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We study the relationship between the dynamics of the action $\alpha$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_\alpha G$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $L(G) \subseteq M \rtimes_\alpha G$ such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorems for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras., Comment: 61 pages. Several minor revisions to improve exposition after referee comments. To appear, Adv. in Math

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between $W^{\ast}$-dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem $\mathfrak{M}_{K}$ if and only if $\mathfrak{N}\cong\mathfrak{M}_{K}$. This generalizes recent work of Duvenhage, which established the result for $G$ abelian.

Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{\alpha,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{\alpha,r}G$, where $A\rtimes_{\alpha,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\alpha,\sigma)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^\sigma_{\alpha,r}G$., Comment: As pointed out to us by Prof. Yuhei Suzuki, there is an error in Lemma 2.9 of v1. Lemmas 2.8 and 2.9 in v1 are replaced by Lemmas 2.8-2.12 in v2. The main results of the paper are unchanged. A correction will be sent to Canad. J. Math

This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which are absolutely continuous but not Lebesgue. We also show that there exist uncountably many pairwise non--conjugate mixing masas in the free group factors each with Puk\'{a}nszky invariant $\{1,\infty\}$.

Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_\alpha G$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous maps on $M$-bimodules to $*$-automorphisms of the containing von Neumann algebras., Comment: 27 pages. Minor revisions after referee's comments. Internat. J. Math., to appear

We show that a number of key structural properties transfer between sufficiently close II$_1$ factors, including solidity, strong solidity, uniqueness of Cartan masas and property $\Gamma$. We also examine II$_1$ factors close to tensor product factors, showing that such factors also factorise as a tensor product in a fashion close to the original., Comment: Minor typos corrected. Munster J. Math., to appear. 16 pages, section 2 contains material from arXiv:1209.4116v2 which was removed from the final published version (v3)

In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_\alpha G$, 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 $M \rtimes_\alpha G$ 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 of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain $w^*$-continuous isometric maps on $X$. We establish a similar theorem for bimodules arising from regular inclusions of finite factors, which generalizes the crossed product situation when $G$ acts on a finite factor. In the final section we apply these ideas to provide new examples of singly generated finite factors., Comment: 34 pages

This paper addresses a conjecture of Kadison and Kastler that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N and, moreover, the implementing unitary can be chosen to be close to the identity operator. This is known to be true for amenable von Neumann algebras and in this paper we describe new classes of non-amenable factors for which the conjecture is valid. These are based on tensor products of the hyperfinite II_1 factor with crossed products of abelian algebras by suitably chosen discrete groups., Comment: 12 Pages. This is a shorter, expository version of arXiv:1209.4116, containing heuristic arguments and some sketch proofs

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\geq 3$ and a free ergodic probability measure preserving action of $SL_n(\mathbb Z)$ on a standard nonatomic probability space $(X,\mu)$, write $M=((L^\infty(X,\mu)\rtimes SL_n(\mathbb Z))\,\overline{\otimes}\, R$, where $R$ is the hyperfinite II$_1$ factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal H$ and $N\subseteq\mathcal B(\mathcal H)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal H$ close to the identity operator with $uMu^*=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler's conjecture. We also obtain stability results for crossed products $L^\infty(X,\mu)\rtimes\Gamma$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module $L^2(X,\mu)$. In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when $\Gamma$ is a free group., Comment: 33 pages. Paper restructured. Some of the material removed will appear in a future article. Duke Math. J., to appear

In this note we show that Kadison's similarity problem for C*-algebras is equivalent to a problem in perturbation theory: must close C*-algebras have close commutants?, Comment: 6 Pages, minor typos fixed. C. R. Acad. Sci. Canada, to appear

In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that the given Cartan bimodule isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15], we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our proof does not require the von Neumann algebras generated by the algebras A_i to have separable preduals. This point of view also yields some insights on the von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening of a result of Aoi [1]. We also examine the relationship between various topologies on a von Neumann algebra M with a Cartan MASA D. This provides the necessary tools to parametrize the family of Bures-closed bimodules over a Cartan MASA in terms of projections in a certain abelian von Neumann algebra; this result may be viewed as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient in the proof of our version of Mercer's theorem. Our results lead to a notion of spectral synthesis for weak-* closed bimodules appropriate to our context, and we show that any von Neumann subalgebra of M which contains D is synthetic. We observe that a result of Sinclair and Smith shows that any Cartan MASA in a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith., Comment: 21 pages, paper is a completely reworked and expanded version of an earlier preprint with a similar title

Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical theorem of Halmos on the automorphisms of a compact abelian group. For a finite von Neumann algebra $M$ and von Neumann subalgebras $A$, $B$ of $M$, we introduce a notion of weak mixing of $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$ relative to $A$ is equivalent to the following property: if $x\in M$ and there exist a finite number of elements $x_1,...,x_n\in M$ such that $Ax\subset \sum_{i=1}^nx_iB$, then $x\in B$. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions., Comment: 24 pages, minor typos corrected

The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors., Comment: 25 Pages, Typos corrected and exposition improved

In this paper we prove that for a type II_1 factor N with a Cartan maximal abelian subalgebra (masa), the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual., Comment: 10 pages