Abstract
Abstract
We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II
$_1$
factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C
$^*$
-dynamics. Given a countable discrete group G and an amenable action
$G\curvearrowright M$
on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.
Funder
Bijzonder Onderzoeksfonds UGent
Fonds Wetenschappelijk Onderzoek
Publisher
Cambridge University Press (CUP)