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Let 
$K$ be a function on a unimodular locally compact group 
$G$, and denote by 
${{K}_{n}}\,=\,K\,*\,K\,*\cdots *\,K$ the 
$n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in 
${{L}^{2}}\left( G \right)$, we give estimates of the norms 
${{\left\| {{K}_{n}} \right\|}_{2}}$ and 
${{\left\| {{K}_{n}} \right\|}_{\infty }}$ for large $n$. In contrast to previous methods for estimating ${{\left\| {{K}_{n}} \right\|}_{\infty }}$, we do not need to assume that the function $K$ is a probability density or nonnegative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.
