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.