Group factors of the Haagerup type

Abstract

1o Let N be a type II factor with the canonical trace . We call it a factor of the Haagerup type if there exists a net (P.). of normal linear maps on N which satisfy the following conditions(1) each P. is completely positive on N, (2) each P. is compact (i.e. for any 0, there exists a finite dimensional linear map Q on N such that P(x)-Q(x)[1x I[ for all xeN), and(3) P.(x)-x I]-.0, for all x e N.Here, we put I I x II= r(x*x) / for x e N.

The Haagerup property for locally compact quantum groups

“…We can now generalise Choda's result that the Haagerup property is a von Neumann property of a discrete group from [20] to the quantum setting.…”

“…The Haagerup property can also be defined for von Neumann algebras (see for example [42]) and, for discrete G, the Haagerup property is a von Neumann property: G enjoys the Haagerup property precisely when the group von Neumann algebra V N.G/ does [20].…”

“…Within the theory of quantum groups, V N.G/ is interpreted as the algebra L 1 . O G/, where O G is the dual quantum group of G. Thus, by analogy with the fact that the Haagerup property is a von Neumann property of a discrete group [20], Brannan's result can be viewed as the statement that the discrete dual quantum groups of the free orthogonal and unitary quantum groups have the Haagerup property. It was shown in [75] that the duals of free orthogonal groups satisfy an appropriate version of the Baum-Connes conjecture.…”

Notes on Treeability and Costs for Discrete Groupoids in Operator Algebra Framework

“…The next lemma can be proved in the essentially same way as in [7], where group von Neumann algebras are dealt with. Although the detailed proof is now available in [18, Proposition 3.5], we give its sketch for the reader's convenience, with focusing the "only if" part, which we will need later.…”

“…Assume that M = A ⋊ α G, i.e., M is the crossed-product of A by an action α of a countable discrete group G. Suppose that the action α has an invariant faithful state φ ∈ A * . Then, the inclusion M ⊇ A with the canonical conditional expectation E M A : M → A has Relative Haagerup Property if and only if G has Haagerup Property (see [17], [7]). …”