Table Of ContentINDUCED COACTIONS OF DISCRETE GROUPS ON C∗-ALGEBRAS
8
9 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
9
1
Abstract. Using the close relationship between coactions of discrete groups and Fell
n bundles,we introducea procedure for inducinga C∗-coaction δ:D→D⊗C∗(G/N) of
a a quotient group G/N of a discrete group G to a C∗-coaction Indδ : IndD → IndD⊗
J
C∗(G) of G. We show that induced coactions behave in many respects similarly to
3
induced actions. In particular, as an analogue of the well known imprimitivity theorem
1
for induced actions we prove that the crossed products IndD×Indδ G and D×δ G/N
] are always Morita equivalent. We also obtain nonabelian analogues of a theorem of
A Olesen and Pedersen which show that there is a duality between induced coactions and
O twistedactionsinthesenseofGreen. WefurtherinvestigateamenabilityofFellbundles
corresponding toinduced coactions.
.
h
t
a
m
[ 1. Introduction
1 One of the most important constructions in ergodic theory and dynamical systems is
v
the construction of an induced action (or induced flow): if H is a closed subgroup of
9
6 the group G and Y is an H-space, then the induced G-space G×H Y is defined as the
0 quotient of G ×Y with respect to the equivalence relation (s,hy) ∼ (sh,y) for h ∈ H,
1
and G-action given by translation on the first factor. The induced G-action behaves in
0
almost all respects similarly to the original H-action on Y, and the theory is particularly
8
9 useful when it is possible to identify a given G-space as one which is induced from a more
/ manageable H-space.
h
t The analogue of the induced G-space in the theory of C∗-dynamical systems is the
a
m induced C∗-algebra IndG A together with the induced action Indα, where we start with
H
an action α : H → Aut(A). Needless to say, if A = C (Y), then IndG A = C (G× Y).
: 0 H 0 H
v
As for G-spaces, the importance of this construction comes from the fact that induced
i
X actions enjoy in most respects the same properties as the original ones. The most im-
r portant manifestation of this statement is certainly Green’s imprimitivity theorem (see
a
[11, Theorem 17]), which implies that the crossed product IndG A× G of the induced
H Indα
system is always Morita equivalent to the crossed product A× H of the original system.
α
To see the importance of this result, note that Morita equivalent algebras have naturally
homeomorphic representation spaces and the same K-theory.
In this paper we are concerned with the question whether a similar theory of induced
algebras can be obtained in the theory of coactions of locally compact groups. Recall that
the theory of coactions of a group G (or rather of the group C∗-algebra C∗(G) equipped
with a natural comultiplication) is in a natural way dual to the theory of actions of G: if
Date: December 30, 1997.
1991 Mathematics Subject Classification. Primary 46L55.
This research is partially supported by National Science Foundation Grant No. DMS9401253.
1
2 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
α :G → Aut(A) is an action of G on the C∗-algebra A, then there is a canonical coaction
α of G on A× G such that the double crossed product A× G× G is stably isomorphic
α α α
to A (see [13] and [26]). This generalizes the Takesaki-Takai duality theorem for actions
b b
of abelian groups, where α is an action of the dual group G of G. Conversely, starting
with any coaction δ of G on A, there exists a dual action δ : G → Aut(A × G), and
b b δ
Katayama obtained a similar duality theorem [14], which works for all normal coactions
b
(see thepreliminarysection forthenotation). Ofcourse, inordertodevelop thefullpower
of this duality theory, it is most desirableto have an as complete as possibledualmirrorof
the usual constructions for actions. In particular it would certainly be interesting to have
a working notion of induced coactions and induced C∗-algebras by coactions. At least if
G is discrete we will see here that there is indeed such a theory, and that it enjoys many
properties which are known in the theory of induced actions.
Our results are based heavily on observations due to the second author, which connect
the theory of coactions of discrete groups to the theory of Fell bundles (or C∗-algebraic
bundles) over G [23]. Recall that a Fell bundle (A,G) over G is a family of Banach spaces
As for s ∈ G, together with a multiplication As×At → Ast and an involution As → As−1,
which satisfies some further conditions. Theset Γ (A) of sections of finite supportforms a
c
∗-algebra, and a cross sectional algebra A of (A,G) is a completion of Γ (A) with respect
c
to a given C∗-norm.
If δ : A → A ⊗ C∗(G) is a coaction of a discrete group G on A, then the spectral
subspaces {A : s ∈G} (i.e., a ∈ A ⇔ δ(a ) = a ⊗s)of δ form a Fell bundle(A,G) over
s s s s s
G and A is a topologically graded cross sectional algebra for A (see §2 for more details).
Similarly to other situations in the theory of C∗-algebras, there may exist more than one
C∗-norm on Γ (A), but if we insist that the corresponding completions are topologically
c
graded, then there always exists a maximal and a minimal one. We denote the respective
cross sectional algebras by C∗(A) (for the maximal norm) and C∗(A) (for the minimal
r
one; see [7] for a detailed treatment of this). Note that both algebras, C∗(A) and C∗(A),
r
carry natural coactions δ and δn which are both determined by the property that they
A A
map a ∈ A to the element a ⊗s∈ A⊗C∗(G). Thus any other coaction δ lies “between”
s s s
δ and δn, if A is the bundle associated to δ. Note that just recently, Fell bundles over
A A
discrete groups were studied extensively by several people [23, 25, 7, 1, 6], partly due to
the discovery that many important C∗-algebras appear as cross sectional algebras of Fell
bundles.
Due to the above-described connection between coactions and Fell bundles for discrete
groups, we are able to use Fell bundles, rather than the coactions themselves, in order to
define induced coactions. Starting with a Fell bundle (D,G/N) over a quotient G/N by
a normal subgroup N of a discrete group G, we define the induced coaction simply as the
dual coaction of the maximal cross sectional algebra IndD := C∗(q∗D) of the pull-back
bundle (q∗D,G), where q : G → G/N denotes the quotient map. Note that there is a
certain arbitrariness in our definition, since we could also have taken the dual coaction (if
it exists) of any other topologically graded cross sectional algebra of q∗D (e.g., C∗(q∗D))
r
instead of the maximal one. Thus there is no canonical choice unless q∗D is amenable in
the sense of Exel [7], which roughly means that all topologically graded cross sectional
algebras are the same.
INDUCED COACTIONS 3
We will show that induced coactions behave in almost all respects similarly to induced
actions; for example, if G is abelian, the induced coactions of G correspond exactly to the
induced actions of G under the usual identification between coactions of G and actions of
G (see §2). In §3 we show that crossed products by coactions of discrete groups can be
b
realizedascrosssectionalalgebrasofcertainFellbundlesoverthetransformationgroupoid
b
G×G. In fact if (A,G) is the Fell bundle associated to the coaction δ : A→ A⊗C∗(G),
then A× G is the enveloping C∗-algebra of Γ (A×G), where A×Gis the productbundle
δ c
over the groupoid G×G. Using this result we show in §4 that there is an analogue, for
inducedcoactions, ofGreen’simprimitivitytheorem: ifδ :D → D⊗C∗(G/N)isacoaction
ofG/N,thenthereisanaturalMoritaequivalencebetweenthecrossedproductsD× G/N
δ
and IndD× G. Notice that both crossed products only depend on the underlying Fell
Indδ
bundles D and q∗D, and not on the particular choices of the cross sectional algebras D
and IndD.
In §5 we show that there is an analogue of Olesen and Pedersen’s classical result about
twisted groupactions (see[20,24]): usingavery usefulgeneralcharacterization ofinduced
coactions, whichistheanalogueofthecharacterization ofinducedactionsgivenbythefirst
authorin[3],wewillseethatadualcoaction β ofacrossedproductB× Gisinducedfrom
β
aquotientG/N ifandonlyiftheactionβ istwistedover N inthesenseofGreen[11]. This
b
leads to a negative result concerning the possibility of a “Mackey machine” for coactions:
there is an important feature of induced actions of compact groups which fails for induced
coactions of discrete groups. Namely, if β is an action of a compact group G on a C∗-
algebra B such that B has no proper G-invariant ideals, then (B,G,β) is always induced
from a system (A,H,α) with A a simple C∗-algebra; this follows from [3, Theorem],
since compactness of G guarantees, by [19, Lemma 2.1], that PrimA is equivariantly
homeomorphic to a homogeneous space G/H. For a discrete group G, however, our
characterizations ofinducedcoactionsallowustoshowthatthereexistnumerousexamples
ofG-simplecoactionswhicharenotinduced(evenintheweaksense)fromsimplecoactions!
This drawback of the theory is mainly due to the fact that the theory of coactions (at
least so far) only allows us to look at quotients by normal subgroups, while for actions we
can work with any closed subgroup of G.
Finally, in §6 we investigate under which conditions the pull-back bundles q∗D are
amenable. This question is of particular interest to us, since, as mentioned above, only
if q∗D is amenable do we have a unique choice for our induced algebra IndD. In [7]
Exel introduced a certain approximation property (which we call property (EP)), which
guarantees amenability of a given Fell bundle (A,G). For instance he showed that all
Cuntz-Krieger bundles, which arise from the natural coactions of F on the Cuntz-Krieger
n
algebras O as found in [25], satisfy property (EP), although the free group F with n
A n
generators is certainly not amenable if n > 1. If D is a bundle over G/N, then we will
show that q∗D satisfies (EP) if D satisfies (EP) and N is amenable; the amenability of N
is also necessary for q∗D to satisfy (EP). Note that as an immediate consequence of this
we see that the Cuntz-Krieger algebras are not induced from any nontrivial quotient of
F , since F does not contain any nontrivial amenable normal subgroup.
n n
This research was conducted while the second author visited the University of Pader-
born, and he thanks his hosts Siegfried Echterhoff and Eberhard Kaniuth for their hospi-
tality.
4 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
2. Preliminaries and basic definitions
Throughout this paper, G will be (except in certain remarks comparing with other
research)adiscrete group. WeareprimarilyconcernedwithcoactionsofGonC∗-algebras,
andfortheseweadopttheconventionsof[23]and[22]. WecanderiveafewbenefitsfromG
beingdiscrete: acoaction ofGonAisaninjective, nondegeneratehomomorphismδ: A →
A⊗C∗(G) (where here nondegeneracy means spanδ(A)(A⊗C∗(G)) = A⊗C∗(G)) such
that (δ⊗id)◦δ =(id⊗δ )◦δ, where δ : C∗(G) → C∗(G)⊗C∗(G) is the homomorphism
G G
defined by δ (s) = s⊗s for s ∈ G. The spectral subspace of A associated with s ∈ G is
G
A := {a∈ A :δ(a) = a⊗s}. SinceGisdiscrete,AistheclosedspanoftheA . Acovariant
s s
representation of (A,G,δ) in a multiplier algebra M(B) is a pair (π,µ) of nondegenerate
homomorphisms of A and c (G) into M(B) (where, for example, nondegeneracy of π
0
means spanπ(A)B = B) such that
π(a )µ(χ ) = µ(χ )π(a ) for a ∈ A ,t ∈ G,
s t st s s s
where χ denotes the characteristic function of the singleton {t}. The closed span
t
C∗(π,µ) := spanπ(A)µ(c (G)) is a C∗-algebra, and is called a crossed product for
0
(A,G,δ) if every covariant representation (ρ,ν) factors through C∗(π,µ) in the sense that
there is a homomorphism ρ × ν of C∗(π,µ) to C∗(ρ,ν) such that (ρ × ν) ◦ π = ρ and
(ρ ×ν) ◦µ = ν. All crossed products are isomorphic, and a generic one is denoted by
A× G, and moreover the covariant homomorphism generating A× G is written (j ,j ).
δ δ A G
The distinction among the various crossed products is frequently blurred, and any one of
them is referred to as the crossed product. The dual action of G on the crossed product
A×δ G is determined by δs(jA(a)jG(χt)) = jA(a)jG(χts−1) for a ∈ A and s,t ∈ G.
The coaction δ is called normal if j is faithful. In any case, there is always a unique
A
b
ideal I of A such that, with q denoting the quotient map from A to A/I, the composition
(q ⊗id)◦δ factors through a normal coaction δn, called the normalization of δ, on A/I
with the same crossed product as δ, that is, if (j ,jA) and (j ,jA/I) are the canonical
A G A/I G
covariant homomorphisms of (A,G,δ) into M(A× G) and M(A/I × G), respectively,
δ δn
A/I
then (j ◦q)×j is an isomorphism of A× G onto A/I × G. The ideal I coincides
A/I G δ δn
with kerj , as well as with ker(id⊗λ)◦δ, where λ denotes the left regular representation
A
of G.
As shown in [23], for discrete groups coactions are strongly related to Fell bundles and
cross sectional algebras of Fell bundles, for which we adopt the conventions of [9] and [7].
More precisely: if (A,G,δ) is a coaction, then the spectral subspaces {A : s ∈ G} (or,
s
more properly, the disjoint union of these subspaces) form a Fell bundle over G, which we
call the Fell bundle associated to δ.
Conversely, if (A,G) is a Fell bundle, there is a canonical coaction δ , which we will
A
call the dual coaction, of G on the full cross sectional algebra C∗(A), determined by
δ (a ) = a ⊗ s for a in the fiber A of A. Throughout this paper, when we write
A s s s s
something like a for an element of a Fell bundle, we always mean this element is to be
s
understood to belong to the fiber over s ∈ G.
If (A,G) is a Fell bundle over G, then a cross sectional algebra A of (A,G) is simply
a completion of Γ (A) with respect to any given C∗-norm. A cross sectional algebra A is
c
called topologically graded (see [7, Definition 3.4]) if there exists a contractive conditional
INDUCED COACTIONS 5
expectation F :A → A which vanishes on each fiber A for s6= e (where we always view
e s
the fibers A of the bundle as subspaces of A in the canonical way). Exel showed that
s
the full and reduced cross sectional algebras C∗(A) and C∗(A) are maximal and minimal,
r
respectively, among all topologically graded cross sectional algebras of a given Fell bundle
(A,G). To be more precise: if A is any topologically graded cross sectional algebra of
A, then it follows from [7, Theorem 3.3] and the universal property of C∗(A) (see [9,
VIII.16.11]) that the identity map on A determines surjective ∗-homomorphisms
φ :C∗(A) → A, λ :A → C∗(A), and Λ: C∗(A)→ C∗(A)
r r
such that λ◦φ = Λ (the map Λ is called the regular representation of C∗(A)). If k·k ,
max
k·k and k·k denote the norms on Γ (A) coming from viewing Γ (A) as a dense subal-
ν min c c
gebra of C∗(A), A, and C∗(A), respectively, then the above result is of course equivalent
r
to saying that k·k ≥ k·k ≥ k·k . Thus the topologically graded cross sectional alge-
max ν min
bras are exactly the completions of Γ (A) with respect to the C∗-norms which lie between
c
k·k and k·k . Exel calls a Fell bundle A amenable if C∗(A) = C∗(A) in the sense
max min r
that the regular representation of A is faithful on C∗(A). In this case all topologically
graded cross sectional algebras of A are identical.
If (A,G,δ) is any coaction, then for each s ∈ G the map δ := (id⊗χ )◦δ : A → A
s s
(where here χ , the characteristic function of {s}, is regarded as belonging to the Fourier-
s
Stieltjes algebra B(G) = C∗(G)∗, and id⊗χ is then the slice map of A ⊗ C∗(G) into
s
A) is idempotent, with range A and kernel containing every A for t 6= s. In particular,
s t
δ : A → A is a contractive conditional expectation which vanishes on A , for all s 6= e.
e e s
Hence A is a topologically graded cross sectional algebra of the associated Fell bundle
(A,G).
Now, [23, Comment immediately following Definition 3.5] states that the dual coac-
tion δn : a 7→ a ⊗ s on the reduced cross sectional algebra C∗(A) is (isomorphic to)
A s s r
the normalization of the dual coaction δ on C∗(A). However, there is a subtlety: the
A
constructions of C∗(A) in [23] and [7] are not quite the same. So, before we can use the
r
results from both sources, we need to check that their notions of the reduced C∗-algebra
of aFell bundleare compatible. Namely, we need to know that the kernels in C∗(A) of the
regular representations of [23] and [7] coincide. The conditional expectation E := (δ ) of
A e
C∗(A)onto thefixed-pointalgebra C∗(A)δA = A makes C∗(A)into aHilbertA -module,
e e
as in [28, Example 6.7]. Then left multiplication gives a representation of the C∗-algebra
C∗(A) on the Hilbert A -module C∗(A), and this in turn gives a Rieffel inducing map
e
from ideals of A to ideals of C∗(A). In [23] the kernel of the regular representation is the
e
ideal of C∗(A) induced from the zero ideal of the fixed-point algebra C∗(A)δA. But this
coincides with the kernel of the above representation of C∗(A) on the Hilbert A -module
e
C∗(A), which is the kernel of theregular representation of [7] (by the proofof [7, Theorem
3.3]). Hence, the definitions of C∗(A) in [23] and [7] are indeed compatible.
r
The maps φ,λ and Λ considered above are clearly equivariant with respect to the coac-
tionsδ ,δandδn (recallthatif(A,G,δ)and(B,G,ǫ)arecoactions,thenahomomorphism
A A
φ: A→ B is called equivariant if ǫ◦φ= (φ⊗id)◦δ). Thus we can say that any coaction
δ : A → A⊗C∗(G) “lies between” the dual coaction δ on C∗(A) and its normalization
A
δn on C∗(A), if A is the Fell bundle associated to δ. For reference it is useful to state the
A r
following lemma.
6 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
Lemma 2.1. Let δ : A → A ⊗ C∗(G) be a coaction of the discrete group G and let
(A,G) be the associated Fell bundle. Let δ and δn denote the dual coaction and its
A A
normalization on C∗(A) and C∗(A), respectively, and let φ, λ and Λ be as above. Then
r
there are canonical isomorphisms
Indφ: C∗(A)×δA G → A×δ G, Indλ : A×δ G → Cr∗(A)×δAn G,
and IndΛ:C∗(A)×δA G→ Cr∗(A)×δAn G
defined by
Indφ= (jA ◦φ)×jGA, Indλ = (jCr∗(A) ◦λ)×jGCr∗(A),
C∗(A)
and IndΛ= (jCr∗(A)◦Λ)×jGr ,
respectively. In particular, IndΛ = Indλ◦Indφ and δn coincides with the normalization
A
δn of δ.
Proof. It follows directly from the equivariance and surjectivity of the maps φ,λ and Λ
that the maps Indφ, Indλ and IndΛ are well defined surjections. Since Λ = λ ◦ φ we
also have IndΛ = Indλ◦Indφ. Since δn is the normalization of δ , it follows from [22,
A A
Corollary 2.7] that IndΛ is an isomorphism, which then implies that Indλ and Indφ are
also isomorphisms. In particular, it follows that kerjA = kerjC∗(A) ◦ λ = kerλ (since
r
jCr∗(A) is injective by the normality of δAn). Thus δAn coincides with the normalization of
δ.
Remark 2.2. Inviewoftheabovediscussiononecouldguessthatanytopologically graded
cross sectional algebra A of a given Fell bundle (A,G) over the discrete group G carries a
dual coaction δ which satisfies δ(a )= a ⊗s. This is not the case.
s s
Tosee acounter example let Gbeany non-amenablediscretegroup suchthat thedirect
sum V = 1 ⊕λ of the trivial representation 1 and the regular representation λ of
G G G G
G is not faithful. Then V(C∗(G)) is a topologically graded cross sectional algebra of the
Fell bundle (A,G) corresponding to G (i.e., A = C for all s ∈ G), since the kernel of V is
s
contained in the kernel of λ . Let U :C∗(G) → L(H) be any faithful representation of G.
G
If therewere acoaction δ on V(C∗(G)) satisfying δ(a )= a ⊗s, this would implythat the
s s
unitary representation V ⊗U of G factors through a faithful representation of V(C∗(G)),
i.e., ker(V ⊗U)= kerV in C∗(G). But V ⊗U = (1 ⊕λ )⊗U = U⊕(λ ⊗U), is faithful
G G G
on C∗(G), while V is not faithful by assumption.
To see that there are numerous examples of groups satisfying the above property on
1 ⊕ λ , let us first note that any non-amenable group with 1 ⊕ λ faithful satisfies
G G G G
Kazhdan’s property (T) (i.e., the trivial representation is an isolated point in G). Since
the nonabelian free groups F in n generators do not satisfy Kazhdan’s property (T)
n
b
(which follows from the simple fact that (F /[F ,F ]) = Zn = Tn), they all serve as
n n n
specific examples for our counter example. Moreover, by a theorem of Fell [8, Proposition
5.2] it is known that for any subgroup H of a discrete gbroupcG and any representation V
of H, V is adirectsummandof (IndG V)| , whichimplies thatany faithfulrepresentation
H H
of C∗(G) restricts to a faithful representation of C∗(H). Thus, if G is a non-amenable
group with 1 ⊕λ faithful, it follows that (1 ⊕λ )| =1 ⊕λ | , and hence 1 ⊕λ
G G G G H H G H H H
is faithful on C∗(H) for any subgroup H of G, since it follows from [12, Addendum of
INDUCED COACTIONS 7
Theorem 1] that kerλ | = kerλ . In particular, 1 ⊕λ is not faithful for any discrete
G H H G G
group which contains the free group F as a subgroup. This shows that any non-amenable
2
group with 1 ⊕λ faithful must indeed be very exotic, and it is certainly an interesting
G G
question whether there exist such groups. We are grateful to Alain Valette for some useful
comments on this.
We want to definea notion of inducedcoactions, dualto theconcept of inducedactions.
IfN isasubgroupofG,wecaninduceanactionofN toanactionofG,soduallyweshould
expect to induce a coaction from a quotient group to the big group. For this we require
N to be a normal subgroup. We don’t know yet how to induce coactions in general; for
dual coactions of Fell bundles the way seems fairly clear now (given the techniques of the
present paper!), but for arbitrary coactions it seems much more difficult. We will develop
the theory for dual coactions of Fell bundles over discrete groups, where the computations
are so much cleaner than for continuous groups. It will be fairly obvious to the reader
that some of what we will do in this paper can be done for Fell bundles over continuous
groups, and indeed we plan to pursue this.
However, we feel it is valuable to have the machinery laid out for the case of discrete
groups, since the discrete theory has a flavor all its own. We first define the Fell bundle
whichwillbeassociated totheinducedcoaction. Thiswilljustbethe“Banach ∗-algebraic
bundleretraction”bythequotientmapG→ G/N,asin[9,VIII.3.17], butweusedifferent
notation and terminology:
Definition 2.3. Suppose (D,G/N) is a Fell bundle over G/N, where N is a normal
subgroup of the discrete group G, and let q: G → G/N be the quotient map. We define
the pull-back Fell bundle over G as
q∗D = {(D ,s) :s ∈ G}.
sN
The bundle projection is (d ,s) 7→ s, and we denote the fiber over s by q∗D = (D ,s).
sN s sN
Each fiber q∗D is given the Banach space structure of D . The multiplication and
s sN
involution are defined by
(d ,s)(d ,t) = (d d ,st)
sN tN sN tN
(d ,s)∗ = (d∗ ,s−1).
sN sN
It is completely routine to verify that the above operations indeed make q∗D into a Fell
bundle over G.
Definition 2.4. Let (D,G/N,δ) be a coaction, D the associated Fell bundle over G/N,
and q∗D the pull-back Fell bundleover G. We call the fullcross sectional algebra C∗(q∗D)
the algebra induced from D and denote it by IndD, and we call the dual coaction on
C∗(q∗D) the coaction induced from δ and denote it by Indδ.
Remark 2.5. Note that the induced algebra and the induced coaction depend only upon
the Fell bundle D; in general D will be some intermediate algebra between C∗(D) and
C∗(D). In a sense, our definition of the induced C∗-algebra IndD above is somehow
r
artificial: we could have equally well defined IndD as the reduced cross sectional algebra
C∗(q∗D), or any algebra which lies “between” the full and the reduced cross sectional
r
algebras and carries a coaction ǫ which satisfies ǫ(d ,s) = (d ,s)⊗s for all (d ,s) ∈
sN sN sN
8 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
q∗D . The only case where there is really a canonical choice is when q∗D is amenable in
s
the sense of Exel [7], since then all cross sectional algebras are the same. We are going to
study this problem in §6. Anyway, it follows from Lemma 2.1 that the crossed product
IndD× G is always independentfrom thechoice of thecross sectional algebra for q∗D!
Indδ
In view of the above remark it makes sense to give also the following
Definition 2.6. Let(A,G,δ) beacoaction ofthediscretegroupGandletN beanormal
subgroup of G. We say that (A,G,δ) is weakly induced from G/N if there exists a Fell
bundle(D,G/N)suchthat(q∗D,G)isisomorphictotheFellbundleassociatedto(A,G,δ).
Hence a weakly induced coaction is actually induced if and only if A is equal to the full
cross sectional C∗(A), where (A,G) is the Fell bundle associated to (A,G,δ).
Remark 2.7. If G is abelian, then our notion of induced coactions is the same as the
notion of an inducedaction of the dualgroupG of G underthe one-to-one correspondence
between coactions of G and actions of G. In order to explain this recall first that if
b
α :G → Aut(A) is an action, then the corresponding coaction δ of G on A is given by
b α
b δ :A → C(G,A); δ (a) (χ):= α (a),
α α χ
(cid:0) (cid:1)
χ ∈ G. Here we made the identificabtions C∗(G) ∼= C(G) (via Fourier transform) and
A⊗C(G) ∼= C(G,A). Since the Fourier transform of s ∈ C∗(G) is given by the function
b b
χ 7→ χ(s) on G, we see that for s ∈ G the spectral subspace A for δ is given by
b b s α
b A = {a∈ A :α (a) = χ(s)a for all χ ∈ G}.
s χ
[
Suppose now that N is a subgroup of G and let β : G/N → Abut(B) be an action of the
[
subgroup G/N = N⊥ of G. The induced C∗-algebra Ind(B,β) is then defined as
[
Ind(B,β) := {F ∈bC(G,B): F(µχ)= β (F(µ)) for all χ ∈ G/N,µ ∈ G},
χ¯
with induced action Indβ :G →b Aut(Ind(B,β)) given by b
b Indβµ(F) (ν):= F(µ¯ν).
(cid:0) (cid:1)
We claim that the coactions (Ind(B,β),G,δ ) and (Ind(B,δ ),G,Indδ ) are isomor-
Indβ β β
phic. For this it suffices to show that the Fell bundle associated to δ is isomorphic to
Indβ
the pull back of the bundle(B,G/N) associated to δ , since by the amenability of G there
β
is only one topologically graded cross sectional algebra for this bundle [7, Theorem 4.7].
Indeed, we claim that the family of maps
Φ :Ind(B,β) → (B ,s);Φ (F )= (F (1 ),s)
s s sN s s s G
is well defined and gives the desired isomorphism of bundles. To see that it is well defined,
it is enough to show that F (1 ) ∈ B for all s ∈ G. But F ∈ Ind(B,β) if and only if
s G sN s s
Indβ (F) = µ(s)F for all µ ∈ G, so that
µ
βχ(Fs(1G)) =bFs(χ¯) = Indβχ(Fs) (1G) = χ(s)Fs(1G),
[ (cid:0) (cid:1)
for all χ ∈ G/N. Thus F (1 ) ∈ B . Each map Φ is norm preserving since
s G sN s
kF (µ)k = kIndβ F (1 )k = kµ¯(s)F (1 )k = kF (1 )k.
s µ¯ s G s G s G
INDUCED COACTIONS 9
If b ∈ B we can define an element F ∈ Ind(B,β) by F (µ) := µ¯(s)b , and then
sN sN s s s sN
Φ (F ) = (b ,s), thus each Φ is surjective. Finally, since all operations in Ind(B,β) are
s s sN s
pointwise, it follows that the family of maps (Φ ) respects the bundle operations; for
s s∈G
instance
Φ (F F )= (F (1 )F (1 ),st) = (F (1 ),s)(F (1 ),t) = Φ (F )Φ (F ).
st s t s G t G s G t G s s t t
3. Enveloping algebras and discrete coactions
In this section we show that for discrete groups all the C∗-calculations we must do
are basically “pure algebra”. The following terminology shows what we mean by “pure
algebra”.
Definition 3.1. Let B be a ∗-algebra. By a representation of B we always mean a
0 0
∗-homomorphism of B into the bounded operators on a Hilbert space. We say B has
0 0
an enveloping C∗-algebra if the supremum of the C∗-seminorms on B is finite, and in
0
this case we call the Hausdorff completion of B relative to this largest C∗-seminorm the
0
enveloping C∗-algebra of B .
0
Thus, if B is a ∗-subalgebra of a C∗-algebra B, and if every representation of B is
0 0
bounded in the norm inherited from B, then the closure of B in B is the enveloping
0
C∗-algebra of B . For example, we have the well known
0
Lemma 3.2. If (A,G) isa Fellbundle, then C∗(A)isthe enveloping C∗-algebra of Γ (A).
c
Proof. ThisfollowsimmediatelyfromtheobservationthatarepresentationofΓ (A)willbe
c
pointwise continuous into the operator norm topology, hence continuous for the inductive
limit topology.
The notion of enveloping algebras will be very convenient for crossed products by coac-
tions: let (A,G,δ) be a coaction, and let A be the associated Fell bundle. Let A × G
be the product Banach bundle over G×G. Then A×G embeds in the crossed product
A× G by identifying (a ,t) with j (a )j (χ ), and the algebraic operations become
δ s A s G t
(a ,t)(a ,v) = (a a ,v) if t = uv (and 0 else)
s u s u
(3.1)
(a ,t)∗ = (a∗,st).
s s
Thus, A×G acquires the structure of a Fell bundle over the groupoid G×G in the sense
of [15], where G×G is given the transformation group groupoid structure associated with
the action of G on itself by left translation:
(s,tr)(t,r) = (s,r) and (s,t)−1 = (s−1,st).
AlthoughthetheoryofFellbundlesovergroupoidsisstillinitsinfancy,itseemsreasonable
to expect that many of the properties of Fell bundles over groups will carry over. In
particular, the ∗-algebra of finitely supported sections should have an enveloping C∗-
algebra. We only need this for the above special case (see Corollary 3.4 below).
We will need to know that
(3.2) k(a ,t)k := ka k= kj (a )j (χ )k
s s A s G t
10 SIEGFRIED ECHTERHOFF AND JOHN QUIGG
for all (a ,t) ∈ A×G, i.e., that the above described embedding of A×G into A× G is
s δ
isometric. First of all, if E is any subset of G, then χ = χ strictly in ℓ∞(G) =
E t∈E t
M(c0(G)), so P
j (χ ) = j (χ ),
G E G t
Xt∈E
strictly in M(A× G). Consequently, for a ∈ A we have
δ s s
j (a ) = j (a )1 = j (a ) j (χ )= j (a )j (χ ).
A s A s A s G t A s G t
Xt∈G Xt∈G
Moreover, sincethej (χ )areorthogonal projections summingstrictly to1inM(A× G),
G t δ
and since j is faithful on the unit fiber algebra A , for a ∈ A we have
A e e e
ka k = kj (a )k = supkj (a )j (χ )k.
e A e A e G t
t∈G
But kjA(ae)jG(χt)k is independentof t ∈G, since jA(ae)jG(χs)= δs−1t(jA(ae)jG(χt)) and
δ is isometric, being an automorphism of a C∗-algebra. Hence, we get
b
b ka k = kj (a )j (χ )k for all a ∈ A ,t ∈ G.
e A e G t e e
Since a∗a ∈ A for all a ∈ A , Equation (3.2) now follows from
s s e s s
k(a ,t)k2 = ka k2 = ka∗a k = kj (a∗a )j (χ )k
s s s s A s s G t
= k(j (a )j (χ ))∗(j (a )j (χ ))k
A s G t A s G t
= kj (a )j (χ )k2.
A s G t
We are now going to show that Γ (A×G) does have an enveloping C∗-algebra, namely
c
given by the crossed productA× G, if (A,G) is the bundleassociated to a given coaction
δ
δ. We do it in somewhat more generality, namely for (appropriate) dense subspaces of the
fibers. Although we do not need the last whistle in the present paper, we include it since
we feel it will be useful elsewhere.
To prepare for the statement, supposewe have a Fell bundle(A,G), and for each s ∈ G
we have a linear subspace B of A such that
s s
(3.3) BsBt ⊂ Bst and Bs∗ ⊂ Bs−1.
Then B := B is a subbundle of A (although not a Fell bundle, since its fibers
s∈G s
may be incomSplete). Let Γc(B) denote the linear span of B in Γc(A). Then Γc(B) is a
∗-subalgebra, and conversely any ∗-subalgebra B of Γ (A) which is the linear span of the
c
intersections B := B ∩A arises in this way.
s s
Theorem 3.3. Let A be a Fell bundle over the discrete group G. Suppose we have a
subbundle B of A (with possibly incomplete fibers), satisfying (3.3), such that A is the
e
enveloping C∗-algebra of B and C∗(A) is the enveloping C∗-algebra of Γ (B). Then,
e c
regarding Γ (B × G) as a ∗-subalgebra of Γ (A × G) ⊆ C∗(A) × G via the inclusion
c c δA
B ֒→ A, the inductive limit topology on Γ (B × G) is stronger than the norm inherited
c
from C∗(A)× G, and C∗(A)× G is the enveloping C∗-algebra of Γ (B×G).
δA δA c
