Kenneth Dykema, Maria Grazia Viola
We show that for any prime p and for any II$_1$ factor N there exist two $mathbb{Z}_{p^2}$-- actions on the free product factor $*_{1} ^{p}N$ that have the same outer invariant but are not outer conjugate. Therefore, in the case of free product factors, the outer invariant is not a complete invariant for outer conjugacy.
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We will show that any two outer actions of Z^N on O_2 are cocycle conjugate.
Based on the fact that, for a subfactor $N$ of a II$_1$ factor $M,$ the first non-trivial Jones index is 2 and then $M$ is decomposed as the crossed product of $N$ by an outer action of ${\mathbb{Z}}_2,$ we study pairs $ \{N, uNu^* \}$ from a view point of entropy for two subalgebras of $M$ with a connection to the entropy for automorphisms, where the inclusion of II$_1$ factors $ N \subset M$ is given as $M$ is the crossed product of $N$ by a finite group of outer automorphisms and $u$ is a unitary in $M.$
We prove that all strongly outer Z^N-actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable Z^N-actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.
We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z^2 on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z^2 on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy.
Let $(A, α)$ and $(B, β)$ be C*-dynamical systems where $α$ and $β$ are arbitrary *-endomorphisms. When $α$ is injective or surjective, we show that the semicrossed products $A \times_α\mathbb{Z}$ and $B \times_β\mathbb{Z}$ are isometrically isomorphic if and only if $(A, α)$ and $(B, β)$ are outer conjugate. This conclusion also holds in various other cases as well.
We show that Bogoljubov actions of $\mathbb{R}$ on the free group factor $L(\mathbb{F}_{\infty})$ associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate.
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in O_n. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA D_n via Bogolubov automorphisms, that are not inner conjugate to D_n.
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 will show the uniqueness of outer coactions of finite groups on the AFD factor of type II$_1$ along the arguments by Connes, Jones and Ocneanu. Namely, we construct the infinite tensor product type action, adopt it as the model action, and prove that any outer coaction is conjugate to the model action.
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,α)$ and $(B,β)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_αG,\widehatα)$ and $(B\rtimes_βG,\widehatβ)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respect…
Approximate representability of finite abelian group actions on the Razak-Jacelon algebra
该研究刻画了有限阿贝尔群外作用与Razak-Jacelon代数恒等映射的张量积作用近似可表示的充要条件，依共轭和上循环共轭分类了此类作用，并构造了模型作用。
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…
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 …
Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra
本文证明了对于特定无挠顺从群类$\mathfrak{C}$中的群$Γ$，其在Razak-Jacelon代数$\mathcal{W}$上的所有强外作用均彼此上闭链共轭，这是Szabó关于强自吸收C$^*$-代数结果的类似结论。
Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-correspondence $E$. In this paper we construct factorizations of inner-outer type of the elements of $H^{\infty}(E)$ represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of $H^\infty(\mathbb{D})$. Our results also generalize some results that were obtained by several authors in some special cases.
A continuous family of non-outer conjugate aperiodic automorphisms whose crossed-products are all isomorphic is given on every interpolated free group factor. An explicit "duality" relationship between compact co-commutative Kac algebra (minimal) free product actions and free shift actions is also discussed.
Conjugate variables approach to mixed $q$-Araki-Woods algebras: Factoriality and non-injectivity
本文利用共轭变量方法，在最一般情形下证明了与可分实Hilbert空间等结构关联的混合$q$-Araki-Woods von Neumann代数的因子性与非内射性。
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…
Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor
本文研究超有限型II₁因子上几乎周期外流的唯一性，证明其满足Rokhlin性质并唯一到余cocycle共轭。
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)…
On the Product of Coninvolutory Affine Transformations
本文研究了共逆元仿射变换的乘积性质，证明了仿射变换可表示为两个或更多共逆元的乘积，且满足特定条件。
Let $Λ$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $Λ$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $Λ$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement:…
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