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

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.