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.
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.
- 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.
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.
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.
The theory has an equivalent formulation in terms of von Neumann algebras.
- 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).