Rui Okayasu

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

We introduce the notion of the a-Haagerup approximation property (α-HAP) for α ∈ [0, 1/2] using a one-parameter family of positive cones studied by Araki and show that the a-HAP actually does not depend on the choice of α. This enables us to prove the fact that the Haagerup approximation properties introduced in two ways are actually equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the Lp-Haagerup approximation property (Lp-HAP) for a noncommutative Lp-space associated with a von Neumann algebra for p ∈ (1,∞) and show the independence of the Lp-HAP on the choice of p.

We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.

For every p >= 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal l(p). This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Voiculescu's topological approximation entropy is extended to automorphisms on unital simple C*-algebras with tracial rank zero. Several expected properties are shown. We also consider the value of our entropy for a cat map on the non-commutative torus.

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa.

We consider the harmonic measure on the Gromov boundary of a non-amenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III0.

We compute the exact value of Voiculescu's perturbation theoretic entropy of the boundary actions of free groups. This result is a partial answer of Voiculescu's question.

Let Gamma be a Gromov hyperbolic group with a finite set A of generators. We prove that h(top)(Sigma(infinity)) <= k(infinity)(-1)(lambda(A)) <= gr(Gamma, A), where gr(Gamma, A) is the growth entropy, h(top)(Sigma(infinity)) is the Coornaert-Papadopoulos topological entropy of the subshift Sigma(infinity) associated with (Gamma, A), and k(infinity)(-1)(lambda(A)) is Voiculescu's numerical invariant, which is an obstruction to the existence of quasicentral approximate units relative to the Macaev norm for a tuple of unitary operators lambda(A) = (lambda(a))(a is an element of A) in the left regular representation of Gamma. We also prove that these three quantities are equal for a hyperbolic group splitting over a finite group.

We construct a nuclear C*-algebra associated with the fundamental group of a graph of groups of finite type. It is well-known that every word-hyperbolic group with zero-dimensional boundary, in other words, every group acting trees with finite stabilizers is given by the fundamental group of such a graph of groups. We show that our C*-algebra is *-isomorphic to the crossed product arising from the associated boundary action and is also given by a Cuntz-Pimsner algebra. We also compute the K-groups and determine the ideal structures of our C*-algebras.

We obtain the exact value of Voiculescu's invariant k(infinity)(-)(tau), which is an obstruction of the existence of quasicentral approximate units relative to the Macaev ideal in perturbation theory, for a tuple tau of operators in the following two classes: (1) creation operators associated with a subshift, which are used to define Matsumoto algebras, (2) unitaries in the left regular representation of a finitely generated group.

We determine the types of factors arising from GNS-representations of quasi-free KMS states on Cuntz-Krieger algebras. Applying our result to the Cuntz-Krieger algebras arising from the boundary actions of some amalgamated free product groups, we also determine the types of harmonic measures on the boundaries.

We give a construction of a nuclear C*-algebra associated with an amalgamated free product of groups, generalizing Spielberg's construction of a certain Cuntz-Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear C*-algebras can be identified with certain Cuntz-Krieger-Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams' result about amenability of the boundary action for hyperbolic groups. We will also give an explicit formula of the K-groups of our algebras. Finally we will investigate a relationship between the KMS states of the generalized gauge actions on our C* algebras and random walks on the groups.