Amenability of monomial algebras, minimal subshifts and free subalgebras

TL;DR

This paper characterizes amenability in monomial algebras using combinatorial methods, establishes the existence of Folner sequences, and explores implications for modules, subshifts, and free subalgebras, revealing deep structural properties.

Contribution

It provides a combinatorial characterization of amenability in monomial algebras and links this to properties of modules, subshifts, and free subalgebras, answering open questions.

Findings

Existence of monomial Folner sequences established.

Modules over projectively simple monomial algebras are exhaustively amenable.

Non-amenable monomial algebras contain noncommutative free subalgebras.

Abstract

We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Bartholdi. Finally, we show that non-amenable monomial algebras must contain noncommutative free subalgebras. Examples are given to emphasize the sharpness and necessity of the assumptions in our results.