I nternat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982)209-256
209
GENERALIZED GROUP ALGEBRAS
AND THEIR BUNDLES
I.E. SCHOCHETMAN
Department of Mathematical Sciences, Oakland University
Rochester, Mchigan 48063
(Received September 16, 1980, and in revised form July 20, 198])
ABSTRACT.
Our primary objective here is to extend the concept of Banach *-algebraic
bundle to the setting where the bundle product and involution are just measurable,
i.e. not necessarily continuous.
Our secondary objective is to introduce the *-algebra
operations into such a bundle by means of operator fields and study the smoothness of
these operations in terms of the smoothness of the fields.
KEY WORDS AND PHRASES. Fild o f Banach spaces, continuity structures, Banach bundles,
induced continuity structures, bundle morphisms product elds involution eld
up-algebra bundles, bundle identes bundle mtipliers genezed group-algebras, homogeneous bundles, uary factor systems, twisting .p group-extension
bundl
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
22D15, 43A20, 55FI0.
0
Primary: 46H99, 46K99. Secondary:
INTRODUCT ION
Let G be a locally compact group and A a Banach *-algebra.
interested in here is how to construct a Banach *-algebra from
izes the canonical one (with pointwise operations).
we introduce multiplication and involution in
gebra?
LI(G,A)
The question we are
LI(G,A)
which general-
Alternately, how generally can
so as to make it a Banach *-al-
The most general existing responses to this question are quite different.
They are
(I)
The cross-section algebras of Fell [1,2
and their underlving Banach
*-algebraic bundles.
(2)
The generalized
Ll-algebras
of Leptin [3] or the equivalent twisted
I.E. SCHOCHETMAN
210
group algebras of Busby and Smith [4].
The underlying Banach bundles in these con-
structions are trivial (see [5]).
Neither of the previous approaches is more general than the other.
they were designed for the same purpose.
However,
Thus, it seems desirable that they be
unified as part of a more general theory which includes them both--as well as their
bundles.
This is our overall goal in this manuscript.
Since multiplication and involution
are
Just
measurable in the trivial bundles
associated with the Leptin and Busby-Smlth constructions, this should be true of the
bundles in the general theory.
bundles.
This is the main shortcoming of the Fell approach via
On the other hand, since Fell’s approach allows for non-lsomorphlc fibers
in the bundles (i.e. non-trivial
the general theory.
structions.
bundles), this should also be true of the bundles in
This is the main shortcoming of the Leptin and Busby-Smlth con-
Consequently, our primary objective is to extend Fell’s Banach
*-algebraic bundle approach to the setting where the multiplication and involution
need only be measurable.
constructions.
This extension will then include the Leptln and Busby-Smith
However, the method by which we realize the measurable *-algebra
structure in a (Banach) bundIe will be quite different from that of Fell.
it will be more consistent with the other two approaches in that it
means of operator fields.
This is our secondary objective.
ill
In fact,
be done by
There is good reason for
our choosing this approach which is explained very well by some recent remarks of
Rieffel
[6]:
"Fell’s approach in terms of bundles has some great advantages over those used
in various other papers concerned with establishing a general
framework,.., in that
Fell can work everywhere with continuous functions, thus avoiding many measuretheoretic arguments, and he has no need to become entangled in lengthy cocycle computations and the like.
On the other hand, in many specific situations which one may
want to study, the bundle structure is often not entirely evident so that the trans-
lation between the immediately evident structure and Fell’s bundle structure may be
tedious.
Thus while the theory developed by Fell in these notes is of very consider-
able philosophical comfort, more experience will be needed before it will be clear
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
211
exactly how incisive a technical tool it is for dealing with specific examples."
To elaborate on Rieffel’s remarks, observe that in the Fell construction the
bundle structure is introduced axiomatically.
On the other hand, in the Lepti and
Busby-Smlth constructions, the bundle structure is introduced operator-theoretlcally.
The latter approach is characteristic of how the "immediately evldent
Thus, in short, our aim is to
structure arises in specific examples.
’ bundle
develop a
general theory of group algebras via bundles which includes the existing theories
and utilizes their respective advantages.
Taking all the above facts and comments into consideration, we have chosen to
proceed as follows.
G
Replace
A
by a field
{Ax:XeG }
of Banach spaces indexed by
A
In Section i, we review the notion of continuity structure
equivalent notion of Banach bundle.
structures which can be derived from
in
x
and the
In Section 2, we study certain continuity
A--the induced structures.
properties of bundle morphlsm are developed in Section 3.
The notion and
Then the notions of pro-
duct and involution are introduced in Sections 4 and 5 by means of operator fields
P
and
I
having the appropriate algebraic properties.
to be continuous (resp. measurable,
By requiring these fields
ultra-measurable), we obtain a continuous (resp.
measurable, ultra-measurable) Banach *-algebraic bundle.
,A group-algebra bundle is
defined to be a measurable Banach *-algebraic bundle and is studied in Section 6.
Section 7 studies to what extent the various notions and properties of bundle ap-
proximate identity extend from the continuous to the measurable case.
In Section 9, we construct the
does the same for multipliers.
sponding to
G,
{Ax}
A
P
I
Section 8
Ll-algebra
corre-
In Section I0, we investigate the extent to
which bundle homogeneity extends from the continuous to the measurable case.
Unitary factor systems and twisting pairs, as well as their bundles and correspond-
ing algebras, are reviewed and compared in Sections ii and 12,
the bundle corresponding to a locally compact group extension
Next we construct
I-NI
Fell’s
original construction was direct, in that he specified the bundle ingredients and
then verified the equivalence of
LI(H)
with the resulting cross-sectlon algebra.
In Section 13, we show that these ingredients arise naturally as a consequence of
212
our
I .E. SCHOCHETMAN
Ll-induction
procedure for regular representations.
The equivalence of
LI(H)
with the cross-section algebra is then an automatic consequence of inductlon-in-
stages.
In what follows, it will be convenient to use the following general notation.
The symbols
.,
Z, IR, C
will denote (as usual) the natural numbers, integers, real
numbers, complex numbers respectively.
the identity mapping on S and
are Banach spaces, then
operators from
A
into
YS
Hom(A,B)
B
X
is a positive Radon measure on
E
is any set, then
will denote
S
If
its characteristic function.
A
and
is a locally compact Hausdorff space and
then
C(X,A)
(resp. C (X,A))
will denote the
linear space of continuous (resp. compactly supported) A-valued functions on
and
"-a.a.
The phrase "for -almost all
x in
x
in
G" will be abbreviated by
G."
CONTINUITY STRUCTURES.
Let
X
spaces over
by
X
M(X,A,) the linear space of (equivalence classes of) such -measurable
functions.
I.
B
will denote the Banach space of bounded linear
X
If
S
If
be a locally compact Hausdorff space and
X
A
Let
-l(x) Ax,XeX
A
denote the disjoint union of the
Note that
be called a vector field.
Ax
is a linear space.
x
a field of Banach
xeX
Define :,%/X
An element of A
x
will
Of fundamental significance here is the notion of con-
tinuity for a vector field.
Since there is no canonical meaning for this notion in
general, it must be introduced axiomatically.
Godement in [7]
{Ax:XeX
This was first accomplished by
(see also [8]) by means of a "continuity structure."
Actually,
Godement’s original terminology was "fundamental family of continuous vector
fields."
The term "continuity
DEFINITION I.i.
(i)
A
structure"
is taken from Fell
A continuity structure
is a subspace of
(ii)
For each
h
in
A
(iii)
For each
x
in
X
A
in
Ax
[9].
is a subset satisfying:
KAx
the scalar function
the subspace
x-li h(x)II
{h(x):heA} of AX
is continuous on
is dense.
Given a continuity structure, we define continuity for a vector field as
follows
X
213
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
if, given
>o
such that
llf(x)
A-continuous
C(D,A)
Hence,
h(x)ll<e
for
x
X
C(A)
we write
{fec(A):
f has compact
A subset
DEFINITIONI.3.
x
in
LEMMA 1.4.
X
in
o
h
o
A
in
(defined on D) is
f
The field
and
x
Denote the space of such fields by
A _c C(A)
Clearly,
in general.
f
is a continuous vector field means that
f
(See sections 1 and 5 of [8].)
A
F of
Thus, A
Let
F be total
h
x
A-continuous at
is
Let
support}
X
and only if the element
of
C(X,A)
for
C(A)
is an element of the linear space
for each
N
D
if it is so at each point of
f
Then
NQD
in
in this context, to say that
C (A)
c
feDAx
and
o
there exists a neighborhood
D
if
x eDX
Let
DEFINITION 1.2
is total if
{h(x):heF}
is dense in
A
x
itself is total.
in
A
in
A
In 1 2,
f
A-continuous at
is
if
x
F
can be chosen from
Some time after Godement, Fell introduced the notion of continuity into our
A[I]
context in a very different way--by axiomatically topologizing
purpose, it will be convenient to speak of the elements of
and denote the space of them by
(A,)
DEFINITION 1.5.
I[A
For this
as cross-sections
x
S(X,A)
is a Banach bundle over
X
if
.A
is a Hausdorff space,
is a continuous, open surjection and:
(i)
The function
(ii)
The operation
a
a
+
For each
If
xEX
then
Let
(a).= (b)}
A
into
(iv)
A
is continuous from
{(a,b)eAxA
(iii)
is continuous on
CS(X,A)
8
in
{a i}
and
a.1
0
x
(the zero of Ax
denote the subset of
[i, p.10].
LEMMA 1.6.
The space
given by
such that
is a net in
in
S(X,A)
Observe that the relative topology of
norm topology
A
the mapping on
a
flail
8a
is continuous.
and
0
w(a i)
A
consisting of continuous cross-sections.
A
on each fiber
A
x
is precisely the
More importantly, we have the following:
CSX,-
x in X
is a continuity structure in
with
214
I.E. SCHOCHETMAN
equality holding in part (iii) of i.I.
Part (i) of i.i follows from [i, p. ii].
PROOF.
Part (ii) is clear.
(A,)
(iii) is a very recent development and follows from the fact that
Part
has enough
continuous cross-sections [i, Thin. ii].
A--CS(X,A)
Letting
we obtain that
A [8, p. 13].
C(A)
Therefore, start-
A
with a Banach bundle, we obtain a continuity structure in
largest such structure in its equivalence class [8, Prop. 1.23].
AI
A2
C(AI)
if
C(A2) .)
which is the
(Recall that
Conversely, it has been known for some time that this
process is reversible (up to equivalence).
LEMMA 1.7.
Let
A
_unique topology on A making
A
LEMMA 1.8.
section
f
tinuous at
is
x
o
.c CS(X,A);
Let
x
(A,)
C(A)
CS(X,A)
feAx
so that
in fact,
oeD c. X
and
A-continuous at
Hence,
a Banach bundle.
xo
for which
tinuous vector fields in
A
Ax are equivalent, i.e.
A
Then the cross-
f
is
A-con-
Banah bundle
we obtain a
CS(X,A)
C(A)
be a continuity structure in
the unique Banach bundle guaranteed by 1.7.
In particular, if
X
For the re-
KAx and (A,)
is discrete then
A
itself is the essentially unique continuity structure
[8,1.22]
Next we turn to the notion of measurability.
be a positive Radon
measure on
X
element of
e > o
DAx
Let
Then
D
f
there exists compact
Ke
Let
in
First we consider the vector field context.
DEFINITION 1.9.
on
f:D-
This shows that the two methods for obtaining con-
mainder of this paper, we will let
HA
Furthermore,
CS(D,A)
C(D,A)
A~CS(X,A)
Then there exists a
if and only if the vector field
Thus, starting with a continuity structure
(A,v)
Ax
be a continuity structure in
Denote such
f
by
be a locally compact subspace of
is
X
and
(A,)-measurable if, given compact
KE c_ K
M(D,A,)
such that
If
D
(K-Ke)<
X
and
f
f
K c_ X
is
an
and
A-continuous
delete it.
Analogously, we wish to introduce measurability in
S(X,A)
(as in [i0]).
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
measurable
f
by
f:K
and
f
_c
K
if, given compact
(K-K)<
that
D and
Let
DEFINITION 1.10.
be as in
D
and
>o
1.9
f
Then
is
there exists compact
is continuous for the
215
(A,B)-
Kc K
A-topology of A
such
Denote such
MS(D,A)
On p. 22 of [1], Fell defines the notion of measurable cross-
aEMARK I.Ii.
However, as can be verified by the results on p. 23 of
section in a different way.
[i], the two definitions are equivalent.
LEMMA 1.12.
D
If
X
is a locally compact subset of
MS(D,A,)
M(D,A,)
MS(X,A,)
M(A,)
then
In partlcular,
This follows from 1.8.
PROOF.
Since continuity and -measurability depend only on the equlva-
REMARK 1.13.
[8,1,3], we may replace A by C(A)
CS(X,A)
lence class of
A
of generality.
(Of course, this is false if we find it necessary to consider equl-
2]).
continuous families of vector fields [8
A
CS(X,)
C()
they are
no__.
we
may (and wILl1) assume
Actually, this is advantageous because of the last equality,
C(A)
or
hC()}
{h(x)
as well as the fact that
what follows,
Thus
without loss
A
is equal to
Ax
for
x
in
X
In
will be used according to which is appropriate when
assumed to be equal.
The remainder of this section is devoted to separability considerations.
DEFINITION 1.14.
if
[1, p.15]. The Banach bundle
X is second countable and
A
(A,)
countably dense, i.e.
is
is strongly
A
separable
contains a count-
able, total subset [8, 1.1].
LEMMA 1.15.
The bundle
(A,)
is strongly separable if and only if it is
second countable.
PROOF.
This is Prop. 1.8 of [I].
enough cross-sections.
Recall that
(A,)
automatically has
216
I.E. SCHOCHETMAN
A Is arbitrary
Suppose for the moment that
Recall
[8, p.10] also that
is
separable If
AIK
[8, p.9]
implies
assume
A
is separable.
A
then:
c C(A)
in general.
A
is countably dense and
is
restricted continuity structure
A
In this context, we have:
strongly separable
However, if
The converse is false in general.
separable.
C(A)
C(A)
X, the
locally separable if, for each compact K
A
again, i.e.
A
is separable if and only if
we again
is countably dense.
Therefore, under this assumption, we have the following separability summary:
LEMMA I. 16.
A
(i)
(ii)
The following are equivalent
countable, i.e. strongly separable.
is second
is countable dense and
(lii)
is separable and
EXAMPLE i. 17.
X
X
is second countable.
is second countable.
Let
be an exact sequence of locally compact groups.
is
and the group algebra
G
a Banach bundle over
G
LI(N)
If
H
is second
countable, then so
In Section 9, we will construct
is separable.
with fibers isometrically isomorphic to
LI(N)
In this
case, the continuity structure will be countably dense, so that the Baach bundle
will be second countable.
2.
INDUCED CONTINUITY
STRUCTURES.
The objectives of subsequent sections suggest that we study certain continuity
structures "induced" from given ones.
This section is devoted to defining these
structures and establishing their basic properties for use later on.
Suppose
{By:yeY}
Hausdorff space
Y
with
corresponding bundle.
into
Y
is a field of Banach spaces over the locally compact
A
a continuity structure in
Suppose also that
The induced bundle [ii]
B
(B,T)
{(x,b)
(x,b)
:X/Y
x
Y
Let
(B,T) denote the
is a continuous mapping from
over
X
xeX, beB, Y(b)
(x,b) e
B
B
is then given by:
(x)}
X
217
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
We then have the following commutative diagram:
/B
X
-Y
where the top mapping is the projection
regard.)
If
B
The set
B
is one-to-one, then
can be identified with
S(X,B)
f +-
{B(x)
(Ex, f)
xX}
r-l((X))
are in one-to-one correspondence with
f(x)eB(x),XeX}
{f:X-
HB(x)
via the mapping
(Also see [2, p.101] in this
b
(roughly) the disjoint union of the field
is
The cross-sections
(x,b)
For such
f
B
B
/
the diagram
/
(Ex, f)
X
is commutative.
Suppose
B
LEMMA 2.1.
{ExXk
Let
A
Let
feHDB(x)
keA}
Then
A
B
X
is equipped with the relativized topology of
is a continuity structure in
B (x)
LEMMA 2.2.
D c X
for
and
XoeD
Then the following are
equivalent
(1)
The vector field
(li)
The cross-sectlon
(Ex, f
(Ex, f)
is
in
A-contlnuous at Xo
S(D,B) is continuous
at
Xo
From 2:2, it follows that
cs(x,m) c(a)
{(Ex, f):fgC(X,),f(x)E(x)
xgX}
and
MS(X,,U) M(a,U)
EXAMPLE 2.3.
the restriction
to
T
-l(x)
c_
X
If
AIX
and
of
T
c_
y
A
IB
@
and
to
X
is the injection mapping, then
[8, p.9].
In this case,
B@
h
is simply
is homeomorphic to
I.E. SCHOCHETMAN
218
EXAMPLE 2.4.
Of course,
essentially.
EXAMPLE 2.5.
)
C(A
:YX+Y
If
-+ C(A)
can be identified with
3.
ExXT
(Xx,
(Ex,h)+-h
i.e.
[8
4]
and
r
heC(A)
%
is the left projection
continuity structure obtained from A
B
B
is a homemorphlsm, then
:X/Y
If
A%
then
is the lifted
(
The corresponding bundle
(See p.27 of [i].)
BUNDLE MORPHISMS
In later sections, it will be necessary to identify bundles (up to isomorphism)
as well as consider products of bundles in studying multiplication.
Accordingly, we
need to develop suitable notions about mappings from direct products of bundles to
Nevertheless, in going through this section, it would be worthwhile for
bundles.
the reader to give special attention to the case where the domain is a single bundle-and not a product.
Let
be locally compact Hausdorff spaces,
Xl,...,Xn,Y
{By:yY}
{A
-
...
(,’)
(Ax
(Recall [11].)
from the product
satisfying
c
T-
BCx
T:Y
l:Al/Xl,...,n:An/Xn
denote the product bundle over
n
(A,)
XI... Xn
X
n
are not requiring that
(,T)
ino
or
given by
of a mapping
x(Ax) _c B(x)
:X-Y
xeX
will be a pair of mappings
,xn)
and
be continuous.
Note that we
n
In fact, our objective here is to
A
X
i.e.
(,)
The connection between
X
A x...xA
X
X
1
(,)
(,)
may be viewed as a pair
together with a family of mappings
(a)
and
(abstract) bundle morphism
develop the notions of continuity and measurability for
Clearly, a bundle morphism
i ....wAn
A
Let
is fiber preserving in the sense that
x- (x
I
where
the projections.
his context, an
Thus, such
xeX
AI,...,%,
fields of Banach spaces over these base spaces,
:x X
xn n n
the bundle spaces and
1
{Ax1:XlX1}
T(a)(a),
.IAx xeX
aeA
and
#
{ :xEX}
X
conslating
satisfying
is then given by
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
A Banach bundle morphism from
DEFINITION 3.1.
($,)
morphism
(A,w)
into
219
(B,T)
is a bundle
satisfying
x
o." (A, S,(x)
x
(See the Appendix for the definition and required properties of such spaces of bounded n-linear mappings.)
may be viewed as a vector field in the product
Thus,
rxm (x’ B,(x))
_c
Now let
Di
Then
defines a mapping
X
fieDiAxi
i
(x)
#f:D
D-DI...Dn
with
l<i<_n
given by
x.,...,xn(.(x.)’’’" ,n (xn))
x((x))-
xED
(fl(Xl)’’’’’fn(Xn)
Clearly,
Cf
f(x)zAx
so that
Moreover, since
.f
:EiEDIAxi
However,
iDiAxl
#fEEDB(x)
Thus,
+
i.e.
EDB(x)
DAx
may be viewed as
we have
l<_i_<n
fi(xi)eAxl
xED
f(x)eB(x)
f’(fl’"’’fn
and
i.e.
:DAx DB(x)
/
AI,...,An
Next suppose that
,Ax ,B
AXl
n
A
in
A
AIX"
B(x)
Y
continuity structures
Let
tlnuous at a (relative to
(,)
A,A)
AI,..
,An
be as in 3.1 with
if
[L%
in
x
aeA
is continuous at
I
...xIL%x
Then
a
(,)
and
ExAx
n
con-.
is
is continuous
(a).
DEFINITION 3.3.
to
On the other side, we have the product
as in section 2.
xAn of the
in the spaces
is aontlnuous, then it induces a continuity structure
If
DEFINITION 3.2.
at
A are continuity structures
A,A)
field
$
containing
if, for each
h
in
is locally bounded at
x
is continous at
The operator field
the set
A
the mapping
x
in
X
Oh=.h
x
in
X
(relative
if, for each compact subset
The
x
is continuous at
K
of
X
220
I.E. SCHOCHETMAN
{ll
TeK}
(I)
T
of operator norms is bounded in
LEMMA 3.4.
(2)
If
PROOF
@
X.
3
If
(I)
(2)
X
For
h
A
in
If
x
is continuous at
X
is continuous on
if, for each
8
(i)
then
x
3
X
in
is
h
HB(t)
in
is
@
the zero element
continuous at
x
is continuous at
the vector field
x.
then
X
in
if and only
A@-continuous
__(@i
x
A
of
x
at
satisfies:
This follows from 2.2.
f.1 e ILEX.
let
1
{llfi(xi)II:xigXi}
sup
l<i_<n
1
For
PROPOSITION 3.5.
(,)
IlCxl sup{I[x(f(x))ll
PROOF.
x--assuming
PROOF.
f=(f
n}, xeX
IIfiIIx <l,i=l
fn)eHiCc(Ai)
I
The proof of 9.2 of [8] can be adapted to this proposition.
COROLLARY 3.6.
at
as in 32, we have:
If
Ai
x
is continuous at
C(A i)
then it is locally bounded
in X
l_<i_<n
By hypothesis, f:X/B
is continuous at
Cf (’)If
Hence, the scalar functions
x
feiC (A i)
for each
are also continuous.
C
This corollary then
follows from 3.5, together with the fact that the supremum of continuous functions
is lower semi-continuous and
REMARK 3.7.
hence, locally bounded.
If we were not assuming
Ai
C(Ai),l<_iin
then the two versions
#
in 3.3 would be equivalent for locally bounded
of continuity for
[7, p.84].
This is reasonable in view of 3.6.
Let
THEOREM 3 8.
only if
is
a
Then
is continuous at
COROLLARY 39
each
xeX
in
A
X
A.-continuous
1
Let
i.e.
at
x.1
x
(0,)
is continuous at each
a
in
AX
if and
(as in [i, p.32]).
x.eD.cX
1
is continuous at
then for
Suppose
l<i<n
1--i
f
(fl
x
(,)
_(Xl,...,xn)_
,fn
is continuous at
If
the mapping
f.1
in
HD A
1
X.
1
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
221
#f:DlX ...XDn +B
($,)
Next we develop measurability for
X
Radon measure on
Then H
lii<n
i
B
A-topology of
for the
x
is continuous at
Let
#
and
HI Q’’’Hn
Hi
be a positive
is a positive Radon measure on
x [10].
is measurable (relative to
(resp. h)
if
$
X
of
(2)
A
A, A, H)
(resp. operator field $)
h
A
in
(,)
Thus,
the mappin
h
is measurable if and only
(recall 3.8).
(i)
Sh in
the vector field
$
(recall 3.4).
is measurable
is measurable, then
If
is continuous, then
If
if, for each
is measurable.
into
is measurable
LEMMA 3.11.
(,)
The Banach bundle morphism
DEFINITION 3.10.
is measurable if and only if, for each
B(x)
is
(A,)-measurable
There is a stronger notion of measurability for
$
h
in
(recall 2.2).
(,)
and
(as in [8,10])
which is useful for our needs.
DEFINITION 3.12.
is ultra-measurable if, for each compact subset
compact subset
ous on
K
Ke
Thus,
THEOREM 3.13.
(,)
The Banach bundle morphism
of
K
(,)
If
is true if, in addition,
such that
H(K-Ke)<e
K
(resp. operator field
X
of
(,’)
and
>0
there exists a
(resp. $)
is continu-
and
is ultra-measurable if and only if
is.
ultra-measurable, then it is measurable.
is
Ai
is locally bounded and each
The converse
is countably
dense,
lin
PROOF.
This is proved as in Prop. 20 of [7].
THEOREM 3.14.
HiM(Ai,i)
then
REMARK 3.15.
Suppose
$
is measurable.
f=(f!,...,fn)
If
is an element of
fgM(X,,)
Once again, if we were not assuming
there would be two versions of measurability for
$
Ai
C(AI)
given in 3.10.
l<i<n
then
However, in view
of 3.14, these would be equivalent.
REMARK 3.16.
As we indicated at
the beginning of this seciton, the case
n=l
I.E. SCHOCHETMAN
222
Specifically, for this case, this section contains the de-
is of special interest.
finitions and properties of continuous, measurable, and ultra-measurable Banach
bundle morphisms.
4.
PRODUCT FIELDS.
Having established the foundations for our analytical needs, we turn to our
algebraic needs--namely multiplication (i.e. convolution) and involution.
A
section we develop multiplication in
G
Suppose
spaces over
(A3)
G
by means of a product field of operators.
(A :xG}
is a locally compact group and
with
A
Then
as before.
In this
of bounded bilinear operators from
X
is a field of Banach
Hom2(AxXAy,Axy)
Ax xAy
is the Banach space
Axy
into
Consequent-
(x,y)eGG
ly,
{Hm2AxXAy ,Axy)
(x, y) GG
GxG
is a field of Banach spaces over
and
IISom 2 (AxXAy,Axy)
is a linear space of operator fields.
If
(x,y)gGxG
is such a field, then for
P
the mapping
P
/A
AxA
x
x,y
y
A
Let the field of
{E :xgG}
(This is in
is a bounded bilinear operator with range contained in
identity operators on the
{A :xEG}
X
be denoted by
X
HGHOm(Ax).
DEFINITION 4.1.
Let
PEHHom2(AxAy,Axy)
IIPx,yll<l
plication or convolution) field if
P
xy,z
For convenience, let
(Px,yXEz)
P
Then
p
x,yz
is a product
(or multi-
and
(ExXpy,z)
HHom2(AxXAy,Axy)
necessarily a product field.
P
x,y,zG
Note that an element of
Moreover, let p
denote the mapping of
P
AxAA
defined by
p(a,b)
Since the mapping
Pv(a),w(b)(a’b)
:GGG
is
a,bEA
given by group multiplication satisfies
no.__
GENERALIZED GROUP ALGEBRAS AND THEIR BUILES
(x)
p
(A,)
Furthermore, since
LAx
(p,)
it follows that
A
is continuous, if
d-’nA
is a continuity structur in
A
in
A
A’
by
A
by
A’
and
by
We will also require that product fields be measurable relative to
(rit) Haar measure B on G as in
P
Let
(i)
(li)
P
is continuous at
(v)
the mapping
For each h,k in A
Then the
P(h,k):GxG
(as in 3) is
the vector field
P(h,k)
hA
in
A’-continuous
is
(Xo,Y o)
at
(iv)
as above.
p
with
(Xo’Yo)
continuous at
(ill)
P
(Xo,Yo)GxG:
(Xo,Y o)
A
For each h,k in
A and
(p,P,)
be an element of
following are equivalent for
’
It will be instructive to
Section 3.
summarize he results of Section 3 for
THEOREM 4.2.
A the
For each h,k
in
continuous at
(Xo,Yo)
The bundle morphism
cross-section
(p,)
P(h,k)
in
S(GxG,A’)
is
(a,b) in Ax xA
is continuous at each point
o
(vi)
The mapping
PROOF.
p
Combine
PROPOSITION 4.3.
at
(Xo,Yo)
If
then the mapping
at
is continuous at each point
(a,b)
in
A
PROOF.
Xo Yo
1.8, 3.2, 3.3, 3.4, and 3.8.
DI,D 2
Let
fEDIA gED2Ay
x
P(f,g):DIXD2-A
XoED1 YoD2
_c G
are
Suppose
A-continuous at
is continuous at
Xo,Y
(Xo,Y o)
(resp.
P
is continuous
respectively,
A’-continuous
Recall 2.2, 3.9.
THEOREM 4.4.
(ii)
P
PEP
Let
Then the following are equivalent:
is measurable.
For each
h,k
Yo
A
(Xo,Yo)).
(i)
into
nAxy
(x,y)
We will denote
as in Section 2.
(A,’w)
is a Banach bundle morphism of
then we obtain the induced continuity structure
nA,y
223
in
A
the mapping
P(h,k):GG/A
is measurable.
I.E. SCHOCHETMAN
224
For each
h,k
A
in
(EGG,P(h,k))
the vector field
in
IiAxy
is
(A’, @)-measurable.
For each
(iv)
h,k
A’
relative to
PROOF.
P
PeP
Let
Then the following are equivalent:
is ultra-measurable.
Ke
subset
If
P
is measurable
Q
and
K
For each compact subset
(ii)
(EGG,P(h,k))
the cross-section
Combine 3.10, 3.11, and 1.12.
THEOREM 4.5.
(i)
A
in
K
of
A
is countably
there exists a compact
>0
and
Q(K-Ke)<e
such that
is a product field and
GG
of
and
P
Kg
is continuous on
dense, then the previous are
equivalent to:
P
(iii)
is measurable.
PROOF.
Combine 3.12, 3.13, and 4.1.
PROPOSITION 4.6.
P
Suppose
P
in
is measurable and
f,geM(A,)
Then
e (f, g)EM(GXG,A, )
PROOF.
Recall 3.14
5. INVOLUTION FIELDS.
The other algebraic operation we need is involution, which will also be intro-
duced by means of a field of operators.
Of course, such fields will have to be
suitably compatible with product fields in order that the resulting operations yield
a *-algebra structure in
For each
i.e.
A
X
A
X
and
x
in
G
A
let
A
X
denote the Banach space conjugate to
A
X
are identical except for scalar multiplication which is given in
A
X
by:
a’a
Consider the fields
{Hom(A,Ax_l)}
aa
eK
{A :xeG}
X
aeA
X
{Ax_l:XeG}
of Banach spaces over
form a field of Banach spaces over
linear space of operator fields.
G
and
G
The spaces
Hom(Ax,Ax_ I)
is a
Ax
into
Note that the linear mappings from
Ax_ I
225
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
A
are precisely the conjugate linear mappings from
NHom(Ax,Ax_ I)
an element of
I
X
A-+A
X
then, for each
(resp.
I
x
in
Ax_ I
into
X
G
Thus
if
I
is
the mapping
A -hA)
X
X
Ax_ I
is bounded and linear (resp. conjugate linear) with range contained in
For convenience, let
S
x,y
:A A
x
A A
y x
y
Let
DEFINITION 5.1.
Ix-I
Ix_ I
i
P
product field
I
P
in
fv
for
f@
f
If
are not necessarily involution fields.
@:G+G
Av
oSK
Now let
i
(i)
struc6ure A@
Aand
the continuity structure
h-sA-
A(x) flax_ 1
oh(x)
viewed as being in
KAX
A
l(a)(a)
(disjoint)
into itself given by
aeA
(A-,n-)
into
(Av,v)
where:
with conjugate scalar multiplication in the
fibers.
(ii)
Av
A
UA-i
X
HAX
xEG
denote the mapping of
DA-X
A
we have
is a bundle morphism of
A-= A
in
{hV:heA}
i(a)
(i,@)
it is customary to write
i.e.
(oh)(x)
Then
be the inversion homeomorphism
G
is a mapping defined on
Av
will be denoted by
Hence, for
x,ygG
Consequently, the induced continuity
Also, denote by
is an
For convenience, let
For the purposes of this section, let
x-I xeG
I
HHom(Ax,Ax_ I)
Note that the elements of
(x)
Then
and
IxyPx,y Py-l,x-l(lyIx)Sx,y
for each
ygG
x
HHom(Ax,Ax_l)
I be an element of
lllxll
involution field if
aEAx, bEAy
(b,a)
(a,b)
denote the switching mapping
(disjoint)
with
A-I
X
the fiber over
x
in
G
I.E. SCHOCHETMAN
226
(iii)
(a)
(iv)
v (a)
(a)
(a) -I
aeAX
for
x
x
-I
for
A
xeG
x
aeA,v.(x
A
x
xEG
-i
As in the previous section, an involution field will be required to
A and
measurable relative to
THEOREM 5.2.
x
(i)
I
leI
Let
with
i
x
is continuous at
o
l(h-) :G-
(ii)
For each
h
in
A
the mapping
(iii)
For each
h
in
A
the vector field
Av-continuous
(iv)
For each
at
(v)
(vi)
Then the following are equivalent
as above.
G
in
O
Here also, it will be instructive to summarize
(i,l,)
Section 3 for
for
be
h
x
at
l(h-)
in
A-i
x
o
is
o
A
in
x
is continuous at
l(h-)
the cross-section
S(G,Av)
in
is continuous
x
o
The bundle morphism
The pping
(i,)
is continuous at each point
is continuous at
i
eac
point
a
in
A-x
o
ine A
a
o
PROOF.
1.8 3.2, 3.3 3.4,nd 3.8.
Combine
PROPOSITION 5.3.
If
x
fEHA
Y
(resp.
AV-continuous
x
Suppose
then the mapping
in
[
is continuous at
l(f-) :D+A
x
is continuous at
x
Let
le[
Then the following are euqivalent
is measurable.
(ii)
For each
h
in
A
the mapping
(iii)
For each
h
in
A
the vector field
(iv)
I
Recall 2.2 and 3.9.
THEOREM 5.4.
I
at
xeD
DG
is A-continuous at
PROOF.
(i)
Let
For each
h
in
A
l(h-)
the cross-section
G-A
l(h-)
is measurable.
in
Ih-:C--A
v
A -i
X
is (A
is measurable realtive to
(v,)
PROOF.
Recall
THEOREM 5.5.
1.12, 3.10, and 3.11.
Let
lel
v,)-measurable.
Then the following are equivalent:
227
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
I
(i)
is ultra-measurable.
For each compact subset
(ii)
Kg
K
of
such that
(7-)
continuous on
If
I
-i
K
G
of
and
(K-Ke)<g and
-i
(K)
(K)).
A
is an involution field and
g>0
I
there exists a compact subset
is continuous on
KE
(resp.
i
is
is countably dense, then the previous are
equivalent to
(iii)
I
is measurable.
PROOF.
Combine 3.12, 3.13, and 5.1.
PROPOSITION 5.6.
Suppose
I
is measurable and
fgM(A,)
Then
I (f-)EM(G,A,)
PROOF.
6.
Recall 3.14.
GROUP ALGEBRA BUNDLES.
We are now ready to construct a measurable analogue of Fell’s Banach
*-algebraic bundle [i].
Let
the locally compact group
product field in
P
A
PROPOSITION 6.1.
A,A,,
with
as above.
P
is a
To intro-
define
a,bgA
P(a),(b)(a’b)’
We have:
aAx bgAy
(i)
a’bAxy
for
(ii)
For each
x,yEG
(iii)
(a’b)’c
a’(b.c)
[la’bll-< llall" l[bl[
Ax’AyCAxy
i.e
the mapping
(a,b)/a-b
of
x,yG
A A
x y
into
A
xy
is bilinear.
a,b,cgA
a,bgA
To obtain continuity for the product mapping
PROPOSITION 6.2.
Suppose also that
the corresponding product bundle mapping.
p(a,b)
a-b
(iv)
p
once again be a field of Banach spaces over
x
G
with
duce multiplication into
{A :xgG}
The product in
A
p:AxA/A
recall
4. i
is continuous if and only if
P
is
continuous.
One of our goals is to generalize the above to the case where the convolution is
just measurable--and not necessarily continuous.
PROPOSITION 6.3.
The product in
A
In this regard, we have:
is measurable if and only if
P
is
228
I.E. SCHOCHETMAN
measurable.
It is ultra-measurable if and only if
is countably
dense, then these are equivalent to measurability for
A
To introduce involution into
field in
with
a*
PROPOSITION 6.4.
(i)
(ii)
(a*)
For each
into
(iii)
(iv)
(a)
(a’b)*
a**
V(a)
G
in
Ax_l)
a
in
A
A
P (4.5).
is an involution
define
(a)
(a)
aA
(Ax), _c Ax_l
i.e.
the mapping
is conjugate linear
a/a*
xG
A
from
into
x
Ax_ I
A-x
(resp.
(resp. linear).
a,bgA
b*’a*
a
I
If
We have
-I
x
I
i(a)
is ultra-measurable.
suppose further that
For
as before.
i
P
agA
i:A-
To obtain continuity and measurability for the involution mapping
recall 5.1 and 5.4:
PROPOSITION 6.5.
The involution in
A
is continuous if and only if
I
is
The involution in
A
is measurable if and only if
I
is
cont inuous.
PROPOSITION 6.6.
measurable.
It is ultra-measurable if and only if
I
dense, then these are equivalent to the measurability of
DEFINITION 6.7.
By a (measurable) Fell bundle
(A,)
mean a Banach bundle
volution fields
understood to be
that
(A,;P,I)
P
over
I
and
CS(G,A)
G
axiomatically.
is countably
I
(A,; P,I)
G
over
(The underlying continuity structure
as before.)
we will
P
A
will be
and
I
are continuous, we will say
bundle over
G
(equivalently, a Banach
If
G ).
In Section 3 of [i], Fell defines a continuous Fell bundle
to be a Banach bundle
A
together with measurable product and in-
is a continuous Fell
*-algebraic bundle over
If
is.
(A,)
over
G
with product
(A,:-,*)
and involution
Clearly, the corresponding product and involution fields
are then determined as follows:
over
*
G
given
P
and
I
229
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
P
xy
(a,b)
Ix(a)
a’b
aA
a*
bAy,
x
x,yEG
A
Of course, these fields are continuous relative to
A
are continuous operations on
CS(G,A)
and
his construction to that of
To extend
(6.2, 6.5).
since
a measurable Fell bundle, the appropriate measurability requirements on
and
*
would be given by 4.4 and 5.4.
EXAMPLE 6.8 [2 ,p.130].
Let
be the 2-dSmensional Banach *-algebra
B
2
with
-algebra structure given by
(a,b) (a’,b’)= (aa’,bb’)
(a,b)*
(b ,a
and norm given by
(II, Jbl)
ll(,b)ll
Let
G
be the cyclic group
subspaces of
B
P
x,y ((a,xa),
Ix((a,xa))
and
I
G
Let
{i,-i}
B
let
AI,A_I
I
A_I
A
AIA_I
AIXA_I
P
Define
an,d
I
by
(ab, xyab)
(b,yb))
x,ygG,
(a ,xa
Let
D
be the Banach *-algebra
A
Then
A
agAx, bgAy
are continuous.
EXAMPLE 6.9 [2,p.131].
D
the closed
{(a,-a):a}
is discrete, we have
Of course, P
{AI,A_I}
and
{(a,a):ag}
I
A_I
G
2
of order
given by
A
Since
{i,-i}
a,a’ ,b,b’g
{ze:Izl-<l}
C(D)
the ususal boundary of
with pointwise *-algebra structure.
B
be the closed subspaces of
{fgC(D)’f(-z)
f(z)
zg3D}
{fgC(D):f(-z)
f(z)
zgD}
once again and continuous
D
and
P
and
I
given by
are given by
For
I.E. SCHOCHETMAN
230
P
(f,g)
xy
I (f)
xy(f’g)
x,yeG, feA
f
x
geA
y
x
Before leaving this section, we wish to point out one of the main consequences
of replacing a continuous Fell bundle by a measurable one.
[2,p.i15].
DEFINITION 6.10
saturated
if the span of
Axy
dense span in
A measurable Fell bundle
A "A
x y
for all
The span of
Ae "A
(ii)
The span of
Ax’Ax_ I
x
P
Conversely, if
i.e. the range of
xy
(A,;P,I)
If
is dense in
A
x,y
has
is saturated, then:
A
xeG
e
is continuous, then (i) and
Hence, the converse implication
(ii) imply saturation.
is questionable for measurable Fell bundles
since its proof depends on the continuity of the product in
7.
P
is
xeG
x
is dense in
Propositions 11.4 and 11.5 of
G
over
x,yeG
PROPOSITION 6.11 [2,p.i15].
(i)
A
is dense in
(A,:P,I)
[2]
A
This also affects
in the same way.
BUNDLE IDENTITIES.
The purpose of this section is to extend (and rephrase) Fell’s study of approxi-
mate identities (units) in
Let
identity
(A,;P,I)
e
G
in
[1,2]
be such a bundle.
A
e
(i)
(ii)
(iii)
It is clear then that the fiber
A
e
over the
is a Banach *-algebra
DEFINITION 7.1 [i,p.34].
in
to the context of a measurable Fell bundle.
A (bounded) approximate identity in
A
is a net
{u.}
satisfying:
llujll-<
Iluj’a
Ila’uj
In particular,
8
all
all
all
{uj }
for some
j
0
aeA
o
aeA
8>0
Ae
is an approximate identity in
DEFINITION 7.2 [i,p.34].
The net
{u.}
3
in
A
e
in the usual sense.
is a strong
approximate
identity if it is an approximate identity and (ii) and (iii) of 7.1 hold uniformly
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
231
A
on compact subsets of
In Prop. ii.I of [2], Fell showed that these two notions agree in the case of
continuous Fell bundles.
To
This appears to be false in the measurable case.
determine what is true, we proceed as follows.
Let
D-CG
feDAx
agAe
and
The left and right translates of
f
by
a
are
defined by
f(x)
a-f(x)
P
f (x)
f(x).a
P
a
e,x
(a,f(x))
x,e
(f(x),a)
and
a
Then
af fa
DAx
belong to
PROPOSITION 7.3.
F
Let
DAx
which is a linear space, i.e.
C(D A) c
Now consider
in general.
sided) A-module
e
xED
be a total subset of
is a
(two-
DAx
C(A)
Then the following are
equivalent
(i)
(ii)
C(D,A)
is a left (resp. right)
For each
h
A-continuous
in
F
on
D
DEFINITION 7.4.
a
in
We say that
A-module.
e
A
the vector field
e
c(A)
A -module, for each compact subset
e
K
is locally an
of
a
h(resp, h
a
Ae "-module if
is
C(K,A)
is an
G
The following is our measurable Fell bundle version of Prop. ii.I of [2].
THEOREM 7.5.
locally an
Let
Ae-module,
{u.}
3
then
be an approximate identity in
{uj}
particular, if multiplication in
A
If
is a strong approximate identity in
A
is continuous, then
C(A)
C(A)
A
is
In
is actually
(globally) and A -module.
e
REM.ARK 7.6.
Observe that if
C(A)
is locally an A -module, then the
e
corollaries 11.2 and 11.3 of [2] are valid.
8.
MULTIPLIERS.
There are two additional significant distinctions between continuous and
measurable Fell bundles which involve multipliers.
multiplier theory
[1,4]
We will briefly discuss here the
we require in this and succeeding sections.
I.E. SCHOCHETMAN
232
DEFINITION 8.1.
If
for the measurable Fell bundle
x
G
is an element of
x
(A,;P,I)
then a multiplier
is a pair
(ml,m2)
of order
m
satisfying the
following
(i)
are continuous mappings of
ml,m2
A
into itself which are bounded in the
sense that
sup{lira i(a)ll :aeA}
Ilmill
is finite for
For each
(ii)
a-m
yeG
ml(res p. m2)
l(b)
m
2(a).b
a,beA
l(a)-b
a,beA
2(b)
a,beA
(iv)
m
l(a-b)
m
(v)
m
2(a.b)
a-m
For convenience, as usual, we will write
aeA
Let
M
x
(A)
Each
M
x
(A)
for
ma
ml(a)
denote the set of multipliers of order
z:M(A)+G
Also, let
A
is a linear mapping of
into
y
A
xy
Ayx)
(resp.
(iii)
1,2
i
and
x
am
and
for
M(A)
m2(a)
UGMx(A)
be the canonical projection.
is a Banach space under the canonical linear operations and norm
given by
llmll
Thus, (M(A),z )
an involution in
max(llmlll,llm211), mgMx(A)
xeG
is algebraically a Banach bundle.
M(A)
Moreover, there is a product and
given by
(m-ml)a
(am)re’
a(m-m’)
m(m’a)
and
m*a
am*
(a’m)*
for
agA
(i)
If
(2)
The product is bilinear on
(3)
The product is associative.
(4)
If
(5)
m,m’eM(A)
The operations have the following properties:
meMx(A),m’eMy(A)
meM (A)
x
then
(ram*)*
then
m*gM
x
mm’eMxy(A)
Mx(A)XMy(A)
-I(A)
The involution is conjugate linear on
M (A)
x
233
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
(6)
The involution is anti-multiplicative.
(7)
The involution is self-invertible.
llm*ll
and
(9)
ilmll
A
The left and right identity mappings of
M(A)
fon
the identity of
M (A)
e
in
We are now ready to describe one of the distinctions between continuous and
measurable Fell bundles referred to at the beginning of this section.
(A,;P,I)
If the Fell bundle
is continuous, then
A
can be mapped into
by right and left multiplication:
ba
tuba
However,
a,bgA
ab
am
b
in the measurable (non-continuous)
case, this does not seem possible, since
the left and right multiplications may fail to be continuous.
The remaining distinction involves the notion of unitary multiplier.
DEFINITION 8.2.
mm*
m*m
If
meM(A)
U(A)
in
is unitary if
m
the canonical identity in
1
The unitary multipliers
Also, if
then
llmll -<
I
and
M(A)
M(A)
form a group under multiplication.
Ux(A) u(A)OMx(A then
U(A)
UGUx(A)
and
_I(A)_
m*eU
x
x
in
LEMM% 8.4.
spaces
{A :xeG}
x
LEMMA 8.5.
xeG
[2,p.122].
G
Ux(A)#
If
(A,;P,I)
has enough unitary multipliers, then the Banach
[2,p.123].
are all isometrically isomorphic
If
P
is continuous and
enough unitary multipliers, then
REMARK 8.6.
meUx(A)
A measurable Fell bundle has enough unitary multipliers if,
DEFINITION 8.3.
for each
if
In general, if
unitary multipliers even if
P
A
has an approximate unit as well as
A
is saturated
A
is
(6.10 and [2,prop.ll.5]).
saturated, then it may not have enough
is continuous and
A
has an identity [2,p.130].
I.E. SCHOCHETMAN
234
The implication in 8.5 is questionable in the measurable case because its
[2],
proof, namely Prop. 11.4 of
P
is questionable for non-continuous
(Recall
the end of Section 6.)
There is a notion stronger than "enough unitary multipliers" called "homo-
geneity" which we will study in detail in Section i0.
Finally,
[1,5].
M(A)
can be equipped with a topology called the strong
In this topology, a net
am.
ma
m.a
3
{M.}
am
3
in
M(A)
converges to
m
tppology
M(A)
in
if
agA
For this topology, involution is continuous and the product is separately continuous.
The mapping
open.
:M(A)
Consequently,
A
is continuous relative to this topology but possibly not
.vIU(A)
is a continuous homomorphism of
if there exist enough unitary multipliers).
U(A)
G
into
(onto
U(A)
Note that multiplication in
is
separately continuous but possible not jointly continuous.
Let
M
I(A)
{meM(A) :llmllo-<l}
The following will be useful in Section i0.
PROPOSITION 8.7.
A
If
has a strong approximate identity and
(m,a)+ma
measurable, then the mappings
measurable in the sense of 3.10.
and
then these mappings are continuous (compare with
9.
(i.e., p)
P
A
are
is continuous,
[l,Prop.5.1]).
GENERALIZED GROUP ALGEBRAS.
To construct a Banach *-algebra from a measurable Fell bundle
G
is ultra
MI(A)A "into
(m,a)/am of
In particular, if
P
let
LI(A,)
denote the Banach space of (null equivalence classes) of
measurable vector fields
Ilfll 1
the results in Section i,
The subspace
LI(A,)
-measurable integrable
modular function for
(A,)-
./*Gill (x) lld (x)<
Cc(A)
of
C(A)
fields with compact support is well-known to be dense in
of
over
which are -integrable, i.e. for which
f
(See [7,8] for the details.)
LI(A,)
(A,;P,I)
G
consisting of vector
LI(A,)
Also, in view of
is the vector field version of the space
cross-sections
[1,2].
Let
be the (right)
Before proceeding further, let us record an important
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
result for future use.
Once again, let
%:GXG-G
235
be the left projection (recall
2.5).
LEMMA 9.1.
fELI(A%,)
Let
Then for
-a.a.x
in
G
the integral
fGf (x,y)d (Y)
belongs to
A
and the resulting (-a.e. defined)
x
fGf (x,y)d (y)
x
belongs to
L
PROOF.
I(A,)
This is the vector field analogue of Prop. 2.11 of [i].
Now, for
f,g
LI(A,)
in
consider the vector field
f(y-l)’g(yx)
(x,y)
in
vector field
P(f,g)(y-i ,yx)
I[Ak(x,y
PROPOSITION 9.2.
LEMMA 9.3.
at most
The vector field of 9.2 belongs to
LI(A
)
and its
I
L -norm is
llfll l’llgll I
PROOF.
This is a straightforward application of the scalar Fubini Theorem.
COROLLARY 9.4.
For
x
(defined
(A)t,ltt)-measurable.
This vector field is
-a.e.
PROOF.
on
G)
LI(A,) the
/Gf (y-l)- g (yx)d (y)
f,g
in
belongs to
L
I(A,)
Apply 9.1 to the integrand.
As a result of the previous discussion, for
vector field
vector field
f’g
in
L
I(A,)
f,g
in
LI(A,)
we may define a
by
fGf(y-l)’g(yx)d(y)
(f’g)(x)
We thus obtain a binary operation
-a.a.
x
in
(multiplication or convolution) on
G
LI(A,)
which satisfies
<_
REMARK 9.5.
llflll-Ilgll I
f,ggL
I(A,)
Before going any further, observe that Fell defines convolution
I.E. SCHOCHETMAN
236
[i,8] and then extends
C (A)
first in
C
case (that of a continuous Fell
bundle),
C (A)
Hence, for us,
C (A)
multiplying elements of
PROPOSITION 9.6.
L
denote by
Next,
Note that in his
is closed under multiplication as a
C
A
consequence of the continuity of multiplication in
for a measurable Fell bundle.
LI(A,)
it to all of
However, this
there is less advantage in first
C (A)
since the products may not be in
C
Under multiplication,
is not true
LI(A,)
C
is a Banach algebra which we
I(A,;P)
we define involution in
LI(A,;P)
For
f
Ax,
in
,
define
f
in
AX by
6(x)-ll(f -)
6(x)-if(x-l)*
f*(x)
LEMMA 9.7.
For each
f
in
LI(A,;P)
LI(A,;P) in fact, llf*ll I llfll I
from L I(A,;P) into itself.
PROPOSITION 9.8.
denote by
L
6(x)-II
LI(A,;P)
Under involution
xeG
(f(x))
also belongs to
f*
the field
Hence, we obtain
X
*
a mapping
(involution)
is a Banach *-algebra which we
I(A,;P,I)
LEMMA 9.9.
PROOF.
(x)
LI(A,;P,I)
If the conditions of 1.16 hold, then
The underlying Banach space
to Proposition 2.2 of
LI(A,)
is separable by the Corollary
[l,p.20].
Next in this section, we turn to a study of identities in
A
Suppose
has a strong approximate identity.
LI(A,;P,I)
In the proof of Prop. 8.2 of
LI(A,;P,I)
to construct an approximate identity for
[i], Fell shows how
is separable.
observe that this proof does not use the A-continuity of
P or
I
Also
Consequently,
we have
LEMMA 9.10.
LI(A,;P,I)
If
has a strong approximate identity (recall 7.5), then
contains an approximate identity (with the same
PROPOSITION 9.11.
if and only if
PROOF.
A
G
The Banach *-algebra
is discrete and
A
LI(A,;P,I)
bound).
contains an identity
has an identity.
The proof of the corresponding result for classical group algebras
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
[12,310]
u
of
A
In particular, the identity
can be adapted to the vector field context.
of.
f
and the identity
LI(A,:P,I)
237
f(e)
are related by
u
For the remainder of this section, we investigate the correspondence between two
generalized group algebras whose underlying Fell bundles are connected by a "Fell
bundle morphism," i.e. a Banach bundle morphism having the appropriate additional
algebraic properties.
(B,T:Q,J)
Suppose
is another Fell bundle over
{ :xeG}
g
x
where
f
x
[A
x’
xgG
If
f
Bx
is a vector field in
is the
be a Banach bundle
..
=EG)
morphism as in section 3 (with
LI(A,:Q,J)
and
:(A,)+(B,T)
Let
corresponding generalized group algebra.
G
Then we have a field
GHOm(Ax,Bx
is a vector field in
HAx
then it follows that
Furthermore, the correspondence
f
f
is linear
between the underlying linear spaces of vector fields; notationally, we have
HAx+liBx
Now suppose that
is measurable
Then by 3.14, we have a linear mapping
(3.10).
M(A,) M(A,)
If
which is constant on null equivalence classes.
is continuous, then
(3.9)
c(A)+C(A)
In particular,
c c (A)< c (A)
since
supp (f)
_e, supp(f)
For convenience, let
I111 G
sup
We will say that
most
IIII G
isometry.
If
:xgG}
{llxl
is bounded if
PROPOSITION 9.12.
nullity)
denote
..llII G
(i.e.)
<m
is measurable and
is a bounded, linear mapping of
In particular, if each
Then we have:
Cx’XgG
L
l(A,)
into
is bounded, then (modulo
L
l(A,)
is an isometry, then
with norm at
is an
I.E. SCHOCHETMAN
238
In order that
be a *-algebra homomorphism as well, we will have to require
more of
DEFINITION 9.13.
The mpping
(A,;P,I) (,T;Q,J)
: (A,n)-(B,y)
is a Fell bundle morphism if
is a measurable Banach bundle morphism
and
Px, y Qx,y (x)
i.e.
Ix Jx
(i)
(ii)
for
x,y
in
#
i.e.
I
Qx, y (xXy)
P
xy x,y
J
X X
X X
G
THEOREM 9.14.
(as in 9.13)
If
is a bounded Fell bundle morphism, then
LI(A,p;P,I)/LI(A,p;Q,J)
is a Banach *-algebra homorphism.
-I
Conversely, suppose
B+A
exists and is a Banach bundle morphism, so that
9.12 and 9.13 apply.
PROPOSITION 9.15
and
(B,T;Q,J)
If
,-i
as above, then
LI(A,;P,I)
LI(A,;Q,J)
and
I0.
xgG
are
isomorphic
In particular, if each
*-algebras which are equivalent as Banach spaces.
invertible isometry,
(A,;P,I)
are bounded Fell bundle morphisms for
Banach
x
is an
then these algebras are isometrically isomorphic.
HOMOGENEITY.
Our objective here is to extend the main ideas and results of sections 6 and 9
of [i] to the setting of (measurable) Fell bundles.
This will be useful in
Section Ii for comparing Leptin bundles with Fell bundles.
U(A)
Recall that the unitary multipliers
A
for the Fell bundle
form a
group and a topological space with the relativized strong topology.
DEFINITION i0.i.
(i)
(ii)
A
The Fell bundle
A
is (measurably) homogeneous if:
has enough unitary multipliers, i.e.
The mappings
(as in 3.10).
(re,a)
ma
and
(m,a)
(U(A))
am
of
G
U(A)xA
into
A
are measurable
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
For each
REMARK 10.2.
into
is
A
xy
x,y
in
is a linear isoetry.
homogeneous), then the fibers
LEMMA 10.3.
A
able, then
A
If
G
and
in U
m
(A)
o
Therefore, if
{Ax:XgG
x
the mapping
ma
a
of
is onto (in particular, if
are all isometrically .somorphlc.
has a strong approximate identity and
is homogeneous if and only if
239
A
P
ultra-measur-
is
has enough unitary multipliers.
(Compare with Remark 3 of [i,p.49].)
It follows from 8.5 that (ii) of i0.i is automatically satisfied under
PROOF.
the given hypotheses.
REMARK 10.4.
In particular, if
mate identity, then
A
A
is a continuous Fell bundle with approxi-
is homogeneous if and only if
IU(A)
is onto (7.5).
Thus, for the case of such bundles, homogeneity is simply the existence of sufficiently many unitary multipliers.
Consequently, this latter property is really
the crux of the homogeneity property--both technically and intuitively.
In Section 9 of
[I],
Fell bundles
Fell shows that all continuously homogeneous
can be constructed (up to isomorphism) from a given set of "ingredlent’s."
next extend this construction to the measurable case.
We will
Moreover, we will do this in
the setting of vector fields, describing the underlying
con.tinulty structure
specifically.
Of even greater significance--especially for the needs of Section 13--is the
description of the underlying field of Banach spaces.
the same way as is the field in Section 4 of [13].
This field is constructed in
However, the contexts
these constructions take place are different in point of view.
in which
Consequently, we
will also adopt the viewpoint of Section 14 of [13J--namely, that of group repreThis will allow the results of Section 13 to follow immediately from
sentations.
[13] and this section.
Let
be a Banach *-algebra with (bounded) approximate identity.
A
a subgroup of the topological group
also that
H
U(A)
of unitary multipliers on
is a topological group extension of
epimorphism on the space of right cosets.
N
with
Assume also that
q:H-H/N
G=H/N
A
Let
N
be
Suppose
the canonical
is locally
240
I.E. SCHOCHETMAN
compact in the usual quotient topology.
G
The local compactness of
REMARK 10.5.
[i].
is not assumed by Fell in
X
This
i.e. all
is essential for us here, since
G
of Section I must apply to
However, this is not a severe additional assump-
G
G
tion, since Fell assumes
will play the role of a base space
is locally compact for the main purposes of
[l]--for
example, cross-sectional algebras, induced representations, etc.
Given such
A, N
and H, Fell constructs a field of Banach spaces over
N
defining an equivalence relation
HA
in the space
point of view, observe that the topologicl group
R
Such
R
is a
t(a)
a
t-i
aEA
A
by:
and (y,b)
N
on
A
which we
are elements of
define:
(x,a)
if there exists
t
in
N
(y,b)
for which
b
Rt(a
at
This defines an equivalence relation
of equivalence classes
HA
is represented on
tEN
(x,a)
If
by
Consistent with our stated
bounded, strongly continuous representation of
call the right regular r.epresentation.
HXA
N
G
Observe that
A
(x,a)"
by
-i
y
in
A
tx
Denote the space
HxA
(HxA)/~
equipped with the quotient topology from
can be viewed as the orbit space of
HXA
under the right
topological transformation group
(HXA)N
where
((x,a),t)
The projection mapping
:A-G
(tx,Rt(a))
in G
let
non-empty) fibers
A,
q(x)
AS -I().
EG.
J :A
agA, tEN
xgH
is then given by
(x,a)~)
For each
HXA
agA, xEH
Then
If we define
/
A
xe-eG
A
is the disjoint union of the
241
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
by
J (a)
J
then
aeA
(x,a)
x
is a (well-defined) bijection, since
x
Jx(Rt(a))
Jtx(a)
agA
tgN
xgH
to A by
A
Therefore, we may transfer the Banach space structure of
J
such
Specifically,
(x,a + b)
+ (x,b)
(x,a)
.(x,a)
(x,a)
ll(x,a)~ll
llall
{A:eG}
In this way, we obtain a field
xgH
,a,bgA
(A,)
The continuity structure underlying the bundle
A
a
CS(G,A)
C(A)
H
CN(D,A)
a union of
{fEC(D,A)
f~
For
J (f(x))
x
f~
Since
f-f~
mappings, it is also continuous, i.e.
LMMA 10.6.
Furthermore, if
f+f~
The mapping
hgA
then
of
J
x
c
of isometric
CN(D,A)
define
f~:G+A which
CN(D,A)
into
tat
-i
CS(G,A)
C(A)
_l(h(q(x)))
*-automorphisms of
of
into
f
R t(R t(a*))*)
aeA
A
in
is a linear bijection.
CN(H,A)
given by
xeH
H
into the topological group
A having the following "admissibility"
is a group homomorphism.
Yt(a)
in
is a mapping of
properties
(ii)
f
tgN}
xgH
is
is also the composition of continuous
CN(H,A)
Next suppose we are given a mapping
T
D
xeH
for the unique
h=f
f(x)
(i)
For
In particular,
{f~:feCN(H,A)}
AUtl(A)
is
is constant on cosets and hence, defines a’mapping
easily seen to be a cross-section.
CS(q(D) ,A).
G
in the following way.
Rt(f(x))
f(tx)
(x,f(x))~
over
N-cosets, define
C(D,A)
is a linear subspace of
f~(x)
The mapping
D
i.e.
CN(D,A)
Then
C(H,A)
This is obtained from
"saturated" subset of
G
of Banach spaces over
teN
I.E.- SCHOCHETMAN
242
Tx(t
(iii)
M(A)
xtx
-I
teN
xgH
T’x
where
is the unique extension of
T
x
to
defined by:
T’
x
(m)rx(S)
The *-algebra structure of
Tx(ma)
A
meM(A)
-, aeA
is defined by means of
In terms of
T
operator fields, this structure is given by:
where
P N((x a)
(y,b) ~)
I((x,a) ~)
LFLMA 10.7.
P
The fields
(x
a’T
(b))
,
-l,r -l(a)*)
x
I
and
(xy
beA
a
xe
y]
are product and involution fields respec-
t ive ly.
EXAMPLE 10.8.
N
then
is trivial.
U(A)
(One-dimensional fibers [i,p.75])
is the circle group.
Thus, for
H
Also, T
A
If
is the complex numbers
must be trivial, since
AUtl(E)
as above, we have:
{fgC(H)
CN(H,A)
f(x)
f(tx)
P,n((x,a) ~,(y,b) ~)
l((x,a) ~) (x-l,)
xgH}
tN
(xy,ab)
xEG
yqG
e,be
In order to motivate suitable definitions of measurability for
T
(as above),
consider the following characterizations of continuity:
PROPOSITION I0.9.
in
Let
T:H
Aut
I(A)
The following are equivalent for
H
(i)
(ii)
is strongly continuous at x
T
The_ mapping of
HxA
into
A
o
given by
(x,a)
T
x
(a)
is continuous on
{x }A
o
(iii)
For each
(x,y)
f
in
Tx(f(y)
C(H,A)
the mapping of
is continuous on
{x }H
o
HxH
into
A
given by
x
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
For each
(iv)
a
A
in
x
is continuous at
(v)
into
given by
~o
(X,lx(a))
x
A
into
(x a)
given by
(x,Ix (a))
is continuous
{x }XA
on
o
Moreover, under the above conditions, for each
HH
A
o
HA
The mapping of
H
the mapping of
243
A
into
(x y)
given bv
PROOF.
f(x)’Y .g(y)
f,g
in
C(H,A)
{x }H
is continuous on
X
the mapping of
O
All implications are straightforward with the exception of "(iv)
implies (i)"
[l,p.70].
which follows from Lemma 9.1 of
In view of 10.9, we propose the following:
A mapping
DEFINITION i0.i0.
A
the mapping
K
subset
(K-K
a
x-W[
X
G
of
< e
(a)
and
AUtl(A)
:H
is measurable
is measurable in the following
>0
sense:
x
x(a)
I
is continuous on
q
K
of
-I
a
in
For each compact
Kg
there exists a compact subset
and the mapping
if, for each
such that
(K ,a
There is also a stronger notion of measurability as in the case of bundle
morphisms.
A mapping
DEFINITION i0.ii.
K
compact subset
<
(K-Kg)
that
of
G
and
and
T
g
Aut I(A)
:H
> 0
is ultra-measurable if, for
there exists a
is continuous on
q
-i
(Kg)
comDact
to
Aut
Kg
subset
of
each
K
such
for the topology
I(A)
of pointwise convergence, i.e.
T:q
-I
(K)
Aut
I(A)
is strongly continuous.
As before, these two notions of measurability are equivalent in the presence of
separability.
Lh-MMA 10.12.
If
I
is measurable and
A
is separable, then
I
is ultra--
measurable.
PROOF.
This follows from Prop. 2 of [14,p.170] as in the proof of 3.13.
PROPOSITION 10.13.
Let
P
and
I
Suppose
be as above.
I
is an admissible mapping of
H
into
AUtl(A)
I.E. SCHOCHETMAN
244
(i)
lf
is
measurable, then the oerator fields
(2)
If
is
,itr-measurable, then
(3)
If
T
P
is continuous, then
P
and
I
and
I
P
and
I
are measurable.
are ultra-measurable.
are continuous (relative to
A ).
It follows from the above that the ingredients (A,N,H,) yield a Fell bundle
A
when
not
A
T
The obvious next question is whether or
is admissible and measurable.
is homogeneous.
For each
y
H
in
define:
(yx,T (a))
Y
M (x,a)
Y
and
(x,a)~M
(xy,a)
Y
For each
LF.MMA 10.14.
y
in
xgH
A
is a unitary multiplier of
M
Y
H
agA
i.e.
M U(A)
Y
See p. 71 of [i].
PROOF.
The previous lemma shows that
A
(U(A))
will be measurably homogeneous if (ii) of i0.I holds.
(i)
be the case if
identity (10.3).
T
(2) A
is ultra-measurable, and
(Note that if
{u.}
is an approximate identity in
unless
is continuous
in the techncial sense of i0.i.
Consequently,
In particular, this will
has a strong approximate
A
is an approximate identity in
{(e,uj) ~}
(7.5).
A
for the bundle
G
A .)
then
These appear to be false in general--
Hence, it seems unlikely that
A
is homogeneous
However, in view of 10.4, we feel that
A
is as
homogeneous as it can be under the circumstances.
In the opposite direction, Fell shows [l,Thm. 9.1] that every (continuously)
homogeneous Banach *-algebraic bundle is isomorphic to one obtained from ingredients
(A,N,H,r)
be
that
with
continuous.
In particular, the group extension
is chosen to
This is not possible in the measurable setting, since it is not clear
U(A)
U(A)
is a topological group.
Perhaps there is another way of obtaining a
measurable generalization of his Theorem 9.1.
replace
H
U(A)
to Section V of
by its image in
U(LI(A,:P,I))
For example, it may be possible to
[2,pp.137-139].
We refer the reader
[5] for further information regarding this possibility.
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
ii.
245
UNITARY FACTOR SYSTE.
The other two (equivalent) bundle constructions referred to in the introduction
are due to Leptin [3] and to Busby and Smith [4].
These constructions are quite
Although their objects
similar to each other, but very different from that of Fell.
are quite familiar (vector-valued functions), their algebraic operations are not.
In this section, we review Leptin’s approach (as in [5]) and show how it gives rise
to a homogeneous Fell bundle.
G
Let
U(A)
M(A)
Let
identity.
[5] are
Leptin construction
A
DEFINITION ii.i.
AUtl(A)
and
a Banach *-algebra with approximate
A
be a locally compact group and
The ingredients for the
be as above.
G,A together with the following:
unit
(T,W)
factor system
for
(G,A)
is a pair of
strongly measurable mappings
T:G/Aut
W:GG-U(A)
I (A)
satisfying:
aW
(ii)
T
(a)
x,y x,y
W
x,y
W
(iii)
e,x
x,e
T
x,y,z
T
where
(EA,EA)
W
REMARK 11.2.
W,T
Wx,yz.Wy,z
Wxy,z.T z -l(Wx,y)
(i)
As in the case of
y
_i
T _I
x
T-I-i
V
x
for x,v in G, a in A
-i
x,y in G, a in A
EA,
e
T
x,y
(multiplication in M(A)).
G
in
there is a notion of measurability for
T
betwaen strong measurability and strong continuity--namely Bourbaki measurThis is what we will call ultra-measurability for
ability [i0,p.169].
obvious reasons.
LEMA 11.3.
W,T
for
As in i0.12, we have:
For
W(resp. T)
strongly measurable and
A
separable, W(resp. T)
is ultra-measurable.
(A,)
Let
xEG
z(x,a)
x
phic to
A, xSG
be the trivial bundle
aA
A
Let
For each
x
f
EGf
in
GA
over
G
be the Banach space
C(G,A)
GG
GA
let
fG
(product topology)with
{x}XA
canonically isomor-
be the mapping
I.E. SCHOCHETMAN
246
Then
{fG
fgC(G,A)}
A
is a continuity structure
A
Ax
for which
CS(G,A)
C(A)
A
The algebraic structure in
in
is defined as follows in terms of operator
fields:
Px,y((X,a)
(x a)(y b)
Ix(X
(x a)*
(x
a)
-I
(i) The field
PROPOSITION 11.4.
(xy,W
(y,b))
,W*
x
_i
x
T
,yTy
-_I l(a*))
-i
(a)b)
x,ygG
a, bgA.
X
is a measurable product (resp.
P (resp.l)
involution) field.
P,I
(2)
If
W,T
(3)
If
W,T are (strongly) continuous, then so are
PROOF.
are ultra-measurable, then so are
Parts (i) and (3) are straightforward.
P,I
(relative to
% ).
Part (2) is proved as in
Section i0.
The resulting Fell bundle
be denoted by
will be called a Leptin bundle and will
(G,A:T,W)
PROPOSITON 11.5.
PROOF.
(A,;P,I)
Every Lepti bundle has enough unitary multipliers.
[5],
As on p. 329 of
.Iz(X,a
for
z
G
in
(zx,Wz,xa)
(xz’Wx,zT z -l(a))
(x’a)lZ
define:
(x,a) e GXA
Hence, every Leptin bundle is a homogeneous Fell bundle in the weak sense of
having enough unitary multipliers.
Observe that if
{u.}
is an approximate identity in
A
then
3
an approximate identity in
A
{(e,uj)}
is
where:
(e,uj)(x,a)
(x,T
(x,a)(e,uj)
(x,auj)
_l(uj)a)
x
and
PROPOSITION 11.6.
If
T
(x,a) e A
is strongly continuous, then
{(e,u])}
is a stron
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
A
approximate identity for
PROOF.
247
This can be proved directly or obtained as a consequence (by 7.5) of
the following result.
PROPOSITION ]1.7.
(I)
C(A)
(2)
If
Let
(G,A;T,W)
is locally a right
T
If
(G,A;T,W)
Leptin bundle
C(A)
is separable and
is locally a (two-sided)
T
A-module.
is strongly continuous, then the
is homogeneous in the sense of i0.i.
This follows from i0.3 together with ii.3, ii.4, ii.5, and
PROPOSITION 11.9.
PROOF.
A
Then:
A-module.
is strongly continuous, then
COROLLARY 11.8.
PROOF.
be a Leptin bundle.
II.6.
Every Leptin bundle is saturated.
This would normally follow from Prop. 11.5 of
[2].
However, since the
validity of this proposition is questionable in the case of measurable bundles, its
conclusion has to be verified directly.
The generalized
Ll-algebra LI(G,A;T,W)
1
L
bundle is the Banach space
(G,A,)
corresponding to the underlying Leptin
with convolution and involution
ifor
ight
Haar measure) given by:
f*g(x)
f*(x)
-I_
f_W
f(xy
T
x,y -I
Y
(x)-Iw*
x
for
f
and
g
in
to our generalized
f-
LI(G,A,)
,x
T I (f (x-l) *)
xEG
This Banach *-algebra is isometrically isomorphic
group algebra
fG
-I
-1) g (y) d (y)
LI(I\,;P,I)
of Section 9 by the mapping
fLl (G,A;T,W)
Moreover, by 11.6 and 9.11,
LI(G,A’T,W)
will have an approximate identity if
is continuous.
As observed in Section IV of [5], certain homogeneous bundles give rise to
equivalent Leptin bundles.
Suppose
(A,N,H,T)
are the ingredients for a homo-
geneous Banach *-algebra bundle as in Section I0.
corresponding Fell bundle, where
A
HA/
and
Let
(x,a)
(A,;P,I)
q(x)
denote the
xaH
agA
T
I.E. SCHOCHETMAN
248
Assume also that
H/N
G
is locally compact (as before) and that there exists a
G
measurable cross-section
T
H
If we define
TO(_I)
and
,n
(n)
then we obtain a Leptin bundle
isomorphic.
[5,p.333].
(G,A;T,W)
LI(A,;P,I)
LI(A,;P,I)
group algebras
(Oo(n)
and
Leinert also shows that the
LI(G,A;T,W)
are isometrically
This can also be concluded from the following.
Define
GA+A
by
(,a)
Then it is easy to verify that
isomorphism.
12.
(,a)
(o(),a)
GA
g
invertible, bi-measurable Fell bundle
is an
Hence, the algebra isomorphism follows from 9.15.
TWISTING PAIRS.
Let
G
and
A
be as in Section ii.
structing a generalized group algebra from
DEFINITION 12.1.
A twisting
The Busby-Smith approach [4] to con-
LI(G,A,la)
(S,V)
requires the following:
for
(G,A)
GXG
U(A)
is a pair of strongly
measurable mappings
S
Aut I(A)
G
V
satisfying
(i)
s x (Vy ,z )vx,yz
vx,y vxy,z
(multiplication in M(A))
(ii)
(iii)
(SxSy(a))Vx,y Vx,ySxy(a)
V
x,e
V
(EA’ EA)
e,x
LEMMA 12.2.
For
S
V(resp. S)
e
S
A
for
x ,y,zgG
aeA
strongly measurable and
is ultra-measurable, i.e. Bourbaki measurable (recall 11.2).
A
separable,
V(resp. S)
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
(A,)
As in Section ii, the underlying bundle
249
is the product bundle
(GA,)
with continuity structure
A
{fG
feC(G,A)}
A
The algebraic structure in
(x,a)-(y b)
(x,a)*
for
x,y
G
in
and
a
x,y
(x
-i
I
as follows:
(xy,aSx(b)Vx,y)
((x,a) (y,b))
I ((x,a))
x
(G,A)
for
(T,W)
unitary factor systems
P
and
V*
X
-i
S
X
X
-i
(a*))
A
in
(S,V)
Twisting pairs
P
is defined by operator fields
are in one-to-one correspondence with
[5,pp.317-318].
(G,A)
for
This correspondence is
given by
T
S
x
-I
x
S-Iv
xy x,y
W
x,y
S
Thus,
are strongly continuous (resp. measurable) if and only if
V
and
x,ygG
T,W
Moreover:
are.
LEMMA 12.3.
T
mappings
The mappings
W
and
S
and
V
are ultra-measurable if and only if the
are.
PROPOSITION 12.4.
(i)
The field
P(resp.l)
is a measurable product
(resp.
involution) field.
P
(2)
If
S
and
V
are ultra-measurable, then so are
(3)
If
S
and
V
are strongly continuous, then so are
The resulting Fell bundle
and will be denoted by
THEOREM 12.5.
(G,A;T,W)
(A,;P,I)
P
and
I (relative to
will be called a Busby-Smith bundle
(G,A;S,V)
For corresponding pairs
(T,W)
and
(S,V)
is measurably isomorphic to the Busby-Smith bundle
PROPOSITION 12.6.
I
and
The following are equivalent for
(i)
T
is strongly continuous at
x
(2)
S
is strongly continuous at
x
o
o
x
o
the Leptin bundle
(G,A;S,V)
in
G
250
I.E. SCHOCHETMAN
(3)
is continuous at
i.e.
PROOF.
o
(I) is equivalent to (2)
(2) since the mapping
and only if
x
h
S (h(x))
x
x
-i
x
T
x
(3) is equivalent to
xgG
-l’
x
for
o
h
in
C(G,A)
if
o
The Fell bundles
phically isomorphic if
S
is continuous at
x
is continuous at
COROLLARY 12.7.
since
(G,A;T,W)
T (equivalently S)
(G,A;S,V)
and
are homomor-
(Note that the
is strongly continuous.
bundles don’t have to be continuous Fell bundles in this case.)
REMARK 12.8.
Clearly, 11.5 through 11.9 are valid also for Busby-Smith bundles.
LI(G,A;S,V)
space LI(G,A,)
The twisted group algebra
Smith bundle is the Banach
corresponding to the underlying Busbywith product and involution (for right
Haar measure) given by
f’g(x)
fGf(xy-l)s -I(g(y))V(xy-I’y)d(Y)
and
(x)-Iv(x,x-l)*Sx(f(x-i ))*
f*(x)
for
f
algebra
and
L
g
in
LI(G,A,)
I(G,A;T,W)
xgG
LI(G,A;S,V)
The algebra
is isomorphic to the
(9.13) by the mapping
f
[x
S
-l(f(x))]
LI(A,;P,I)
LI(G,A;S,V) will have
fEL
l(G,A;S,V)
Moreover, by 9.11,
and hence, to our algebra
by the same mapping.
11.6, and 12.5,
an approximate identity if
S
is strongly
continuous.
EXAMPLE 12.9 [5,p.330].
with
A
Let
[0,i] under addition modulo i.
V,W:GxG
U(A)
be
LI()
and
Sx Tx
Define
by
V
a
x,y
W
a
x,y
a
a
V
x,y
W
a
x,y
a
x+y<l
I
x+y>_l
where
al(n)
l+n
nE
G
E
A
the circle group identified
xgG
and
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
V
Then
x,y
aV
x,y
a
aeA
(G,A)
a unitary factor system) for
(x,y)
any point
GxG
in
(S,V)
and
The mapping
H/N
N
ll.2).
is ultra-measurable
LI(G,A;S,V)
is isometrically isomor-
H
is a closed normal subgroup of the locally compact group
(right cosets) and
H- G
q
be right Haar measures for
on
is not strongly continuous at
EXTENSIONS.
Suppose
G
is
LlR)
phic to the group algebra
GROUP
V
However, V
1
In the next section (13.2), we will see that
13.
(T,W)
is a twisting pair (equivalently,
x+y
where
251
N,H
the quotient mapping.
0,
be the right Haar measure
Let
respectively.
Let
with
defined [15] by
G
IHf(X)dg(x)
for
f
H,G,N
in
Cc(G)
Of course,
On p.77 of
[i],
fG /N f(tx)dO(t)d(q(t))
Let
6H G 6N be the respective modular
6HIN 6N
Fell shows how to construct ingredients
homogeneous Banach *-algebraic bundle
bundle.
Specifically,
LI(N,D)
A
N
called the
(A,N,H,T)
for a
(H N) -group extension
U(A) and
is identified with a subgroup of
6(x)a(x-ltx)
(a) (t)
x
(A,n;-,*)
functions for
where
d0(xtx-l)/dO(t)
6(x)
Fell then verifies directly
xeH
aeA
teN
that the cross-section algebra
LI(A,;’, *)
is
LI(H,))
isometrically isomorphic to the group algebra
In this section, we will accomplish these two tasks in very different ways.
will show that the mapping
space representations
arises naturally from a certain induction of Banach
([16,3]
[13,4]).
and
it will then be automatic that the previous
As a consequence of this approach,
Ll-spaces
I
Transferring the *-algebra structure from L (H,)
LI(A,;P,I))
Let
We
to
are isometrically isomorphic.
L I (A,;’, *)
(i.e. to
then completes the picture.
G,H,N
be as above with
identity representation of
E
E
{e}
and
iE(e)
i
Then
and the regular representation of
i
E
E
on
is both the
I
L (E)
The
252
I.E. SCHOCHETMAN
Ll-induction
R
N
tation
N
of
H
up to
[16] of
N
i
[16].
is also
E
from
I
L (N,0)
on
N
up to
[16,4.5].
FI(H,RN)
I
Ll-induction
I
F (H,RN)
and
I
L (H,)
on
R
N
of
from
[16,6.4].
The
of
i
R
H
E
from
E
directly up to
[16,4.5]
T
L
I(H,)
F
denotes the isomorphism, then it is given
(Tf)(x)(t)
In particular, if
Therefore, the re-
must be isometrically isomorphic Banach
Since
fEC (H)
I(H,RN)
[16,p.72]
f(tx)
xEH
tEN
fEL
l(H,)
and the images of such
Tf
then so is
by
f
by
T
FI(H,)
are dense in
LI(H,)
has a *-algebra structure, it may be transferred to
FI(H,RN)
by requiring:
(Tf)’(Tg)
T(f-g)
(Tf)*
In the case of involution, for
If, for
y
in
H
and
T
Y
a
f
f,gELI(H,)
T(f*)
in
C (H)
we have:
H(X)-l[rf(x-l)]*(x-l(.)x)
(rf)*(x)
LI(N,o)
in
we define
(y-l)a(y-lsy)
a(s)
xEH
sEN
then, combining the above, we obtain [15]
(Tf)*(x)
G (q (x) -l) T x (Tf(x-l) *)
In the case of multiplication, for
(Tf)’(Tg)(x)
where
H
If
spaces.
T
H
of
(modulo equivalence) the representation
L (H,)
Ll-induction
Consequently,
(by this induction) is given as in Section 3
R
H
of
On the other hand, the
presentation spaces
via
is then the (right) regular represen-
is the regular representation
representation space
of
E
*N
f,g
C (H)
in
fGTf(y-I)*NT Y-l(Tg(yx))d(q(Y))
Next observe that the representation
t(a)
we have:
c
a(’t)
LI(N,0)
A
denotes the usual convolution in
R
xgH
R
N
aEL
of
N
I(N,0)
on
A
tEN
which is given by
253
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
is also given by
R (a)
t
a’t
-I
U(LI(N,0))
where the right side is the action of the multiplier in
to
t
-i
on the function
[13]),
of
(i)
Hence, by the beginning of Section i0 (or Section 4
a
there exist:
{A
a field
eG} of Banach spaces over G
each isomorphic to
e I(N,0)
A
(2)
corresponding
A
a continuity structure
A
of the
H
in
as in 10.6.
A-topology is precisely
with the
A
Moreover, the disjoint union
HXA/~
(as in Section i0)
with the quotient topology.
PROPOSITION 13.1.
homomorphism of
(i)
T
(ii)
T
t
(t)
PROOF.
into
-I
tat
(a)
x
H
xtx
defined above is a strongly continuous
The mapping
AUtl(A)
aEA
teN
-I
satisfying:
xEH
tEN
Compare with example 3 on p. 77 of [i].
Left to the reader.
(H,N,A,T)
As a consequence of the above,
is the set of ingredients of a
(continuously) homogeneous Banach *-algebraic bundle
P
product and involution fields
obtain the generalized group algebra
{A
EG}
[13],
it follows that
and
Banach spaces.
A above are
If
S
f
in
FI(H,RN
FI(H,)
V
ST
are given as in Section i0.
LI(A,;P,I)
(x,f(x))
In particular, for
(Vf)*()
)
in
V(f*)()
C (H)
c
as
xEEG
may in turn be transferred to
f, g
LI(A,)
[13,4.12]
continuous with compact support module
LI(H
We thus
Since the constructions of
is isometrically isomorphic to
is the isomorphism, then
-algebra structure of
The underlying
just special cases of those given in Section 4 of
Sf()
for
I
and
(A,n; ,*)
N
Thus, the
LI(A ,)
by
we obtain:
S(T(f*))()
(x,Tf*(x))
xE
and
Vf’Vg
V(f*Hg)
S(T(f*HG))
We may then verify that the two *-algebra structures on
LI(A,)
are
254
I.E. SCHOCHETMAN
LI(H,)
{Vf
Note that
isomorphic.
LI(A,;P,I)
and
EXAMPLE 13.2.
feC
G
H
T
X
The quadruple
Consequently,
l /G /0
7/.
(a)
aEA
xgH
a
since
[0,i).
H
Then
A
LI(.
is abelian and hence,
N
.
6 is trivial.
forms the ingredients for a homogeneous Banach *-algebraic
(A,N,H,I)
as above.
Observe also that
H
O:G
exists a measurable cross-section
[0,i)
LI(A,)
Consider the locally compact group extension
(A,;-,*)
bundle
is dense in
is the circle group identified with
and
]R
(H)}
are isometrically isomorphic Banach *-algebras.
0
where
c
is locally compact and there
G
given by the canonical injection of
It then follows from the results of Section ii that there also
into
(G,A;T,W)
exists an isomorphic Leptin bundle
LI(G,A;T,W)
such that the corresponding algebras
LI(A,V;P,I) are (isometrically) isomorphic.
LI(H) Finally, we leave it to the reader to
and
isomorphic to
In turn, these are
verify that the
Leptin bundle for this example is precisely the one described in 12.9.
APPENDIX
In this Appendix, we state those properties of multi-linear mappings which will
All proofs are left to the reader.
be used in Section 3.
Let
AI,...,An,
B
:A I x...x
) (a
sup
PROPOSITION AI.
(i)
ll(a I
a
n
Suppose
n) <-
sup{If (a I
(ii)
(iii)
a
I
(a I
An
B
Define
is an n-linear mapping.
+
Suppose
be Banach spaces.
an)-@(b I
aigA i
$
l__n}
I1<
aieAi,llaill
n
oo.
Then:
llanll
an)
+llblll lla2-b211 lla311
0
is bounded, i.e.
lla III
b
ai#
II-<
i
l-<i<_n}
llal-blll" lla211--- llanll +
llanll + + llblll llbn_lll llan-bnll]
GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES
for
al,bleAl;...;an
LEMMA A2.
Let
m
255
b eA
n n
be a positive integer,
i
{a k
i _< i
k
_<
k
c
m}
and
{ek
i
i
k
_<
m}
k
i,
n
Then
all
THEOREM A3.
[
i
ennan
n
The space
i
i
i
I
Homn(Al,...,An;B)
an
(al
an
n
of n-linear mappings
as above
which are bounded is a Banach space relative to the norm
ACKNOWLEDGEMENT.
The author was partially supported by an Oakland University
Research Fellowship.
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