We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains at least one factor of X, then X embeds into Y. This result partially extends the classical result of Krieger when $G = \mathbb {Z}$ and the results of Lightwood when $G = \mathbb {Z}^d$ for $d \geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.

A probability measure-preserving action of a discrete amenable group G is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb {Z}$, an action is dominant if and only if it has positive entropy and that for any G, positive entropy implies dominance. In this paper, we show that the converse also holds for any G, that is, that zero entropy implies non-dominance.

In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy $h(X)> 0$, then the entropies of the SFT subsystems of X are dense in the interval $[0, h(X)]$. In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs Z for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic G-shifts.

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math. 397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys. 39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math. 308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

Motivated by the recent result in Samei and Wiersma (2020, Advances in Mathematics 359, 106897) that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Suppose G is an amenable locally compact group with lattice subgroup $\Gamma $. Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc. 288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$, $\Gamma ={\mathbb Z}^d$. In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd , this result was first proven by the current set of authors using completely different techniques.

We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$ -envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

Erdős conjectured that for any set $A\,\subseteq \,\mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,\,C\,\subseteq \,\mathbb{N}$ such that $B\,+\,C\,\subseteq \,A$ . We verify Erdős’ conjecture in the case where $A$ has Banach density exceeding $\frac{1}{2}$ . As a consequence, we prove that, for $A\,\subseteq \,\mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,\,C\,\subseteq \,\mathbb{N}$ such that $B\,+\,C$ is contained in the union of $A$ and a translate of $A$ . Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

Let $\mathcal{A}$ be a Banach algebra with a bounded right approximate identity and let $\mathcal{B}$ be a closed ideal of $\mathcal{A}$ . We study the relationship between the right identities of the double duals ${{\mathcal{B}}^{*}}^{*}$ and ${{\mathcal{A}}^{**}}$ under the Arens product. We show that every right identity of ${{\mathcal{B}}^{*}}^{*}$ can be extended to a right identity of ${{\mathcal{A}}^{**}}$ in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra $A\left( G \right)$ of a locally compact group $G$ , an element $\phi \in A{{\left( G \right)}^{**}}$ is in $A\left( G \right)$ if and only if $A\left( G \right)\phi \subseteq A\left( G \right)$ and $E\phi =\phi$ for all right identities $E$ of $A{{\left( G \right)}^{**}}$ . We also prove some results about the topological centers of ${{\mathcal{B}}^{**}}$ and ${{\mathcal{A}}^{**}}$ .

For a locally compact group $G$ and $1\,<\,p\,<\,\infty $ , let ${{A}_{p}}\left( G \right)$ be the Herz-Figà-Talamanca algebra and let $P{{M}_{p}}\left( G \right)$ be its dual Banach space. For a Banach ${{A}_{p}}\left( G \right)$ -module $X$ of $P{{M}_{p}}\left( G \right)$ , we prove that the multiplier space $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ is the dual Banach space of ${{Q}_{X}}$ , where ${{Q}_{X}}$ is the norm closure of the linear span ${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$ in the dual of $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ . If $p\,=\,2$ and $P{{F}_{p}}\left( G \right)\subseteq X$ , then ${{A}_{p}}\left( G \right)X$ is closed in $X$ if and only if $G$ is amenable. In particular, we prove that the multiplier algebra $M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$ is the dual of $Q$ , where $Q$ is the completion of ${{L}^{1}}\left( G \right)$ in the $||\cdot |{{|}_{M}}$ -norm. $Q$ is characterized by the following: $f\,\in \,Q$ if an only if there are ${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$ and ${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$ with $\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $ such that $f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$ on $M{{A}_{p}}\left( G \right)$ . It is also proved that if ${{A}_{p}}\left( G \right)$ is dense in $M{{A}_{p}}\left( G \right)$ in the associated ${{w}^{*}}$ -topology, then the multiplier norm and $||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$ -norm are equivalent on ${{A}_{p}}\left( G \right))$ if and only if $G$ is amenable.

Let G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

Let with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L ∞(G) which are mutually singular to every element of TLIM(L ∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L ∞(G). It follows that the cardinalities of LIM(L ∞(G)) ̴ TLIM(L ∞(G)) and LIM(L ∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G 1 ⨯ G 2, where G 1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G 2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.