similar existence theorem is proved for Borel probability measures of $$\mathbb T$$, but for $$t \geq 2$$. Some results about atoms and regularity of $$\mu^{\boxplus^t}$$ as well as $$\mu^{\boxtimes^t}$$ are also given. The paper also contains a number of results about analytic and meromorphic functions of one complex variable which are very interesting on their own.

##### MSC:
 46L54 Free probability and free operator algebras
##### Keywords:
free probability; free convolution
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