✦ LIBER ✦

The Unitary Implementation of a Locally Compact Quantum Group Action

✍ Scribed by Stefaan Vaes

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 315 KB
Volume: 180
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.2000.3704

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✦ Synopsis

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When : is an action of the locally compact quantum group (M, 2) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N : /N / Ä M : _ N is a basic construction.

Here N : denotes the fixed point algebra and M : _ N is the crossed product. When : is an outer and integrable action on a factor N we prove that the inclusion N : /N is irreducible, of depth 2 and regular, giving a converse to the results of M.

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