Ambient Dependence and Extreme Extension for Exchangeable Free-Product States

A proposed independence relation declares two subalgebras independent when their joint state is realized in two different copies of an extreme exchangeable state on the infinite unital free product of the ambient algebra. We show that this relation depends on the ambient algebra. The diagonal inclusion \(\mathbb C^2\) into \(M_2(\mathbb C)\) gives an explicit example; a four-point version has distinct proper subalgebras with scalar intersection. Downstairs, the joint law forces a nontrivial mixture of two invariant characters. Upstairs, a pure vector state provides a pure folded witness. In contrast, every extreme invariant state has an extreme invariant extension along a unital \(C^*\)-algebra inclusion. The latter standard compact-face argument rules out the particular forward-extension obstruction suggested in the original problem, without implying ambient invariance. The examples distinguish extremality among all invariant states from extremality at a fixed marginal.