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

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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

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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

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

See also

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References

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  • 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).