The unitary implementation of a locally compact quantum group action

By S. Vaes. 2000

148 citations recorded by the source corpus.

Abstract 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 α is an action of the locally compact quantum group ( M ,  Δ ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N α ⊂ N ↪ M α ⋉ N is a basic construction. Here N α denotes the fixed point algebra and M α ⋉ N is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion N α ⊂ N is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal. 137 , 466–543; 1998, J. Funct. Anal. 154 , 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand. 84 , 297–319): every integrable outer action with infinite fixed point algebra is a dual action.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d74443d30c6ede86262679bbce8926cd1b1396f5

How researchers see this paper

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In Algebra and Number Theory

1. Martijn Caspers
2. K. De Commer
3. S. Vaes
4. L. Vainerman
5. Saad Baaj
6. Pierre Fima
7. J. Kustermans
8. A. V. Daele

In Geometry and Topology

1. Martijn Caspers
2. K. De Commer
3. S. Vaes
4. Saad Baaj
5. L. Vainerman
6. Pierre Fima
7. J. Kustermans
8. Matthew Daws

In Mathematical Physics

1. Martijn Caspers
2. K. De Commer
3. S. Vaes
4. L. Vainerman
5. Saad Baaj
6. Pierre Fima
7. J. Kustermans
8. Sorin Popa

Citations in the corpus

• Kazhdan's Property T for Discrete Quantum Groups (2008, background): “Because Ĝ is unimodular we know from [16], Proposition 4.”
• Inclusions of Von Neumann Algebras and Quantum Groupoı̈ds II (2000, unknown): “1. Introduction 1.1. In two articles ([V1], [ V2 ]), J.-M.”
• Extensions of locally compact quantum groups and the bicrossed product construction (2001, methodology): “Proposition 1.20.”
• DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS (2002, methodology): “12 in [13] to conclude that ut ∈ N ⊗N and, analogously, wt ∈ N ⊗N for all t ∈ R.”
• Strictly Outer Actions of Groups and Quantum Groups (2002, unknown): “The faithfulness of α follows immediately.”
• Locally compact quantum groups in the von Neumann algebraic setting (2000, background): “This stronger invariance property is needed whenever an action of a von Neumann algebraic quantum group on a von Neumann algebra appears: see [5] and [13], but also [2] for Kac algebra actions, and it will certainly be useful in future investigations as well.”
• Galois objects and cocycle twisting for locally compact quantum groups (2008, methodology): “If m ∈ Nφ̂op and z ∈ NφN , then ρ((1⊗m)α(z)) ∈ Nφ̃2 , and we can make a GNS-map Λφ̃2 for φ̃2 in p(L 2(N)⊗ L 2(M)) by Λφ̃2(ρ((1 ⊗m)α(z))) = p(ΛN⋊M ((1⊗m)α(z))) = p(ΛN (z)⊗ Λ̂ (m)), since by the results of [23], the linear span of the (1⊗m)α(z) forms a σ-strong∗-norm core for ΛN⋊M .”
• Locally compact quantum groups (2017, background): “Firstly we recall the definition of the crossed product, see [33].”