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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.

46L54 Free probability and free operator algebras
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