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 action $G$ on $X,$ there is maximal, closed $G$-invariant subgroup $Y$ $X$ so weakly contained Bernoulli shift. This also minimal any shift $G\curvearrowright X/Y$-ergodic presence X$. give several applications, including major simplification proof measure entropy equals topological principal actions whose associated convolution operator injective. deduce from our techniques square summable homoclinic group dense have completely positive when acting sofic.