To study outer actions $\a$ of a group $G$ on a factor $\sM$ of type {\threel}, $0<\la<1$, we study first the cohomology group of a group with the unitary group of an abelian {\vna} as a coefficient group and establish a technique to reduce the coefficient group to the torus $\T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a {\cdag} on an AFD factor of type {\threel}, sharpening the result in \cite{KtT2: \S4}. The periodicity of the flow of weights on a factor $\sM$ of type {\threel} allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\Aut(\sM)_\txm$ relative to the modulus homomorphism $\txm=\mod: \Aut(\sM)\mapsto \rt'z$. We then discuss the reduced {\mhjr}, which allows us to describe the invariant of outer action $\a$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $\tH_{\txm, \fs}^\out(G, N, \T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an {\AFD} factor $\sM$ of type \threel.
We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.
In this last article of the series on outer actions of a countable dicrete amenable group on AFD factors, we analyze outer actions of a countable discrete free abelian group on an AFD factor of type $\text{III}_λ$, $0<λ< 1$, and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a $\Bbb T$-valued cocycle of a torsion free abelian group, which might have been known by the group cohomologists. As the asymmetrization technique gives us a very handy criteria for coboundaries, we present it here in detail in the second section.
Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$ are finite and that their cardinality is bounded. We show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). If $\partial_eT(A)$ is …
We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γis elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.
We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case tha…
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a prec…
Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a countable amenable group $G$. If the trace space $T(A)$ is a Bauer simplex and the action of $G$ on $\partial_eT(A)$ has finite orbits and Hausdorff orbit space, we show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, then these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2)…
Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ up to approximately inner automorphisms then they are outer conjugate by an approximately inner automorphism, in the operator norm topology, which makes $A$ invariant. In addition, when $A$ is unital, simple, and with a unique tracial state and $α$ is an automorphism of $A$ we also show that the aperiodicity of $α$ on the von Neumann algebra is equivalent to…
In this paper we consider a bootstrap class $\mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$\mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Γ\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly …
We show that if $M$ is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then $N$ is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if $M$ is a full factor and $σ\colon G \curvearrowright M$ is an outer action of a compact group $G$, then $σ$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in $M$. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closin…
We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in part…
We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $φ_0$ and any discrete group $Γ$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,φ_0)^{\mathbin{\bar{\otimes}}Γ}\rtimes Γ$ is solid relatively to $\mathcal{L}(Γ)$. In particular, if $\mathcal{L}(Γ)$ is solid then $M$ is solid and if $Γ$ is non-amenable and $A_0 \neq \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obta…
We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann alg…
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…
We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a B…
In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_αG$, where $G$ is a discrete group acting on a factor $M$ by outer automorphisms. The second is a regular inclusion $M \subseteq N$ of finite factors. In the case of crossed products, we characterize the $M$-bimodules $X$ that lie between $M$ and $M \rtimes_αG$ and are closed in the Bures topology, in terms of the subsets of $G$. We show that this characterization also holds for $w^*$-closed bimodules when $G$ has the approximation property ($AP$), a class …
Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…
For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that …
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgro…
