Bernoulli disjointness

Abstract

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 $\mathfrak{A}(G)$ is a proper subalgebra of ${\ell }^{\infty }(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $\beta G$. 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 icc group, then it admits a free, minimal, proximal flow.

28 pages; some details added, minor corrections

References in corpus (2)

• Boundaries of reduced C*-algebras of discrete groups
• Minimal flows with arbitrary centralizer

Cited by in corpus (3)

• On the virtual automorphism group of a minimal flow
• Minimal subdynamics and minimal flows without characteristic measures
• $C^{\ast }$-irreducibility of commensurated subgroups