We introduce a notion of entropy for automorphisms of discrete groups which admit amenable actions on a compact space. This entropy is dual to classical topological entropy in the sense that if G is discrete and abelian then our notion of entropy agrees with the topological entropy of the induced automorphism on the (compact) dual group of G.
We prove a number of basic properties of this dual entropy and give a few calculations. In particular, we are able to give precise calculations for arbitrary automorphisms of crystallographic groups.
In a prior paper, the author generalized the classical factor theorem of Sinai to actions of arbitrary countably infinite groups. In the present paper, we use this theorem and the techniques of its proof in order to study connections between positive entropy phenomena and the Koopman representation. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. By combining this result with the generalized factor theorem of the p…
This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely …
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related…
We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…
Operator amenability of Fourier-Stieltjes algebras
算子顺从性的概念是由Z. J. Ruan，1995年。他表明，一个局部紧群G是顺从的当且仅当其傅立叶代数A（G）是经营者顺从。本文研究了Fourier-Stieltjes代数B（G）和约化Fourier-Stieltjes代数B_r（G）的算子顺从性。自然的猜想是，这些代数中的任何一个都是算子顺从的当且仅当G是紧的。我们部分地证明了这个猜想与纯粹的运营商顺从性取代运营商C-顺从性，某些常数C < 5。在这个过程中，我们得到了B（G）的一个新的分解，它可以解释为M（G）分解为离散测度和连续测度的非交换对应。我们进一步引入了算子顺从性的一种变体--算子Connes顺从性，它也考虑了B（G）和B_r（G）上的对偶空间结构.证明了B_r（G）是Connes-affected的当且仅当G是affected的.令人惊讶的是，B（F_2）是算子Connes顺从的，尽管两个生成元中的自由群F_2不顺从.
We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simp…
