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
![{\displaystyle D_{z}:=\{a\in A\mid {\text{There exists a norm-regular}}~f:I(z)\to A~{\text{such that}}~f(t)={\alpha _{t}}(a)~{\text{for all}}~t\in \mathbb {R} \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6663342d882cf061a01eeadf4e5751d33a0603b7)
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.
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).