Partially defined semigroups relative to multiplicative free convolution.

*(English)*Zbl 1092.46046The 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.

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.

Reviewer: Debashish Goswami (Kolkata)

##### MSC:

46L54 | Free probability and free operator algebras |