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We give an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is ITPFI. Our proof is given in parallel with each type of factors and it is based on the strategy of Haagerup. As a consequence, the uniqueness theorem of injective factors except type III$_0$ follows from Araki-Woods' result., Comment: The paper will appear in the Publications of Research Institute for Mathematical Sciences, Kyoto University

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 is recently generalized for 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 partly 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., Comment: Since it is not clear whether H_mix is a subspace, we modified the definition of the strict mixing property. Also, we made comments on amenability

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalized to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in terms of the standard form and the other in terms 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., Comment: To appear in C. R. Acad. Sci

We introduce the notion of the $\alpha$-Haagerup approximation property for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-Haagerup approximation property actually does not depend on a choice of $\alpha$. This gives us a direct proof of the fact that two characterizations of the Haagerup approximation property are equivalent, one in terms of the standard form and the other in terms of completely positive maps. We also discuss the $L^p$-Haagerup approximation property for a non-commutative $L^p$-spaces associated with a von Neumann algebra ($1

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., Comment: To appear in Publ. Res. Inst. Math. Sci

For every $p\geq 2$, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to the positive linear functionals on the free group $C^*$-algebra associated with the ideal $\ell_p$. This is a generalization of Haagerup's characterization for the case of the reduced free group $C^*$-algebra. As a consequence, the associated $C^*$-algebras are mutually non-isomorphic, and they have a unique tracial state.

We consider the harmonic measure on the Gromov boundary of a nonamenable 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 III_0., Comment: 21 pages

We give a construction of a nuclear $C^\ast$-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 $C^\ast$-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 the relationship between the KMS states of the generalized gauge actions on our $C^\ast$ algebras and random walks on the groups.