To a real Hilbert space and a one-parameter group of orthogonal transformations we associate a C∗-algebra which admits a free quasi-free state. This construction is a freeprobability analog of the construction of quasi-free states on the CAR and CCR algebras. We show that under certain conditions, our C∗-algebras are simple, and the free quasi-free states are unique. The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the Araki-Woods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For non-trivial one-parameter groups, these von Neumann algebras are type III factors. In the case the one-parameter group is nontrivial and almost-periodic, we show that Connes’ Sd invariant completely classifies these algebras.

Abstract. We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a measure preserving action Γ � X of Γ on a probability space (X, µ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action Γ � [0,1] Γ) is cohomologous to a group morphism of Γ into V. We use the case V discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between Γ � X and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. There has recently been increasing interest in the study of measure preserving actions of groups on (non-atomic) probability spaces up to orbit equivalence (OE), i.e. up to isomorphisms of probability spaces taking the orbits of one action onto the orbits of

The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K.

Let A ⊇ D ⊆ B be two von Neumann algebras together with a common Cartan subalgebra. Then the amalgamated free product M = A ∗D B with respect to the unique conditional expectations from A, B onto D can be considered. In our previous paper [U1], the questions of its factoriality and type classification

Abstract. Let Od be the Cuntz algebra on generators S1,..., Sd, 2 ≤ d < ∞, and let Dd ⊆ Od be the abelian subalgebra generated by monomials S α S ∗ α = S α1 · · · S α k S ∗ α k · · · S ∗ α1 where α = (α1... αk) ranges over all multi-indices formed from {1,..., d}. In any representation of Od, Dd may be simultaneously diagonalized. Using Si (SαS ∗ α) = ( SiαS ∗) iα Si, we show that the operators Si from a general representation of Od may be expressed directly in terms of the spectral representation of Dd. We use this in describing a class of type III representations of Od and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups.

Abstract. Z.-J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper we extend this work to the case of discrete quantum groups with a quite different method. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it satisfies a certain condition, which is called strong-Voiculescu amenability in the case of Kac algebras. 1.

Abstract. We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a Bernoulli Γ-action is cohomologous to a group morphism of Γ into V. We use this result to prove that if in addition Γ has no non-trivial finite normal subgroups, then any orbit equivalence between a Bernoulli Γ-action and a free ergodic measure preserving action of some group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. Bernoulli actions of Kazhdan groups were shown in ([P2,3]) to have sharp rigidity properties, to the extent that both the group and the action (up to conjugacy) can be completely recovered from the isomorphism class of the associated group von Neumann algebra. When applied to algebra isomorphisms coming from orbit equivalence (OE)

Abstract. We classify compact abelian group actions on injective type III factors up to conjugacy, which completes the final step of classification of compact abelian group actions on injective factors. The purpose of this paper is to provide a classification, up to conjugacy, of actions of a (separable) compact abelian group on injective factors of type III (Theorem 3.1). Studying automorphism groups has been a powerful and fruitful approach to

Abstract. Let A be a separable unital C*-algebra and let π: A → L(H) be a faithful representation of A on a separable Hilbert space H such that π(A) ∩ K(H) = {0}. We show that OE, the Cuntz-Pimsner algebra associated to the Hilbert A-bimodule E = H ⊗C A, is simple and purely infinite. If A is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to OE. Hence by the Kirchberg-Phillips Theorem the isomorphism class of OE only depends on the K-theory of A and the class of the unit. In his seminal paper [Pm], Pimsner constructed a C*-algebra OE from a Hilbert bimodule over a C*-algebra A as a quotient of a concrete C*-algebra TE, an analogue of the Toeplitz algebra, acting on the Fock space associated to E. There has recently been much interest in these Cuntz-Pimsner algebras (or Cuntz-Krieger-Pimsner algebras), which generalize both crossed products by Z and Cuntz-Krieger algebras, as well as the associated Toeplitz algebras. The structure of these C*-algebras is not yet fully understood, although considerable progress has been made. For example, Pimsner found a six-term exact sequence for the K-theory of OE which generalizes the Pimsner-Voiculescu exact sequence (see [Pm, Theorem 4.8]); conditions for simplicity were found in [Sc2, MS, KPW1, DPW] and for pure infiniteness in [Z].

Abstract. A continuous one-parameter group of unitary isometries of a right-Hilbert C*bimodule induces a quasi-free dynamics on the Cuntz-Pimsner C*-algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner C*-algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, Cuntz-Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial.