Partially defined semigroups relative to multiplicative free convolution

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 exi...

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Published inInternational Mathematics Research Notices Vol. 2005; no. 2; pp. 65 - 101
Main Authors Belinschi, S. T., Bercovici, H.
Format Journal Article
LanguageEnglish
Published Hindawi Publishing Corporation 01.01.2005
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Abstract 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.
AbstractList 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.
Author Belinschi, S. T.
Bercovici, H.
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Title Partially defined semigroups relative to multiplicative free convolution
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