Locally compact quantum group

A locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems.
One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group.
Definitions.
Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems.
Definition (weight). Let formula_1 be a C*-algebra, and let formula_2 denote the set of positive elements of formula_1. A weight on formula_1 is a function formula_5 such that
Some notation for weights. Let formula_11 be a weight on a C*-algebra formula_1. We use the following notation:
Types of weights. Let formula_11 be a weight on a C*-algebra formula_1.
Definition (one-parameter group). Let formula_1 be a C*-algebra. A one-parameter group on formula_1 is a family formula_40 of *-automorphisms of formula_1 that satisfies formula_42 for all formula_43. We say that formula_44 is norm-continuous if and only if for every formula_45, the mapping formula_46 defined by formula_47 is continuous (surely this should be called strongly continuous?).
Definition (analytic extension of a one-parameter group). Given a norm-continuous one-parameter group formula_44 on a C*-algebra formula_1, we are going to define an analytic extension of formula_44. For each formula_51, let
which is a horizontal strip in the complex plane. We call a function formula_53 norm-regular if and only if the following conditions hold:
Suppose now that formula_61, and let
Define formula_63 by formula_64. The function formula_65 is uniquely determined (by the theory of complex-analytic functions), so formula_66 is well-defined indeed. The family formula_67 is then called the analytic extension of formula_44.
Theorem 1. The set formula_69, called the set of analytic elements of formula_1, is a dense subset of formula_1.
Definition (K.M.S. weight). Let formula_1 be a C*-algebra and formula_5 a weight on formula_1. We say that formula_11 is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on formula_1 if and only if formula_11 is a "proper weight" on formula_1 and there exists a norm-continuous one-parameter group formula_79 on formula_1 such that
We denote by formula_87 the multiplier algebra of formula_88.
Theorem 2. If formula_1 and formula_90 are C*-algebras and formula_91 is a non-degenerate *-homomorphism (i.e., formula_92 is a dense subset of formula_90), then we can uniquely extend formula_94 to a *-homomorphism formula_95.
Theorem 3. If formula_96 is a state (i.e., a positive linear functional of norm formula_97) on formula_1, then we can uniquely extend formula_99 to a state formula_100 on formula_101.
Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is an ordered pair formula_102, where formula_1 is a C*-algebra and formula_104 is a "non-degenerate" *-homomorphism called the co-multiplication, that satisfies the following four conditions:
From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight formula_114 is automatically faithful. Therefore, the faithfulness of formula_114 is a redundant condition and does not need to be postulated.
Duality.
The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups.
Alternative formulations.
The theory has an equivalent formulation in terms of von Neumann algebras.