We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

We study certain q-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group (Z=2Z)^k and shown to be a maximal abelian subalgebra which is singular and with Pukanszky invariant [∞]. Further all nonequal generator masas in the q-deformed Gaussian von Neumann algebras are shown to be mutually nonintertwinable.

We discuss just infiniteness of C^*-algebras associated to discrete quantum groups and relate it to the C^*-uniqueness of the quantum groups in question, that is, to the uniqueness of a C^*-completion of the underlying Hopf C^*-algebra. It is shown that duals of q-deformations of simply connected semisimple compact Lie groups are never C^*-unique. On the other hand, we present an example of a discrete quantum group which is not locally finite and yet is C^*-unique.