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A note on regularity for free convolutions. (English) Zbl 1107.46043
Summary: Let \(\mu\boxplus\nu\) and \(\mu\boxtimes\nu\) denote the free additive convolution and the free multiplicative convolution of the Borel probability measures \(\mu\) and \(\nu\), respectively. We analyze the boundary behavior of the functions \(G_{\mu\boxplus\nu}(z)=\int\frac {1}{z-t}d(\mu\boxplus\nu)(t)\) and \(\psi_{\mu \boxtimes\nu}(z)=\int \frac{zt}{1-zt}d(\mu\boxtimes\nu)(t)\). We prove that, under certain conditions, these functions extend continuously to the boundary of their natural domains as functions with values in the extended complex plane \(\mathbb{C} \cup\{\infty\}\). As a consequence, we obtain that \(\mu\boxplus \nu\) (respectively, \(\mu\boxtimes\nu)\) can never be purely singular, unless \(\mu\) or \(\nu\) is concentrated in one point.

MSC:
46L54 Free probability and free operator algebras
30D40 Cluster sets, prime ends, boundary behavior
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