Locally compact quantum groups in the von Neumann algebraic setting

By J. Kustermans, S. Vaes. 2000

363 citations recorded by the source corpus.

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a $C^*$-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper [8]. We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.

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

View Paper: https://www.semanticscholar.org/paper/34396448e85faaafae07a64083b773ff3c865104

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Julien Bichon
2. T. Banica
3. K. De Commer
4. Martijn Caspers
5. S. Vaes
6. Saad Baaj
7. Matthew Daws
8. L. Vainerman

In Geometry and Topology

1. Julien Bichon
2. Martijn Caspers
3. T. Banica
4. K. De Commer
5. S. Vaes
6. Saad Baaj
7. Matthew Daws
8. L. Vainerman

In Mathematical Physics

1. Martijn Caspers
2. Julien Bichon
3. K. De Commer
4. T. Banica
5. S. Vaes
6. Saad Baaj
7. Matthew Daws
8. L. Vainerman

Citations in the corpus

• HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS (2008, background): “These notions appeared, under various degrees of generality, in a number of independent investigations: vertex models and related quantum groups [6, 11, 12], locally compact quantum groups and their outer actions [20, 27].”
• Strictly Outer Actions of Groups and Quantum Groups (2002, neutral): “The faithfulness of α follows immediately.”
• The unitary implementation of a locally compact quantum group action (2000, result): “We have (M α⋉N) α̂ = α(N) α(N) = {z ∈M ⊗N | (ι⊗ α)(z) = (∆⊗ ι)(z)}.”
• Locally compact quantum groups (2017, methodology): “Then there are the original papers [22] and [23] of which [22] contains all the essential proofs.”
• Extensions of locally compact quantum groups and the bicrossed product construction (2001, methodology): “After the appearence of quantum groups in the eighties a lot of efforts were spent in order to construct their general theory in operator algebraic framework that would cover known examples and that would be as elegant and symmetric as the one of Kac algebras.”
• Approximation properties for free orthogonal and free unitary quantum groups (2011, neutral): “Furthermore, Dr extends to a normal -homomorphism Dr : LyðGÞ ! LyðGÞnLyðGÞ: The CQG Gr 1⁄4 CðGÞ;Dr is called the reduced version of G, and LðGÞ;Dr is a von Neumann algebraic CQG, in the sense of [25].”
• Galois objects and cocycle twisting for locally compact quantum groups (2008, methodology): “We refer to [13] and [28] for further definitions and formulas.”
• Amenability and the bicrossed product construction (2001, methodology): “The theory of locally compact quantum groups has been introduced by J. Kustermans and the third author in [9, 10 ], unifying compact quantum groups and Kac algebras.”
• DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS (2002, background): “Finally, the theory of locally compact quantum groups was proposed by Kustermans and the second author in [8,9].”
• A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS (2009, methodology): “There are several different definitions of locally compact quantum groups given by Baaj and Skandalis [1], Woronowicz [33], and Kustermans and Vaes in the C∗-algebra setting [15] and in the von Neumann algebra setting [16].”
• Locally Compact Quantum Groups. A von Neumann Algebra Approach (2006, background): “in an earlier stage of the development. This in turn will yield other simplifications. Finally, our approach in this paper is also self-contained. In their paper on the von Neumann algebraic approach [9], Kustermans and Vaes do not really use results from the earlier paper on the C-algebra approach ”
• Multipliers, Self-Induced and Dual Banach Algebras (2010, methodology): “We shall quickly sketch the theory of locally compact quantum groups, in the sense of Kustermans and Vaes, [29, 31, 32].”
• Non-Semi-Regular Quantum Groups Coming from Number Theory (2002, background): “In fact, a multiplicative unitary can really be quite singular: in Remark 4.5 below, we give an example of a multiplicative unitary which should certainly not be considered as the quantum analogue of a locally compact group.”