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Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

Authors: Yves Guivarc'h and Riddhi Shah
Journal: Trans. Amer. Math. Soc. 357 (2005), 3683-3723
MSC (2000): Primary 60B15, 60F05, 60G50; Secondary 43A05, 22D25, 22D40
Published electronically: February 4, 2005
MathSciNet review: 2146645
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Abstract: We consider a locally compact group $G$ and a limiting measure $\mu$of a commutative infinitesimal triangular system (c.i.t.s.) $\Delta$of probability measures on $G$. We show, under some restrictions on $G$, $\mu$ or $\Delta$, that $\mu$ belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. $\Delta$ on any locally compact group $G$. It is also valid for a limiting measure $\mu$ which has `full' support on a Zariski connected $\mathbb{F}$-algebraic group $G$, where $\mathbb{F}$ is a local field, and any one of the following conditions is satisfied: (1) $G$ is a compact extension of a closed solvable normal subgroup, in particular, $G$ is amenable, (2) $\mu$ has finite one-moment or (3) $\mu$ has density and in case the characteristic of $\mathbb{F}$ is positive, the radical of $G$ is $\mathbb{F}$-defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group $G$, we show that it is always positive for any probability measure on $G$, and it is also multiplicative in case of symmetric commuting measures.

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Additional Information

Yves Guivarc'h
Affiliation: IRMAR, Université de Rennes1, Campus de Beaulieu, 35042, Rennes Cedex, France

Riddhi Shah
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

Keywords: Infinitesimal triangular systems of measures, embeddable measures, spectrum of convolution operators, Lyapunov exponents, algebraic groups
Received by editor(s): July 30, 2003
Received by editor(s) in revised form: February 12, 2004
Published electronically: February 4, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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