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Stochastic operator on $\ell^1$ has dense range
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
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