Bernoulli disjointness

Generalizing a result of Furstenberg, we show that, for every infinite discrete group G, the Bernoulli flow 2^G is disjoint from every minimal G-flow. From this, we deduce that the algebra generated by the minimal functions A(G) is a proper subalgebra of ℓ^∞(G) and that the enveloping semigroup of the universal minimal flow M(G) is a proper quotient of the universal enveloping semigroup βG. When G is countable, we also prove that, for any metrizable, minimal G-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable G-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if G is a countable group with infinite conjugacy classes, then it admits a free, minimal, proximal flow.