Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is $G\curvearrowright X\mathrm{/}Y$-ergodic “in the presence of $G\curvearrowright X$”. We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal algebraic actions whose associated convolution operator is injective. We also deduce from our techniques that algebraic actions whose square summable homoclinic group is dense have completely positive entropy when the acting group is sofic.