Download Locally compact quantum groups

Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
Locally compact quantum groups
A. Van Daele
Department of Mathematics
University of Leuven
September 2009 - Seminar Trondheim
References
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Outline
Introduction.
The coproduct (and counit).
The coinverse (or antipode).
C∗ -algebras versus von Neumann algebras.
Locally compact quantum groups.
Conclusion.
References.
Conclusion
References
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Introduction
What is a locally compact quantum group?
Consider a locally compact space X . Then C0 (X ) is an
abelian C∗ -algebra and any abelian C∗ -algebra is of this
form.
Therefore, we think of a non-abelian C∗ -algebra as ’the
space of continuous functions’, ’tending to 0 at infinity’ on a
(non-existing) ’locally compact quantum space’.
Definition (preliminary)
A locally compact quantum group is ’locally compact quantum
space’ with a ’product’, an ’identity element’ and an ’inverse’ for
any element.
The main problem is the ’inverse’ (the antipode).
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The coproduct and the counit
The coproduct
Let G be a locally compact group. Consider A = C0 (G) and
identify A ⊗ A = C0 (G × G) and M(A ⊗ A) = Cb (G × G). Define
∆ : A → M(A ⊗ A) by
∆(f )(p, q) = f (pq).
Then ∆ is a coproduct on A.
Definition
Let A be (any) C∗ -algebra. A coproduct is an injective
non-degenerate ∗ -homomorphism ∆ : A → M(A ⊗ A), satisfying
coassociativity (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆.
Usually, it is also assumed that the sets (A ⊗ 1)∆(A) and
∆(A)(1 ⊗ A) are subsets of A ⊗ A.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The coproduct and the counit
The counit
Consider again G and C0 (G). Define ε : A → C by ε(f ) = f (e)
where e is the identity in G. Then ε is a ∗ -homomorphism
satisfying
(ε ⊗ ι)∆(f ) = f
and (ι ⊗ ε)∆(f ) = f .
This suggest the following definition.
Definition (preliminary)
Let A be a C∗ -algebra with a coproduct ∆. A counit on A is a
∗ -homomorphism from A to C satisfying the above equation for
all a in A.
The problems start here. In many interesting cases, such a
counit does not exist. However, the problem is not serious.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The coinverse or antipode
The abelian case
In the case of A = C0 (G), define S : A → A by S(f )(p) = f (p−1 )
for all p ∈ G. Then S is a ∗ -automorphism of A satisfying
m(S ⊗ ι)∆(f ) = ε(f )1
and m(ι ⊗ S)∆(f ) = ε(f )1
for all f ∈ A where m denotes multiplication, defined on
M(A ⊗ A) by m(f )(p) = f (p, p) for p ∈ G.
This also suggests the definiton of a coinverse in the general
case of a coproduct ∆ on a C∗ -algebra A, but the problems are
obvious and much more serious now.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
The coinverse or antipode
Problems with the passage from the abelian to the non-abelian case
What are the problems?
The antipode is expected to be a anti-automorphism.
The square of the antipode is not necessarily the identity
map.
The antipode is not expected to be a ∗ -map but rather it
should satisfy S(a∗ ) = S −1 (a)∗ .
The antipode is not bounded in general.
The multiplication map m is not well-defined on A ⊗ A
when A is not abelian.
So ... there is no hope to give a meaning to expressions like
m(S ⊗ ι)∆(a)
in the general situation.
and m(ι ⊗ S)∆(a)
References
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The coinverse or antipode
Towards a definition of the antipode
Let A be a C∗ -algebra with a coproduct ∆.
How to define and/or characterize the antipode? The idea is to
look at other properties of the antipode in the abelian case that
can be translated to the non-abelian case.
Proposition
Let G be a locally compact group and let ϕ be the weight on
C0 (G) obtained by integration over the left Haar measure. Then
we have the following formula. For any two functions
f , g ∈ K (G) we have S(u) = v if
u = (ι ⊗ ϕ)(∆(f )(1 ⊗ g))
and v = (ι ⊗ ϕ)(1 ⊗ f )(∆(g).
This idea was basically used in most of the earlier approaches
to locally compact quantum groups.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The coinverse or antipode
An approach, not depending on the Haar weights
Proposition
Let (A, ∆) be a Hopf algebra with antipode S. Then S(a) = b if
X
X
a⊗1=
∆(pi )(1 ⊗ qi ) and b ⊗ 1 =
(1 ⊗ pi )∆(qi ).
i
i
In the group case, we have a similar result.
Proposition
If we approximate
f (r ) = f (rs · s−1 ) ≃
we also get
X
i
X
pi (rs)qi (s),
pi (s)qi (rs) ≃ f (s(rs)−1 ) = f (r −1 ).
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
The C∗ algebra vs the von Neumann algebra approach
Definition
Let A be a C∗ -algebra with a coproduct. Assume that the
spaces ∆(A)(A ⊗ 1) and ∆(A)(1 ⊗ A) are dense in A ⊗ A. The
pair (A, ∆) is a locally compact quantum group if there exist a
left and a right Haar weight.
The weights are supposed to be central: If extended to the
double dual, they have central supports.
Now, it turns out to be relatively simple to obtain the following
property, without the need to develop the full theory on this
C∗ -algebra level:
Proposition
If ϕ is a left Haar weight and if ψ is a right Haar weight, then
they have the same central support in the double dual.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally compact quantum groups
Definition
Recall the definition (in the von Neumann algebraic context):
Definition
Let M be a von Neumann algebra and ∆ a coproduct on M.
The pair (M, ∆) is called a locally compact quantum group if
there exists a left and a right Haar weight.
A coproduct ∆ on M is a normal, unital ∗ -homomorphism
from M to the von Neumann tensor product M ⊗ M
satisfying coassociativity (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆.
A left Haar weight ϕ on (M, ∆) is a faithful, normal,
semi-finite weight on M that satisfies left invariance:
(ι ⊗ ϕ)∆(x) = ϕ(x)1 when x ≥ 0 and ϕ(x) < ∞.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally compact quantum groups
The antipode: definition and properties
Let (M, ∆) be a von Neumann algebraic locally compact
quantum group.
Proposition
Let D(S) be the space of elements x ∈ M such that there is an
element y ∈ M with the property that we can jointly approximate
X
X
x ⊗1≃
∆(pj )(1 ⊗ qj∗ ) and y ⊗ 1 ≃
∆(qj )(1 ⊗ pj∗ ).
j
j
The space D(S) is dense in M. The element y is uniquely
determined by x and we can define a linear map S : D(S) → M
by S(x) = y ∗ .
The linear map S is called the antipode of (M, ∆).
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally compact quantum groups
The antipode - properties
Remark
The right Haar weight is used to show that S is
well-defined (i.e. that y is uniquely determined by x).
The left Haar weight is used to show that S is densely
defined.
It is immediate from the definition that S is a closed linear
map.
If x ∈ D(S), then S(x)∗ ∈ D(S) and S(S(x)∗ )∗ = x.
The domain is a subalgebra and S is a
anti-homomorphism.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally complact quantum groups
The antipode - first properties
We mentioned already that the left Haar weight ϕ is used to
show that the domain D(S) is dense. More precisely, we have
the following property.
Proposition
If x, y ∈ M are such that ϕ(x ∗ x) < ∞ and ϕ(y ∗ y ) < ∞ and if
z = (ι ⊗ ϕ)(∆(x ∗ )(1 ⊗ y ))
then z ∈ D(S) and
S(z)∗ = (ι ⊗ ϕ)(∆(y ∗ )(1 ⊗ x)).
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally compact quantum groups
The antipode - more details
The main idea behind this approach to the antipode goes as
follows.
Consider the left ideal Nψ of elements x ∈ M satisfying
ψ(x ∗ x) < ∞.
Take the G.N.S. construction for ψ. Denote the Hilbert
space with Hψ and use Λψ for the canonical map from Nψ
to Hψ . Let M act directly on Hψ .
Construct the right regular representation V on Hψ ⊗ Hψ
defined by
V (Λψ (x) ⊗ Λψ (y )) = (Λψ ⊗ Λψ )(∆(x)((1 ⊗ y )).
Use this map to define the operator G : Λψ (x) 7→ Λψ (S(x)∗ )
in a similar way as is done for x 7→ S(x)∗ .
The operator G implements the map x 7→ S(x)∗ essentially
because S(xy )∗ = S(x)∗ S(y )∗ .
... etc. ...
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Locally compact quantum groups
Further steps
In this approach to the theory of locally compact quantum
groups, the further development is based very much on this
definition of the antipode:
The uniqueness of the left and right Haar weights is
obtained in an early stage.
The polar decomposition of the Hilbert space operator G is
used to get the polar decomposition S = Rτ− i of the
2
antipode S on M.
All sorts of equations are quickly found, e.g.
∆(σt (x)) = (τt ⊗ σt )∆(x)
∆(σt′ (x)) = (σt′ ⊗ τ−t )∆(x)
′ )∆(x)
∆(τt (x)) = (τt ⊗ τt )∆(x)
= (σt ⊗ σ−t
... etc. ...
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Conclusion and final remarks
We finish with a few important remarks:
Remark
It is well-known that there are two versions of locally
compact quantum groups: A C∗ -algebraic one and a von
Neumann algebraic version. They are completely
equivalent.
It is possible to pass from the concept of a locally compact
quantum group in the C∗ -algebraic framework to the one in
the von Neumann algebraic setting in an early stage of the
development.
And it is much easier to develop the theory further in this
von Neumann algebra framework.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
Conclusion and final remarks
Remark
By defining the antipode independently from the Haar
weights, it is possible to obtain the uniqueness of the Haar
weights rather quickly.
The construction of the dual locally compact quantum
group is more or less standard.
An important technical tool is the Tomita-Takesaki theory.
Introduction
The coproduct
C*- versus VN-algebras
Locally compact quantum groups
Conclusion
References
References
Relevant references for the talk
1. J. Kustermans & S. Vaes Locally compact quantum
groups, Ann. Sci. Éc. Norm. Sup. 33 (2000), 837–934.
2. J. Kustermans & S. Vaes, Locally compact quantum
groups in the von Neumann algebra setting, Math. Scand.
92 (2003), 68-92.
3. T. Masuda, Y. Nakagami & S.L. Woronowicz, A C∗
-algebraic framework for the quantum groups, Int. J. of
Math. 14 (2003), 903-1001.
4. A. Van Daele, Locally compact quantum groups. The von
Neumann algebra versus the C∗ -algebra approach,
Bulletin of the Kerala Mathematics Association, Special
Issue (2006), 153-177.
5. A. Van Daele, Locally compact quantum groups. A von
Neumann algebra approach, Preprint K.U.Leuven (2006),
Arxiv math.OA/0602.212