Partially defined semigroups relative to multiplicative free convolution

• Published in: International Mathematics Research Notices Vol. 2005; no. 2; pp. 65 - 101
• Main Authors: Belinschi, S. T., Bercovici, H.
• Format: Journal Article
• Language: English
• Published: Hindawi Publishing Corporation 01.01.2005
• ISSN: 1073-7928; 1687-1197
• DOI: 10.1155/IMRN.2005.65

Nica and Speicher have shown that every probability measure on the line belongs to a partial semigroup {μt : t ≥ 1} relative to additive free convolution (i.e., μt+s=μt⊞μs for t, s≥1). We prove analogous results for multiplicative free convolution on the positive half-line and on the circle. The existence of semigroups is derived from certain global inversion results for analytic functions defined on the disk or on the slit complex plane. A close analysis of the global inverses yields regularity results for the measures in these semigroups. We also use this analysis to improve the known results for the additive situation.