Locally compact quantum group
In mathematics and theoretical physics, 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  be a C*-algebra, and let  denote the set of positive elements of . A weight on  is a function  such that
  for all , and
  for all  and .

Some notation for weights. Let  be a weight on a C*-algebra . We use the following notation:
 , which is called the set of all positive -integrable elements of .
 , which is called the set of all -square-integrable elements of .
 , which is called the set of all -integrable elements of .

Types of weights. Let  be a weight on a C*-algebra .
 We say that  is faithful if and only if  for each non-zero .
 We say that  is lower semi-continuous if and only if the set  is a closed subset of  for every .
 We say that  is densely defined if and only if  is a dense subset of , or equivalently, if and only if either  or  is a dense subset of .
 We say that  is proper if and only if it is non-zero, lower semi-continuous and densely defined.

Definition (one-parameter group). Let  be a C*-algebra. A one-parameter group on  is a family  of *-automorphisms of  that satisfies  for all . We say that  is norm-continuous if and only if for every , the mapping  defined by  is continuous (surely this should be called strongly continuous?).

Definition (analytic extension of a one-parameter group). Given a norm-continuous one-parameter group  on a C*-algebra , we are going to define an analytic extension of . For each , let
,
which is a horizontal strip in the complex plane. We call a function  norm-regular if and only if the following conditions hold:
 It is analytic on the interior of , i.e., for each  in the interior of , the limit  exists with respect to the norm topology on .
 It is norm-bounded on .
 It is norm-continuous on .
Suppose now that , and let

Define  by . The function  is uniquely determined (by the theory of complex-analytic functions), so  is well-defined indeed. The family  is then called the analytic extension of .

Theorem 1. The set , called the set of analytic elements of , is a dense subset of .

Definition (K.M.S. weight). Let  be a C*-algebra and  a weight on . We say that  is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on  if and only if  is a proper weight on  and there exists a norm-continuous one-parameter group  on  such that
  is invariant under , i.e.,  for all , and
 for every , we have .

We denote by  the multiplier algebra of .

Theorem 2. If  and  are C*-algebras and  is a non-degenerate *-homomorphism (i.e.,  is a dense subset of ), then we can uniquely extend  to a *-homomorphism .

Theorem 3. If  is a state (i.e., a positive linear functional of norm ) on , then we can uniquely extend  to a state  on .

Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is an ordered pair , where  is a C*-algebra and  is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions:
 The co-multiplication is co-associative, i.e., .
 The sets  and  are linearly dense subsets of .
 There exists a faithful K.M.S. weight  on  that is left-invariant, i.e.,  for all  and .
 There exists a K.M.S. weight  on  that is right-invariant, i.e.,  for all  and .

From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight  is automatically faithful. Therefore, the faithfulness of  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.

See also 
 Locally compact space
 Locally compact field
 Locally compact group

References 
Johan Kustermans & Stefaan Vaes. "Locally Compact Quantum Groups." Annales Scientifiques de l’École Normale Supérieure. Vol. 33, No. 6 (2000), pp. 837-934.
 Thomas Timmermann. "An Invitation to Quantum Groups and Duality - From Hopf Algebras to Multiplicative Unitaries and Beyond." EMS Textbooks in Mathematics, European Mathematical Society (2008).

C*-algebras
Functional analysis
Quantum groups
Harmonic analysis
Representation theory