Examples of non-compact quantum group actions

0.25 MB·English

by  Piotr M. Sołtan

EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS
PIOTRM.SOL TAN
Abstract. Wepresenttwoexamplesofactionsofnon-regularlocallycompactquantumgroups
0
on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these
1
examples,butthedefinitionsweusehavetheadvantageofbeingexpressedinpurelyC∗-algebraic
0
language. We also discuss continuity of the obtained actions. Finally we describe in detail a
2
generalconstructionofquantumhomogeneousspacesobtainedasquotientsbycompactquantum

n subgroups.
a
J

4

] 1. Introduction
A
The main aim of this paper is to present two examples of quantum homogeneous spaces for
O
non-regular quantum groups. We will construct quotients of the quantum “az+b” and quantum
.
h E(2)groups(from[27,18,17]and[21]respectively)bytheirclassicalsubgroups. Inbothcasesour
at definition of the quantum homogeneous space will be deeply rooted in the particular form of the
m C∗-algebras related to these quantum groups. On the other hand, our definitions of quantum ho-
mogeneousspaceswillbegiveninpurelyC∗-algebraiclanguage, withoutanyuseofvonNeumann
[
algebras. Let us also remark that non-regular quantum groups do not fit into the elaborate and

1
powerful framework of [20]. We will prove that in both cases the action of the quantum group on
v
its homogeneous space is continuous in an appropriate sense. The second example (presented in
0
2 Section 4) can also be used as basis of a simple definition of a quantum homogeneous space of the
5 form / , where is a quantum group and is a compact quantum subgroup of . We develop
0 this idGeaKin the laGst section. The notion of Ka quantum subgroup we will use is mGore restrictive
1. than that of [20]. On the other hand, our construction works for general bisimplifiable Hopf C∗-
0 algebras,notonlyforlocallycompactquantumgroups. Inparticularwecanuseuniversalversions
0 of quantum groups ([8], [19, Section 5]).
1
Wewillprecedeourexamplesofquantumhomogeneousspaceswithabriefdiscussionofactions
:
v of classical locally compact groups on C∗-algebras. We shall provide a description of such actions
Xi in language appropriate for generalization to quantum groups. Continuity of actions is discussed
andadefinitionofacontinuousactionofaquantumgroupisproposedandcomparedwithexisting
r
a approaches.
The tools used to prove our results will range from the notion of G-product and Lanstad
algebra to C∗-algebras generated by unbounded affiliated elements. The material on crossed
products by of C∗-algebras by actions of abelian groups (including the notion of a G-product
and Lanstad algebra) can be found in [11, 13] and [7]. For the notions of multiplier algebra of a
C∗-algebra, strict topology, morphisms of C∗-algebras and elements affiliated to C∗-algebras we
referto[24,10,25]. Letusonlymentionherethatbyamorphism formaC∗-algebraAtoanother
C∗-algebra B we shall mean a ∗-homomorphism ϕ form A to the multiplier algebra M(B) of B
which is nondegenerate, i.e. one whose image contains an approximate unit for B. The set of all
morphisms from A to B will be denoted by Mor(A,B). In [25] the reader will also find a detailed
exposition of the concept of a C∗-algebra generated by unbounded elements affiliated with it. We
will also use some earlier work on one of our our examples from [16] as well as a wide range of
results related to the considered quantum groups ([22, 21, 23, 27, 18, 17]).
Date:January4,2010.
2000 Mathematics Subject Classification. 46L89,46L55,46L85.
Key words and phrases. quantumgroup,quantumgroupaction,C∗-algebra,quantumhomogeneousspace.
ResearchpartiallysupportedbyPolishgovernmentgrantno.N201177033.
1
2 PIOTRM.SOL TAN
The definition of a locally compact quantum group is due to Kustermans and Vaes ([9]), but
we will not be making any use of the Haar weights of the quantum groups under consideration.
On the other hand our quantum groups will have a continuous counit. For the theory of compact
quantum groups we refer to [26].
Let us briefly describe the contents of the paper. We begin with a section which carefully
rephrasesthetheoryofactionsoflocallycompactgroupsonC∗-algebrasintthelanguagesuitable
forquantumgroups(muchlikevariousaspectsoflocallycompactgroupactionsonlocallycompact
spacesweretreatedin[5]). Therearenonewresultsinthatsection,butwefeelthatsomeconcepts
present in the literature require clarification. At the end of this section we define continuous
actions of quantum groups on C∗-algebras (quantum spaces) and provide a short explanation of
the standard problems encountered in construction of quantum homogeneous spaces. In Section
3 we introduce the quantum “az+b” groups for various values of the deformation parameter and
definethehomogeneous spaceobtained asthequotient byaclassicalsubgroup. Thisconstruction
was already performed in [16]. Then we analyze the action of the quantum “az+b” group on its
homogeneous space to show that it is continuous. The example from Section 4 is very similar in
spirit, but the methods of analyzing this example are significantly different.
MotivatedbytheExamplefromSection4, inthelastsectionwegeneralizeknownconstruction
ofaquotientquantumhomogeneousspaceforcompactquantumgroup(cf.[14,15])tothesituation
wherethegroupisnolongercompact,butthesubgroupis. Weshowthattheactionoftheoriginal
quantum group on the resulting quantum homogeneous space is continuous.
2. Continuity of classical actions
Let (B,G,α) be a C∗-dynamical system. Then for each x∈B we have the function
α(x):G∋t7−→α (x)∈B. (2.1)
t
This function has constant norm and is norm-continuous by assumption. In particular it does not
vanish at infinity on G unless x=0. It follows that α(x)6∈C (G)⊗B for non-zero x. However,
0
it is easy to see that α is a ∗-homomorphism from B to M C (G)⊗B , where the last algebra
0
is naturally identified with the space of all norm-bounded fu(cid:0)nctions G→(cid:1)M(B) continuous in the
strict topology.
This raises the question how to describe the special property that for each x∈B the function
α(x) has its values in B (not merely in M(B)) and is norm-continuous (not only strictly contin-
uous). The well known answer is that these properties are equivalent to the fact that for each
f ∈C (G) the element
0
(f ⊗ )α(x)∈C (G)⊗B
0
1
(consider a function f constant and non-zero on a neighborhood of a given point t∈G).
Let us note here the fact which is well known, but difficult to find in the literature:
Proposition 2.1. Let (B,G,α) be a C∗-dynamical system and for each x∈B let α(x) be defined
by (2.1). Then the linear span of the set
(f ⊗ )α(x) f ∈C (G), x∈B
0
1
(cid:8) (cid:9)
is dense in C (G)⊗B.
0
We will give a very short and easy proof of this fact once appropriate structure on C (G) has
0
been defined (in the proof of Proposition 2.3). Nevertheless we want to include an elementary
proof of Proposition 2.1 which does not use the additional structure of C (G).
0
Proof of Proposition 2.1. We need to approximate any element of C (G)⊗B by functions of a
0
specificform. ItisenoughtoapproximateelementsfromadensesubspaceC (G,B)⊂C (G,B)∼=
c 0
C (G)⊗B. Therefore let F be a fixed continuous function G → B with compact support. For
0
t∈suppF let
Ut = s∈G F(s)−αst−1 F(t) <ε .
(cid:8) (cid:13) (cid:0) (cid:1)(cid:13) (cid:9)
(cid:13) (cid:13)
EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 3
n
By compactness of suppF there exist t ,...,t such that suppF ⊂ U . Let (χ ) be a
1 n ti i i=1,...,n
iS=1
partition of unity subordinated to the covering (U ) of suppF and define
ti i=1,...,n
n
X(t)= χj(t)αt αt−1 F(tj) .
Xj=1 (cid:16) j (cid:0) (cid:1)(cid:17)
In other words
n
X =Xj=1(χj ⊗1)α(cid:16)αt−j1(cid:0)F(tj)(cid:1)(cid:17).
Take now t∈suppF. We have
n
(cid:13)X(t)−F(t)(cid:13)=(cid:13)(cid:13)Xj=1χj(t)αtt−j1(cid:0)F(tj)(cid:1)−F(t)(cid:13)(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) n n (cid:13)
=(cid:13)(cid:13)Xj=1χj(t)αtt−j1(cid:0)F(tj)(cid:1)−Xj=1χj(t)F(t)(cid:13)(cid:13)
(cid:13) (cid:13)
(cid:13)n (cid:13)
≤ χj(t) αtt−1 F(tj) −F(t) .
Xj=1 (cid:13) j (cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
Now the j-th term of the last sum is non-zero only if t ∈ U . It follows that each term of this
tj
sum is either zero or less than χ (t)ε.
j
Wehavethusshownthatforeachiwehave sup X(t)−F(t) ≤ε. Ontheotherhand,outside
t∈Uti(cid:13) (cid:13)
n (cid:13) (cid:13)
U both X and F are zero. (cid:3)
ti
iS=1
An immediate corollary of Proposition 2.1 is that the linear span of
(f ⊗y)α(x) f ∈C (G), x,y ∈B
0
(cid:8) (cid:9)
is dense in C (G)⊗B. Indeed, if f ⊗y is a simple tensor in C (G)⊗B then we can approximate
0 0
it by finite sums of the form
(f ⊗ )α(x ).
i i
1
X
Now if (e ) is a (bounded) approximate unit for B then, by the diagonal argument, we can
λ
approximate f ⊗y by sums of the form (f ⊗e )α(x ) because
i λ i
P
(f ⊗e )α(x )−g⊗y ≤ ( ⊗e ) (f ⊗ )α(x )−g⊗y + g⊗(e y−y) .
i λ i λ i i λ
(cid:13) (cid:13) (cid:13) 1 (cid:18) 1 (cid:19)(cid:13)
(cid:13)X (cid:13) (cid:13) X (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Sinc(cid:13)etheapproximationworks(cid:13)fors(cid:13)impletensors,italsoworksforgenera(cid:13)lelementsofC0(G)⊗B.
Corollary 2.2. Let (B,G,α) be a C∗-dynamical system and for each x ∈ B let α(x) be defined
by (2.1). Then α∈Mor B,C (G)⊗B .
0
(cid:0) (cid:1)
Letusnotehereoneotherdensityconditionappearingintheliterature. Quiteclearlythelinear
span of
ϕ(t)α (x)dt ϕ∈L1(G), x∈B
t
(cid:26)Z (cid:27)
is dense in B (any x∈B is the limit of such integrals with δ-like net of integrable functions). In
order to rewrite this condition in a more convenient way let us introduce for each ϕ∈L1(G) the
continuous functional
ω :C (G)∋f 7−→ ϕ(t)f(t)dt.
ϕ 0
Z
Then we see that the linear span of
(ω ⊗id)α(x) ϕ∈L1(G), x∈B
ϕ
(cid:8) (cid:9)
is dense in B.
4 PIOTRM.SOL TAN
Let us introduce the standard comultiplication ∆∈Mor C (G),C (G)⊗C (G) :
0 0 0
(cid:0) (cid:1)
∆(f)(s,t)=f(st). (2.2)
Then one easily checks that (∆⊗id)◦α=(id⊗α)◦α.
Clearly one can also see that (ǫ⊗id)◦α=id, where ǫ is the evaluation functional
C (G)∋f 7−→f(e)∈ .
0
C
We will now state and prove a simple proposition dealing with various conditions defining
continuity of a group action in the language adaptable to the more general context of quantum
groups.
Proposition 2.3. Let G be a locally compact group and let B be a C∗-algebra. Let
α∈Mor B,C (G)⊗B
0
(cid:0) (cid:1)
be such that
• with ∆ defined by (2.2) we have
(∆⊗id)◦α=(id⊗α)◦α (2.3)
• for any f ∈C (G) and x∈B we have
0
(f ⊗ )α(x)∈C (G)⊗B,
0
1
Then the following conditions are equivalent
(1) (ǫ⊗id)◦α=id, where ǫ is the evaluation at the neutral element of G,
(2) kerα={0},
(3) the linear span of
(f ⊗ )α(x) f ∈C (G), x∈B (2.4)
0
1
is dense in C (G)⊗B,(cid:8) (cid:9)
0
(4) the linear span of
(ω ⊗id)α(x) ϕ∈L1(G), x∈B (2.5)
ϕ
is dense in B. (cid:8) (cid:9)
Moreover, if the equivalent conditions (1)–(4) are satisfied then then there exists a continuous
action α of G on B such that α is defined by (2.1).
Proof. Let us begin with defining a family (αt)t∈G of maps B →B by
α (x)=α(x)(t)=(δ ⊗id)α(x),
t t
where δ is the evaluation functional C (G) ∋ f 7→ f(t). One immediately find that each α is
t 0 t
an endomorphism of B and that for each x∈B the mapping G∋t7→α (x) is norm continuous.
t
Moreover, it follows from (2.3) that
α ◦α =α
t s ts
for all t,s∈G. In particular α is an idempotent mapping B →B which commutes with all α ’s.
e t
The range of all α ’s is equal to the range of α and their kernels are all equal to kerα .
t e e
It follows that (1)⇔(2). Indeed, (1) clearly implies (2), but (2) means that for any non-zero
x∈B thefunctiont7→α (x)isnonzero. Thusifx∈kerα thenα (x)=0forall tandsox=0.
t e t
In other words, α is an idempotent with zero kernel, so α =id. This is exactly (1).
e e
In particular if (1) is satisfied then all α ’s are automorphisms and α becomes a continuous
t
actionofGonB. Toseethatthisimplies(3)withoutusingProposition2.1notethatthemapping
C (G)⊗ B ∋(f ⊗x)7−→(f ⊗ )α(x)∈C (G)⊗B (2.6)
0 alg 0
1
extends to an automorphism of C (G)⊗B. Indeed, the map (2.6) is bounded and multiplicative
0
(because C (G) is commutative). The inverse mapping is given by
0
(f ⊗x)7−→(f ⊗ )(id⊗κ) α(x) ,
1
where κ is the automorphism of C (G) given by κ(f)(t) =(cid:0) f(t−(cid:1)1). Therefore the linear span of
0
(2.4) is dense in C (G)⊗B as the image of C (G)⊗ B under an automorphism of C (G)⊗B.
0 0 alg 0
EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 5
The fact that (3)⇒(4) is standard: approximate a simple tensor f⊗y ∈C (G)⊗B by sums of
0
the form
(f ⊗ )α(x ).
i i
1
X
Then take ϕ∈L1(G) such that ω (f)=1. Then the elements
ϕ
(ω ⊗id)α(x )
ϕfi i
X
approximate y and consequently the linear span of (2.5) is dense in B.
Finally let us see that (4) implies (1). If (1) is not satisfied then we know that the ranges of
all α ’s are equal to the range of α which is strictly contained in B and closed (as the image of a
t e
C∗-algebra under a ∗-homomorphism). Then it follows that for any ϕ∈L1(G) the element
(ω ⊗id)α(x)= ϕ(t)α (x)dt
ϕ t
Z
also lies in the range of α and we see that (4) is not satisfied. (cid:3)
e
Inwhatfollowswewilldealwithquantumgroupsoftheform =(A,∆)withcontinuouscounit
ǫ. Wewillsaythat actscontinuouslyonaC∗-algebraBifthereGisamorphismα∈Mor(B,A⊗B)
G
such that
• (∆⊗id)◦α=(id⊗α)◦α,
• for any c∈A and x∈B we have (c⊗ )α(x)∈A⊗B,
1
• (ǫ⊗id)◦α=id.
In the literature the last condition is often replaced by either requirement that the linear span
of
(c⊗ )α(x) c∈A, x∈B
1
be dense in A⊗B or that (cid:8) (cid:9)
(ω⊗id)α(x) ω ∈A∗, x∈B
bedenseinB (whereA∗ isthesp(cid:8)aceofnormallinearfunctiona(cid:9)lsonA, cf.[19]). In[3, Proposition
5.8] it is shown that these conditions are equivalent for regular locally compact quantum groups
and that the second condition is strictly weaker for semi-regular, but non-regular locally compact
quantum groups.
If α ∈ Mor(B,A⊗B) is such that (∆⊗id)◦α = (id⊗α)◦α and kerα = {0} then the
established terminology is that α is a reduced action of on B (see e.g. [20, 6]). As Proposition
G
2.3 shows, in case when is a classical locally compact group, all these conditions are equivalent.
G
Let =(A,∆) and =(C,∆ ) be quantum groups. We say that a morphism π ∈Mor(A,C)
C
G K
identifies as a closed quantum subgroup of if π is a surjective ∗-homomorphism A → C and
K G
(π⊗π)◦∆ = ∆ ◦π. As we mentioned in Section 1, such a definition of a quantum subgroup
C
is more restrictive than that of [20]. In the following sections we will give examples of situations
where this notion of quantum subgroup applies.
Letusexplainthedifficultywhichwecomeacrossaswetrytogeneralizethestandardconstruc-
tion of a quotient quantum homogeneous space from [15]. Let us assume that the C∗-algebras
A and B are commutative, so that A = C (G) and C = C (K), where G is a locally com-
0 0
pact group and K is a closed subgroup of G. In this situation π is the restriction morphism
A ∋ f 7→ f ∈ M(C). The range rarely coincides with C. In the language of noncommutative
K
topology th(cid:12)e homogeneous space G/K is described by the C∗-algebra B = C (G/K). It is a
(cid:12) 0
nontrivial question howtoidentify thisC∗-algebraasasubalgebraof M(A)=C (G). Clearly any
b
element x∈B should satisfy
(id⊗π)∆(x)=x⊗ , (2.7)
1
but this is certainly not enough. In fact all elements of M(B) also satisfy this condition. In the
examples of Sections 3 and 4 = (A,∆) will be a non-regular locally compact quantum group
G
with a subgroup =(C,∆ ) (which in both cases will be a classical group) and we will produce
C
a candidate for BK. It will be a C∗-subalgebra of M(B) consisting of elements x satisfying (2.7)
andsomeadditionalconditionstailoredtothesespecificexamples. Thenextquestionwhicharises
naturallyafterthedefinitionofahomogeneousspaceiswhethertherestrictionof∆toB actually
6 PIOTRM.SOL TAN
defines an action of on / . We will have to check that α = ∆ is a morphism from B to
G G K B
A⊗B and that algebraic conditions of a (co)action are satisfied. Mo(cid:12)re importantly we would like
(cid:12)
to show that the action α is continuous. None of these facts is totally obvious and we prove them
by application of the theory of C∗-algebras generated by unbounded elements ([25]).
3. The action of the quantum “az+b” group on its homogeneous space
Theconstructionofthequantum“az+b”groupstartswithchoosingthevalueofthedeforma-
tion parameter q. There are at least three possibilities to choose q. They are described in detail
in [17]. One of these possibilities is to take q ∈]0,1[ (cf. [27, Appendix A]). It is not important for
our purposes to dwell on this choice since the aspect of quantum “az+b” groups we shall need
are the same regardless of the value of the deformation parameter. All details can be found in
[27, 18, 17].
We let Γ be the subgroup of the multiplicative group \{0} generated by q and {qit t ∈ }
C R
with an appropriate fixed choice of logarithm of q. The group Γ is then self-dual and we denote
by χ the nondegenerate bicharacter Γ×Γ→ such that
T
′ ′
χ(γ,γ )=χ(γ ,γ),
χ(γ,qit)=|γ|it,
χ(γ,q)=Phaseγ
for all γ,γ′ ∈Γ and t∈ .
R
Then we let Γ be the closure of Γ in and let β be the action of Γ on Γ by multiplication. We
define A = C (Γ)⋊ Γ. An affiliated eClement bηA is introduced as the image of the canonical
0 β
generator of C (Γ) in A, while aηA as the unique affiliated element such that Spa ⊂ Γ, a is
0
invertible (with a−1ηA) and χ(a,γ) is the canonical family of unitary elements of M(A)
γ∈Γ
implementing the action β on C(cid:0) (Γ). (cid:1)
0
By the results of [27, 18] (the actual reference depends on the value of q, cf. [17]) there exists
a unique ∆∈Mor(A,A⊗A) such that
∆(a)=a⊗a, ∆(b)=a⊗b+˙ b⊗
1
and =(A,∆) is a quantum group (in fact a non-regular locally compact quantum group cf. also
G
[28]).
LetusfirstseethatΓisasubgroupof inthesensedescribedattheendofSection2,i.e.there
G
exists a morphism π ∈Mor A,C (Γ) such that
0
(cid:0) (cid:1)
∆◦π =(π⊗π)◦∆
and π is a surjection onto C (Γ). Indeed, since the C∗-algebra A is generated by unbounded
0
elements a,a−1 and b affiliated with it (cf. [27, 18, 17]), it is enough to define π on the generators
by putting π(b) = 0 and letting π(a) be the canonical generator of C (Γ) (the identity function
0
Γ∋γ 7→γ ∈ ).
C
Let us remark here that in [16] it is the dual group of Γ which is identified with a subgroup of
which we are describing. This is correct since Γ∼=Γ, but the distinction has no bearing on our
G
construction of quantum homogeneous space, so we will stick with the choice of Γ as a subgroup
b
of .
GWe shall now define a quantum homogeneous space /Γ. By definition the C∗-algebra B of
G
continuous functions vanishing at infinity on /Γ is the set of those x∈M(A) for which
G
(1) (id⊗π)∆(x)=x⊗ ,
(2) for any y ∈C∗(Γ)⊂1M(A) the element yx∈A,
(3) the mapping Γ∋γ 7→χ(a,γ)∗xχ(a,γ) is continuous.
Condition (1) selects elements of M(A) which are “constant along the fibers” of the fibration
→ /Γ. However, they cannot correspond to functions vanishing at infinity on /Γ (e.g. the
G G G
unit satisfies this condition). Therefore condition (2) is introduced to select those “continuous
1
bounded functions on ” which not only are constant on cosets of Γ, but in addition vanish
G
at infinity “in the direction transversal to the cosets.” Condition (3) does not have a classical
EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 7
analogue. This definition of the algebra of “continuous functions vanishing at infinity on /Γ”
G
was first introduced in [16].
It is not difficult to check that B = f(b) f ∈ C (Γ) . This is because conditions (1)–(3)
0
are really the Lanstad conditions for the(cid:8)Γ-product struct(cid:9)ure on A = C (Γ)⋊ Γ (cf. [11, 13]).
0 β
Therefore the homogeneous space /Γ is a classical space, i.e. one described by a commutative
C∗-algebra. G
It was pointed out already in [16] that the restriction of ∆ to B ⊂ M(A) defines a morphism
α∈Mor(B,A⊗B) and that
(∆⊗id)◦α=(id⊗α)◦α.
Moreover one can easily see that (ǫ⊗id)◦α=id, where ǫ is the counit of defined by ǫ(a)=1,
G
ǫ(b) = 0. Therefore, in order to say that the action of on the homogeneous space /Γ is
G G
continuous we only need to prove the following:
Theorem 3.1. Let =(A,∆)bethequantum“az+b”groupandletB ⊂M(A)betheC∗-algebra
G
offunctions vanishing at infinity onthequantum homogeneous space /Γ. Let α∈Mor(B,A⊗B)
G
be the morphism defined above. Then for any c∈A and any x∈B the element
(c⊗ )α(x)∈A⊗B. (3.1)
1
Proof. We shall identify B with C (Γ) and A⊗B with A⊗C (Γ)∼=C (Γ,A). The element bηB
0 0 0
corresponds to the function Γ∋z 7→z ∈ and α(b) to
C
Γ∋z 7−→(az+˙ b)ηA.
(cf. [25, Formula 2.6]).
It follows that for any f ∈C (Γ) the image of f(b) under α is identified with the element
0
Γ∋z 7−→f(az+˙ b)
of Cstrict Γ,M(A) . Therefore with x=f(b) the element (3.1) corresponds to the function
b
(cid:0) (cid:1)
Γ∋z 7−→cf(az+˙ b)
which is an element of C (Γ,A). We want to show that this is in fact an element of C (Γ,A) =
b 0
C (Γ)⊗A.
0
According to results [22, 27, 18] (again the reference depends on the choice of q) there is a
continuous function F :Γ→ such that az+˙ b=F (z−1ba−1)∗azF (z−1ba−1), so that
q q q
T
f(az+˙ b)=F (z−1ba−1)∗f(az)F (z−1ba−1). (3.2)
q q
Our aim is to show that for any fixed c∈A the norm of
cF (z−1ba−1)∗f(az)F (z−1ba−1)
q q
tends to 0 as z goes to ∞ of Γ. Clearly this norm is equal to the norm of
cF (z−1ba−1)∗f(az).
q
Since ba−1ηA, it is known (see e.g. [27, Theorem 5.1]) that F (z−1ba−1) tends to strictly in
q
1
M(A) az z → ∞ (note that F (0) = 1). Therefore for any ε > 0 there exists a constant R such
q ε
that
cF (z−1ba−1)∗−c <ε.
q
(cid:13) (cid:13)
for all z ∈Γ with |z|>Rε. Thus th(cid:13)e norm (cid:13)
cF (z−1ba−1)∗f(az) ≤ cf(az) + cF (z−1ba−1)∗−c f(az) < cf(az) +εkfk.
q q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13)(cid:0) (cid:1) (cid:13) (cid:13) (cid:13)
Tosee(cid:13)that the normofcf((cid:13)az)(cid:13)goes to(cid:13)zero(cid:13)as z →∞we can assume t(cid:13)hat f(cid:13)has co(cid:13)mpact support
and that c is of the form
c=f˜(b)g(a)
for some f˜∈C (Γ) and g ∈C (Γ) (the set of such elements is linearly dense in A). Then
0 0
cf(za)=f˜(b)G (a),
z
8 PIOTRM.SOL TAN
where G (z′) = g(z′)f(zz′). Since g vanishes at 0, it is easy to see that kG k −−−→ 0 and it
z z
z→∞
follows that cf(za) <ε for sufficiently large |z|. (cid:3)
(cid:13) (cid:13)
(cid:13) (cid:13)
4. The action of the quantum E(2) group on its homogeneous space
ForthedescriptionofthequantumE(2)groupwewillfixaHilbertspaceHwithanorthonormal
basis(ei,j)i,j∈ . Wewillchangeournotationslightlyinordertobeinagreementwithconventions
established inZthe literature [21, 23]. Let q ∈]0,1[ be a parameter and let q be the subgroup of
\{0} generated by q and {qit t∈ } — this is Γ from Section 3 for the caCse of real q. As before
C R
we let q be the closure of q in .
C C C
We will now define two operators on H. We let v be the unitary operator defined uniquely by
vei,j =ei−1,j for a i,j ∈ and let n be the closed linear operator on H such that the linear span
of the orthonormal basisZ(ei,j)i,j∈ is a core of n and nei,j =qiei,j+1 for all i,j ∈ . Then n is a
normaloperatorwithSpn⊂ q. WZedefineAastheclosureofthesetoffinitelineaZrcombinations
C
f (n)vl, (4.1)
l
X
where f ∈ C ( q) for all l ∈ . With this definition A is a non-degenerate C∗-subalgebra of
l 0
B(H) isomorphiCc to C ( q)⋊ Z, where β is the natural action by multiplication by q (cf. [21]).
0 β
C Z
Moreover v ∈M(A) and nηA and A is generated by n and v ([25, Example 4]).
It is shown in [21] that there exists a unique ∆∈Mor(A,A⊗A) such that
∆(v)=v⊗v, ∆(n)=v⊗n+˙ n⊗v∗
and = (A,∆) is a non-regular locally compact quantum group ([1]) called the quantum E(2)
G
group.
The classical group is a subgroup of , i.e. we have a quantum group morphism π ∈
T G
Mor A,C( ) defined uniquely by putting π(n) = 0 and letting π(v) be the canonical unitary
T
gene(cid:0)rator of (cid:1)C( ). We wish to describe the quantum homogeneous space / .
T G T
Tothatendletusdefinestructureofa -productonAdifferentfromthestandardone(arising
Z
from the fact that A is the crossed product of C ( q) by ). We define a representation
0
C Z
V : ∋n7−→V =vn ∈M(A)
n
Z
and action of on A by automorphisms (β˜µ)µ∈ such that
T T
β˜ (v)=µv, β˜ (n)=µ−1n
µ µ
for all µ∈ . The triple (A,V,β˜) is a -product. This -product structure on A is not a “twist”
of the standTard -product structure onZA as defined inZ[7], but let us note that if we denote by βˆ
Z
the action of on A dual to β then for each µ∈ .
T T
β˜ (y)=U βˆ (y)U∗,
µ µ µ µ
where U is the unitary operator such that U e =µ−je .
µ µ i,j i,j
Following the example from Section 3 we define the C∗-algebra of B of continuous functions
vanishing at infinity on / as the set of those x∈M(A)
G T
(1) (id⊗π)∆(x)=x⊗id,
(2) for any y ∈C∗( )⊂M(A) the element yx∈A.
Z
The above conditions are, as in Section 3, precisely the Lanstad conditions for the -product
structure we have defined (in particular B really is a C∗-algebra). Condition 1 is exZactly the
condition of being β˜-invariant, while condition 2 says that the “function” x vanishes at infinity
in the direction transversal to that of the subgroup . In fact condition (2) is equivalent to
demanding that x ∈ A (because C∗( ) is unital). TheTthird Lanstad condition is empty because
Z
is a discrete group.
Z Before proceeding let us define a certain C∗-algebra which will turn out to be the algebra
of continuous unctions vanishing at infinity on / . Let C ( ∪{+∞}) be the C∗-algebra of
0
sequences (xn)n∈ for which lim xn = 0 andGlTim xn exisZts and is finite. We let C be the
Z n→−∞ n→+∞
EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS 9
crossed product C=C ( ∪{+∞})⋊ , where the action of is by translation. The C∗-algebra
0
C is an extension of K byZC( ). More pZrecisely we have Z
T
x y
C∼= x∈T, y,z,u∈K
(cid:26)(cid:20)z u(cid:21) (cid:27)
(where T is the Toeplitz algebra) and the extension is
0 //K∼=M (K) //C ρ //C( ) //0
2
T
with ρ sending a matrix xy to the symbol of x.
z u
(cid:2) (cid:3)
Proposition 4.1. Let B be the algebra of continuous functions vanishing at infinity on / .
G T
Then
(1) The C∗-algebra B is the closure of the set of finite sums of the form
f (n)vk (4.2)
k
X
where for each k ∈ the function f ∈C (Γ) is such that
k 0
Z
f (µz)=µkf (z)
k k
for all z ∈Γ and µ∈ .
T
(2) The operator vn is affiliated with B,
(3) B is generated by vn.
(4) B is isomorphic to C.
Proof. We shall first prove point (1). Let B be the closure of the set of finite sums described in
0
Statement (1). Since each term f (n)vn is invariant under β˜ and is already contained in A (as
k
opposed to M(A)) we have B ⊂ B. Moreover B is invariant under conjugation with v. Finally
0 0
let us note that
∗ ∗
C ( )B C ( ) (4.3)
0
Z Z
is dense in A (as before, we identify C∗( ) with the C∗-subalgebra of M(A) generated by v). This
Z
is because the functions f appearing in (4.2) are trigonometric monomials along the individual
k
circles of q. Therefore we can uniformly approximate on each circle sufficiently regular functions
C
(e.g. twice differentiable) and those are dense in all continuous functions on the circle. This can
be done on each circle separately. As a result all elements of the form (4.1) lie in the closure of
(4.3). By [7, Lemma 2.6] B =B.
0
Now we shall deal with points (2)–(4). The operator vn is affiliated with B. This is because
thez-transformofvnclearly belongs toB andtheelement −z∗ z =( +n∗n)−1 isastrictly
0 vn vn
1 1
positive element of B (cf. [25, Section 1]). Let us denote by X the operator vn and let X =U|X|
be the polar decomposition of X. We have Uei,j =ei−1,j+1 and |X|ei,j =qiei,j. One easily finds
that for f ∈C ( q) such that f (µz)=µkf (z) we have
k 0 k k
C
f (n)vk =f |X| Uk. (4.4)
k k
A moment of reflection shows now that B ∼=C and(cid:0)poin(cid:1)t (4) follows.
Let ρ be a representation of B on a Hilbert space L. Then ρ(vn) is a closed operator on
L. Moreover by (4.4) the image under ρ of any element x ∈ B is determined by the polar
0
decomposition of ρ(vn). This means that vn separates representations of B and it follows that B
is generated by vn by [25, Theorem 3.3], since +(vn)∗(vn) −1 ∈B. (cid:3)
1
(cid:0) (cid:1)
Before proceeding let us emphasize the fact that B ⊂A. Moreover since B ⊂A we have that
B = x∈A β˜ (x)=x for all µ∈
µ
T
(cid:8) (cid:9)
= x∈A (id⊗π)∆(x)=x⊗ .
1
(cid:8) (cid:9)
We also have
Lemma 4.2. A⊗B = X ∈A⊗A (id⊗β˜ )(X)=X for allµ∈ .
µ
T
(cid:8) (cid:9)
10 PIOTRM.SOL TAN
Proof. The inclusion “⊂” is obvious. For the converse one let us take X ∈ A ⊗ A such that
(id⊗β˜ )(X)=X for all µ∈ . X is a limit of finite sums of simple tensors:
µ
T
Nk
X = lim a(k)⊗b(k).
k→∞ i i
Xi=1
Let dµ be the normalized Haar measure on . We have
T
Nk
X = (id⊗β˜ )(X)dµ= lim a(k)⊗ β˜ (b(k))dµ
Z µ k→∞ i Z µ i
Xi=1
Clearlytheelementsc(k) = β˜ (b(k))dµareβ˜-invariant, sothattheybelongtoB. Itfollowsthat
i µ i
X ∈A⊗B. R (cid:3)
Now let us describe the action of on its homogeneous space / . The image of vnηA under
G G T
∆ is
v2⊗vn+˙ vn⊗ .
1
By[29,Theorem6.1]and[22,Theorem5.1]thiselementisaffiliatedwithA⊗B. Sincevngenerates
B, we see that the map α=∆ is a morphism form B to A⊗B. Clearly we have
B
(cid:12)
(cid:12) (∆⊗id)◦α=(id⊗α)◦α
and (ǫ⊗id)◦α=id, where ǫ is the counit of (defined by ǫ(v)=1 and ǫ(n)=0).
G
Since B ⊂A we have α(B)⊂A⊗A by [23, Formula (57)]. Note also that
(id⊗β˜ )◦∆=∆◦β˜ . (4.5)
µ µ
In view of Lemma 4.2, this implies that in fact α(x)∈A⊗B for any x∈B. In particular for any
c∈A and x∈B we have (c⊗ )α(x)∈A⊗B. Thus we obtain
1
Corollary 4.3. α∈Mor(B,A⊗B) is a continuous action of on / .
G G T
Remark 4.4. Let us also note that the set of all elements (c⊗ )α(x) with c∈A and x∈B is not
1
only contained (as shown above), but also linearly dense in A⊗B. Indeed if we denote by E the
conditional expectation A∋c7→ β˜ dµ∈B used in the proof of Lemma 4.2 then it follows from
µ
(4.5) that (id⊗E)◦∆=∆◦E.RUsing the fact that the span of (a⊗ )∆(b) a,b∈A is dense
1
in A⊗A we see that (cid:8) (cid:9)
span (c⊗ )α(x) c∈A, x∈B
1
(cid:8) (cid:9)
=span (c⊗ )∆(x) c∈A, x∈B
1
(cid:8) (cid:9)
=span (c⊗ )∆ E(d) c,d∈A
1
(cid:8) (cid:0) (cid:1) (cid:9)
=span (c⊗ )(id⊗E) ∆(d) c,d∈A
1
(cid:8) (cid:0) (cid:1) (cid:9)
=span (id⊗E) (c⊗ )∆(d) c,d∈A
1
(cid:8) (cid:0) (cid:1) (cid:9)
is dense in A⊗B (cf. Lemma 4.2).
5. Quotients by compact subgroups
A pair = (A,∆) consisting of a C∗-algebra A and a coassociative ∆ ∈ Mor(A,A⊗A) such
G
that
span ∆(a)( ⊗a′) a,a′ ∈A =A⊗A, (5.1a)
1
(cid:8) (cid:9)
span (a⊗ )∆(a′) a,a′ ∈A =A⊗A. (5.1b)
1
is usually called a bisimplifiable H(cid:8)opf C∗-algebra (see e.g(cid:9). [2]), while in [12] the such objects are
called“properC∗-bialgebraswithcancellationproperty.” Wewillsticktotheformerterminology.
This section is devoted to the proof of the next theorem which is a direct generalization of
the construction in [14][Section 6], [15, Section 1] (cf. also [4]) to the situation where the original
quantumgroup(orHopfC∗-algebra) isnotcompact. Weincludetheassumptionthat possesses
G G