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arXiv · 1901.03406

Bernoulli disjointness

Eli Glasner·Todor Tsankov·Benjamin Weiss·Andy Zucker
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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.

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Eli Glasner, Todor Tsankov, Benjamin Weiss, Andy Zucker. 2019-01-10. Bernoulli disjointness. https://doi.org/10.1215/00127094-2020-0093

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