Locally Compact Quantum Groups. A von Neumann Algebra Approach

Abstract

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von Neumann algebra framework. At various points, we also do things differently. We have a different treatment of the antipode. We obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the $C^{\ast }$-approach and the von Neumann algebra approach eventually yield the same objects.

References in corpus (1)

• The multiplicative unitary as a basis for duality

Cited by in corpus (25)

• Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres
• Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms
• Compact and discrete subgroups of algebraic quantum groups I
• Completely positive definite functions and Bochner's theorem for locally compact quantum groups
• Around Property (T) for quantum groups
• Fourier transform on locally compact quantum groups
• Multiplier Hopf algebras imbedded in C${}^{\ast }$-algebraic quantum groups
• On positive definiteness over locally compact quantum groups
• Amenability of locally compact quantum groups and their unitary co-representations
• Twisting of the Quantum double and the Weyl algebra
• Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
• A class of C*-algebraic locally compact quantum groupoids Part II: Main theory
• I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type
• Ergodic theory for quantum semigroups
• Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups
• A (2n+1)-dimensional quantum group constructed from a skew-symmetric matrix
• The approximation property for locally compact quantum groups
• Generating functionals for locally compact quantum groups
• On a Morita equivalence between the duals of quantum SU(2) and quantum E(2)
• Ergodicity and mixing of W*-dynamical systems in terms of joinings
• Quantum information theory and Fourier multipliers on quantum groups
• Amenability and co-amenability in non-abelian group duality
• On cocycle twisting of compact quantum groups
• Isometries between quantum convolution algebras
• I-factorial quantum torsors