  We introduce an endofunctor $H$ on the category $bal$ of bounded archimedean
$\ell$-algebras and show that there is a dual adjunction between the category
$Alg(H)$ of algebras for $H$ and the category $Coalg(V)$ of coalgebras for the
Vietoris endofunctor $V$ on the category of compact Hausdorff spaces. We also
introduce an endofunctor $Hu$ on the reflective subcategory of $bal$ consisting
of uniformly complete objects of $bal$ and show that Gelfand duality lifts to a
dual equivalence between $Alg(Hu)$ and $Coalg(V)$. On the one hand, this
generalizes a result of \cite{Abr88,KKV04} for the category of coalgebras of
the Vietoris endofunctor on the category of Stone spaces. On the other hand, it
yields an alternate proof of a recent result of \cite{BCM20a}.
