I nternat. J. Math. & Math. Sci.
Vol. 4 No. 4 (1981) 625-640
625
INTEGRAL OPERATORS IN THE THEORY OF
INDUCED BANACH REPRESENTATIONS II.
THE BUNDLE APPROACH
I.E. SCHOCHETMAN
Mathematics Department, Oakland University
Rochester, Michigan 48603
(Received March 4, 1981)
ABSTRACT.
Let G be a locally compact group, H a closed subgroup and L a Banach rep-
Suppose U is a Banach representation of G which is induced by L.
resentation of H.
Here, we continue our program of showing that certain operators of the integrated
form of U can be written as integral operators with continuous kernels.
we show that:
(i)
the representation space of a Banach bundle;
(2)
Specifically,
the above oper-
ators become integral operators on this space with kernels which are continuous cross-
sections of an associated kernel bundle.
Locally compact group, anach reprentation, rnduced repre-
KEY W@RDS AND PHRASES.
senton, integrated form, integral operator, vector field, cross-section, continuity structure, anach unle, quotient topology, kernel unle.
190 MATHEMATIC SUBJECT CLASIFICATION CODE.
22D12, 28B99, 6B99 5415.
Primary: 22D20, 22D30, 47G05, 54A70
Secondary:
i.
INTRODUCTION.
Let G be a locally compact group with right Haar measure dx, H a closed subgroup
of G with right Haar measure dt, and :Cr+X, the canonical projection onto the right
coset space X
Banach space E.
space F
U
G/H.
Suppose L is a (strongly continuous) representation of H on the
Suppose also that U is a representation of G on a certain Banach
which is induced by L in the sense of [i, sec. 3].
is the representation of
LI(G)
The integrated form of U
determined by the bounded operators U() on FU, where
U()
(x)U(x)dx
|
G
I. E. SCHOCHETMAN
626
for
a continuous function on
G
e C (G)
with compact support, i.e.
It is
well-known [2] that these operators can be written as integral operators with continuous kernels in the following sense:
in a certain dense subspace of
f
for
F
U
we have
Ix I
U()f(x)
d((y))
where
of
G
G
(x,y)f(y)d((y))
X
is a quasi-invariant measure on
Horn(E)
into t’he bounded operators
is determined by the fact that for each
constant on cosets.
Thus, the kernel
I
in
for
I
and
G
is a continuous mapping
The existence of the integral
E
I (x,.)f(-)
the mapping
U()
Moreover, the mapping
X
the integration is over
x
on
x e G
is defined on
I
G
G
is
while
is not constant on cosets
in general.
Our primary objective in [2] was to represent the operators
U()
as integral
operators with continuous kernels in a consistent fashion, i.e. where the kernels and
There are two canonical choices for
integration are defined over the same space.
this space
namely
G
and
In Chapter II of [2] we acomplished our objective
X
over each of these spaces.
In section 3 of [2], we constructed a representation V
function space
V()
operator
F
V
certain dense subspace of
F
V
G
on a Banach
In particular, each
U
which was isometrically equivalent to
was written in the following form:
of
for each continuous
g
in a
we have
J-
V()g(x)
J (x,y)g(y)dy
G
x
G
J
where
is a continuous mapping from
G
G
into
Hom(E)
Although this result
is satisfactory from the consistency viewpoint, there is a significant shortcoming.
The space
F
V
LP-space.
is a proper closed subspace of a vector-valued
Thus, many
of the important existing results for integral operators cannot be used with this
model of the integrated form of
U
In section 4 of [2], we next turned our attention to the quotient space
constructed a representation
{E :
W
of
G
on a continuous sum
X} which was also isometrically equivalent
to
U
F
W
X
of Banach spaces
Each operator
W()
We
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
for each "continuous" vector field
was written as follows:
subspace of
F
W
h
627
in a certain dense
we have
W()h()
f K(,n)h(n)d]
g
X
X
where
K
is a "continuous" field of bounded operators in
K(,) Hom(EA,E)T!
LP-theory
expressly for this purpose.)
(The latter was developed by the author
Since the space
does not have the shortcoming that
V
has.
FW
is a full
LP-space,
W
this model of
In fact, it proved to be quite useful
U
in studying certain compactness properties of the integrated form of
However,
and
for vector fields [3,4], as well as the theory of kernels and integral
operators for such Lebesque spaces [4].
U
,E)
This model requires the knowledge of continuity structures
g
and
Hom(E
[2, Ch. IV].
has two minor shortcomings which are more akin to mathematical utility
and aesthetics than to mathematical substance.
First, a continuity structure (and
its implications) is quite complicated and is a less intuitive object to work with
than is the more fundamental and familiar notion of topological continuity.
although the kernels
K
do belong to a continuity structure in
Hom(E
structure partially loses a desirable property in the transition from
G
Second,
,E)
to
this
X
(See pp. 25-29 of [2] for a rigorous explanation.)
[2],
Since the writing of
it has been discovered
[5] that the theory of
uity structures is equivalent to the theory of Banach bundles [6].
contin-
Moreover, in the
bundle context, the appropriate mappings are cross-sections, so that continuity is
simply topological continuity.
(i)
The representation
(2)
The kernels
K
W
These facts suggest:
can be reconstructed in the setting of Banach bundles.
should be continuous cross-sections for a suitable bundle.
The main objectives of this paper are to show exactly how to accomplish (i) and (2).
Section 2 is devoted to recalling the necessary preliminaries.
In sections 3
and 4, we construct the Banach bundles corresponding to the continuity structures in
NE
K
and
NHom(E
,E)
respectively.
Finally, in section 5, we show that the kernels
are continuous cross-sections for the bundle of section 4.
628
I. E. SCHOCHETMAN
PRELIMINARIES.
2
[2].
The following is a brief summary of sections i and 2 of
Let
A
and
G
A
be the modular functions for
H
AH/(AGIH)
quotient function
G
p(tx)
satisfying
X
L
The representation
(p,q)
exists a pair
Let
6
6(t)p(x)
quasi-invariant measure on
H
E
on
is
d((x))
be the
p
G"
"Banach inducible up to
<_ b6(t)
if there
and
0 < q <_
i <_ p <_
such that
into the positive
d
Also let
I/p
t e H
linear space of all continuous mappings
H
i.e.
f:G
E
f(tx)
6(t)i/qL(t)f(x)
denote the
has compact
is (L,q)-homogeneous,
f
(ii)
is compact.
f
(i)
such that:
(supp(f))
C (G,L)
q
Let
on a certain Banach function space as follows.
support modulo
H
Given such a pair, we can construct [i] an isometric representa-
b > i
G
tion of
G
corresponding to
of
The
respectively.
be a continuous, non-negative function
p
H,x
t
8(t)i/qllL(t)l
for some
H
and
is a continuous homomorphism of
reals which we shall denote by
on
G
ioeo
LP-norm
For a suitable
f fx
by an element
F
tion
mapping
of
U
U
x
C (G L)
q
of
G
x
g
G
[i, sec. 3], we find that right translation
C (G,L)
q
on
t g H
Hence, its extension to the comple-
is an isometry.
U(x)
is also an isometry which we denote by
is then an isometric representation of
induced Banach representation corresponding to
G
(L,p,q)
on
F
The resulting
which we call the
U
We refer the reader to
[i] for a thorough development of such representations.
U(), g C (G)
In this setting, the operators
(x-lty)L(
8(t)
p(y)
may be written as follows:
dt f(y)d((y))
X
for
f
in
C (G L)
q
(where
i/q- i/p)
q:p
Thus, the mapping
in the introduction is given by
I (x,y)
AG (x)-I P(Y)
-i
f
H
8(t)
q:
I (x-lty
)L(t)dt
x,y
g
G
I
referred to
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
Before concluding this section, observe that if we let
H
a bounded representation of
E
on
with bound
629
6q:PL
M
Hence, we may renorm
b
M
then
E
is
by
defining
[Ivll M
sup{[IM(t)v[[:t s H}
We thus obtain a Banach space
which is equivalent to
llv[l
Horn(E)
The Banach spaces
v e E
<_
[IVllM <_ bl[vl]
Hom()
and
E
since
v s E
are also equivalent with
I[T[I <_ IIT[I M <_ bl[Tll
T s Horn(E)
The following result will be useful in what follows:
LEMMA 2.1.
PROOF.
(t,v)
The mapping
L(t)v
of
This follows from the fact that
L
H
E
into
E
is continuous.
is strongly continuous and locally
bounded [1,3.3].
It will also be convenient to fix in advance a compact neighborhood
identity
3.
e
in
G
Z
of the
for use later on.
THE VECTOR FIELD BUNDLE
In order to construct the bundle version of
W
we begin as in section 4 of
[2]
Consider the space
x,y s G
y
tx
and
and
v,w s E
w
:
let
X
then
L(t)v
quotient topology) by
G x E
(x,v)
(y,w)
E
and let
o: G x E- E
G x E-+
-l(o(A
x
B))
t
in
H
if
such that
Denote the resulting space of equivalence classes (with
then have the following composition:
PROOF.
if there exists
-
be the canonical projection.
be the (well-defined) projection given by
LEMMA 3.1.
defined as follows:
with equivalence relation
(o(x,v))
(x)
X
The mapping
o
is continuous and open.
The openness of
o
follows from the fact that the saturation
of a basic open subset
A x B
of
G x E
is of the form
Also
We
630
I. E. SCHOCHETMAN
U {(tA)
LEMMA 3.2.
The mapping
is continuous and open.
follow from the continuity and open-
The continuity and openness of
ness of
T e H}
G x E
which is open in
PROOF.
(L(t)B)
x
and
PROPOSITION 3.3.
PROOF.
Let
o(x,v)
8’
is Hausdorff.
E
be a net in
{8i }
and
E
The space
o(x’,v’)
converging to
Since
o(xi,v i)
1,2,
i
8i
(xi,v i) (x,v)
net {(xi,v ’)}
i
and
assume there exists a corresponding
o(x.,v[.)
t
i
H
g
8
x’i
such that
element of
in
(x.,v[.)-
and
i
G
H
t.x.
since the net
(x’ ,v’)
and
v’i
(xi,vi) }
L(ti)v i
E
G
Consequently,
x’x
converges to
8’
8
and
E
-i
()
is uniquely of the form
equivalent to
E
(E,X,)
there exists
x’x -i
t
H
and
v’
is an
is closed
L(t)v
X
For fixed
(x,v)
v e E
x
"
in
E
Thus,
-i
()"
each element of
becomes a Banach space
under the following (well-defined) operations:
and
=(x,v)
llo(x,v)ll
For each
x e G
llo(x,v)ll-PROPOSITION 3.5.
As on p. 12 of [2],
into a Banach bundle.
(x,v) + (x,w)
LEMMA 3.4.
such
is Hausdorff.
The next step is to make
define
E
such that
o(x.,v i)’
-i
G
Similarly, we may
Moreover, by I.i and the triangle inequality, we see that
Thus,
[6]).
in
in
E
G
in
o(x i,vi
Since
I}
x’.x
1 1
{t i
where
in
is open, passing to a subnet if neces-
o
sary, we may assume there exists a corresponding net
that
8’
and
8
The bundle
(x,v+w)
(x,=v)
p(x)q:Pllvll M
v,w
E
v e E
sup{p(y)q:Pllwll
(,X,)
(y,w) (x,v)}
is a Banach bundle
(as in section i of
E
631
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
CS(X,)
A
NE
in
given by
C(A)
we then have the space
was denoted by
q
As in sections 1 and 5 of [4],
Aq.)
C (A)
HE
T
S(X,)
C
which in turn is in
Also,
Sf
f
via the above mapping
A [2, 4.15], so that
S(Hq(G,L)
C(A)
The continuity structure
A
A-topology) on
yields a topology (the
continuous cross-sections relative to the
space
N C(G,E))
COl)
A-topology [6, Prop. i. 6].
is also the space of quotient-continuous cross-sections.
THEOREM 3.6.
The quotient and
CS (X,E) --C (A)
C
C
Furthermore
if
is the same as the space
LP-completion
CS (X )
of
However
C
F
representation space
U
of
is the measure
LP((,X,)
LP(A,)
where
h
o(x,v)
H (G,L)
q
C(G,E)
be an element of
d
X
on
then
{19 s E:’T(E)) e A
A
LP(A,)
[6, sec. 2], which is the
)
W
will be
lle
Sh,’.’(e))ll
is an open subset of
W(h,A,e)
and
F
W
of
LP((E,X,)
Hence,
W
)
is of the form
A basic A-open subset of
W(h,A,e)
[7, sec. i]
is isometrically isomorphic to the
the representation space of the bundle version of
PROOF OF 3.6.
Actually, the
C(A)
CS(X,E)
and is the representation space
U
are the
A-topologies are the same.
Before proving this theorem, observe that we then have
[4, sec. 6]
making
C(A)
a Banach bundle having the property that the elements of
(E,X,)
8
(L,q)-homogeneous
is the space of
Note that
H (G,L)
bijective correspondence with
G
x
A-continuous vector fields and the subspace
of
of compactly supported such fields.
C(A)
H (G L)
q
where
(In [2], A
G
functions on
where
o(x,o(x)-I/qf(x))
Sf(x)
H (G L) N C(G E)
q
in
the continuous cross-sections with
C
{Sf}
A
Sf((x))
f
E
into
Analogously, recall [2, sec. 4] that there is a continuity
compact support.
for
CS (X,)
those that are continuous and
structure
X
denote the linear space of cross-sections from
S(X,E)
Let
so that
(e)
X
and
<
e}
e > 0
(x) s A
and
Let
I. E. SCHOCHETMAN
632
llo(x,v)
Sh(ie))ll
lle
p(x)-i/qh(x))ll
llo(x,v
p(x)-i/qh(x)llM
p(x)q:Pllv
< g
The function
y
N
G
x
y
of
g
N
in
p(y)-i/qh(y)ll M
_+p(y)q:Pllv
is continuous on
is open in
G
which is a neighborhood of
x
where this function is less than
x
so that
g
N N Zx 0
-I(A)
this set contains an open neighborhood
Hence, the set
G
B
of
e/2
max{P(y) q:p
C
{w s E
in
G
Thus,
For
x
a
Also,
y
Zx}
g
define
E
is an open subset of
C
Then
in
G
and
C
x
B
v
C
r > 0
There exists
C
is contained in
e
Let
E
open in
Choose
{h(y)
h
g
Cq(G,L)}
[i, 3.8] and h- p-1/qh
C)
o(B
is a basic
g
C)
o(B
E
o(x,v)
Then
min{p(y)
E
is a bijection of
q:p
B
for
open
for some
vll< r/a}
llw
E
{w
such that the ball
is dense in
W(h,A,g)
which can be shown to be in
sufficiently large so that
c
m
Since
e/2ab}
be a basic quotient-open subset of
C)
o(B
Conversely, let
e
<
Thus,
v
containing
o(x,v)
quotient-open neighborhood of
llv- wll
c > 2a/m
where
y e Zx} > 0
(and hence in
C (G,L)
q
)
y
for each
h
there exists
in
G
C (G,L)
q
in
such that
llv
By the
continuity
N c Zx
that
V
of
h
there exists an open neighborhood
A
o(x,v)
(N)
0
in
are the same.
N
of
x
in
G
such
and
llv
Let
p(x)-i/qh(x)llM < r/ac
and
E
g
r/c
Then
p(y)-i/qh(y)llM < r/ac
W(h,A,e)
which is contained in
o(B
is a basic
C)
y
g
N
A-open neighborhood of
Therefore, the two topologies
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
4.
633
THE KERNEL FIELD BUNDLE
Our next objective is to construct another Banach bundle for which the kernel
s C (G)
c
K
fields
Let
T’
L(r)TL(s)
X
X
H
Let
G
G
LEMMA 4.1.
PROOF.
((A
H
Hom(E)
C))
x’
such that
rx
y’
sy
and
the canonical projection.
Also, let
((x,y,T))
((x),(y))
We then have the following composition:
The mapping
H
Horn(E)
G
X
X
is continuous and open.
The openness of
B
defined by:
denote the space of equivalence classes (with quotient
G
-i
H
in
Hom(E)
G
be the (well-defined) projection given by
T s Horn(E)
x,y s G
r,s
if there exist
-i
topology) and
:H
G
be the equivalence relation on
(x’ ,y’ ,T’)
(x,y,T)
are continuous cross-sections.
follows from the fact that the saturation
A
of a basic open subset
B
C
of
Horn(E)
G
G
is of the
f orm
U {rA
which is open in
LEMMA 4.2.
subset
n
,
A
r,s s H}
G x G x Horn(E)
:-+ X
The mapping
To see that
PROOF.
L(r)CL(s) -i
sB
of
X
is continuous and open.
is open, observe that
(A)
(
)(
The lemma then follows from the facts that
-i
(A))
for any
is continuous and
is open.
PROPOSITION 4.3.
PROOF.
The space
H
is Hausdorff.
Similar to that of 3.3.
The next step is to make
define
H
-i(,)
uniquely of the form
(,X
For fixed
a(x,y,T)
for
X,)
x
in
T
in
into a Banach bundle.
y in
Horn(E)
For
,
each element of
Define:
sX
H
is
I. E. SCHOCHETNAN
534
a(x,y,T + T’)
a(x,y,T) + a(x,y,T’)
a(x,y,cT)
ca(x,y,T)
and
Ila(x,y,T)l]--
(p(x)/p(y))q:PllTI] M
where the norm is well-defined
T,T’
e Horn(E)
Under these operations
c e
H,D
is a Banach space.
In fact:
THEOREM 4.4.
PROOF
Fix
The space
a(x,y,T)
H, is isometrically isomorphic to Hom(E,E)
in
H,D and define T’ :ED- E by
T’ (o(y,v))
Then
T’
g(x,Tv)
lIT’ll
is well-defined, linear and
v s E
lla(x,y,T)ll as
we shall next verify
We have:
lIT’If
where
ll(y,v)ll
sup{llT’(a(y,v))ll
p(y)q:Pllvll
M
v e E
Thus,
Ila(y,v)ll
ll(y,v)ll
i
i}
llvll M
if and only if
p(y)P:q-
Also,
[[T’ ((y,v))[[
[[g(x,Tv)[[
p(x) q:p[ITvl] M
(p(x)/9(y))q:P][T(p(y)q:Pv)[[M.
Therefore,
sup{llT’(o(y,v))ll
ilT’ll
v s E
llVlIM p(y)P:q}
sup{(p(x)/p(y))q:PllT(p(y)q:Pv)ll M v g
sup{(9(x)/p(y))q:PllTwllM w m IIwll M
llvll M
E
O(y)P:q}
i}
(p(x)/p(y))q:PllrlIM
II(x,y,T)
Hence,
T’
Hom(E,E)
s
Hom(E_,E)T]
and we have a mapping
a(x,y,T) T’
which is clearly a linear isometry.
this mapping is onto.
of
H
E
into
We will be done once we show that
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
Recall that
E,
Thus, the same is true of
operator
T’
s X
are all equivalent Banach spaces.
and the Hom(E
Hom()
Horn(E)
T s Horn(E)
particular, if
E
and the
c(x,y,T)
,
,E)
Hom(E
then the corresponding element of
which is the image of
635
s X
In
,E)
is the
x s
under our identification,
YS
{Hom(E
Consequently, modulo the isomorphisms, the field
Banach spaces is the same as the field
REMARK 4.5.
{,
,
,E):,
X}
of
x}
Recall that in 3.4 we verified that
sup{o(y)q:Pllwll
llo(x,v)ll
(y,w)}
(x,v)
The analogous question here is: does
lla(x,y,T)ll
sup{(9(x’)/p(y’))q:PllT’ll
(x’,y’,T’)}
(x,y,T)
?
The answer depends on the question: does
IITIIM sup{llM(r)TM(s)-lll
r,s
g
?
T s Horn(E)
H}
We believe the answer to both questions is yes; however, we have been able to only
partially verify the latter.
PROPOSITION 4.6.
(,X
The bundle
X,)
is a Banach bundle.
X,H)
S(X
As in section 3, we obtain the spaces
sections and continuous cross-sections respectively.
CS(X
and
The space
X,H)
CS(X
of cross-
X,)
is then
the space of continuous kernels.
The vector field analogue of this space requires the existence of a continuity
structure in
as follows.
B:G
Hom(E
Let
G- Hom(E)
This was essentially accomplished in section
,E)
G,Hom(E))
(G
satisfying
B(rx,sy)
For each such
B
denote the linear space of mappings
define
L(r)B(x,y)L(s) -i
RB(,)
RB(,)((y,v))
E
E
,
(x,B(x,y)v)
x,y s G
r,s s H
s X
by
x s ,Y s
v s E
4 of [2]
I. E. SCHOCHETMAN
636
Then
RB(,)
Hom(%E)
e
B(x,y)’
RB(,)
NHom(E ,E
and we thus have a mapping
which is the element corresponding to
under the bijection between
LEMMA 4.7.
g
G,Hom(E)) NHom(E ,E
R:HL(G
Note that
RB
so that
R
The mapping
(4. 4)
,E)
Hom(E
and
(x,y,B(x,y))
is a bijection.
The onto property is proved using a (possibly non-measurable) cross-
PROOF.
section.
Now consider the linear space
L(G x
C
RB:B e
The set
in
{RB(,):RB
the
CL(G
G,Hom(E))
was observed to be a, precontinuity structure
,
Hom(E ,E
If this is not the case, then
e X
the latter is
is not the same as the quotient topology
ftopology on
Hausdorff while the former is not.
Moreover, our proof in 3.6 (which we would like
here) required this density property.
to duplicate
x
This means that we were unable to verify that
is dense in
e }
G,Hom(E)) N C(G
G,Hom(E)) }
2, 4.32].
Hom(E,E)
(G x
G,Hom(E))
This is not a significant
problem since we can bypass it by simply restricting our attention to the appropriate
portion of
Before doing so, the following result shows to what extent the
H
density property does hold and (more importantly) suggests why it may not hold in
general.
THEOREM 4.8.
Hom(Erl, E)
The subspace
(After section 3 of [i]).
Hem(E)
element of
G
is strongly dense in
x
G
Let
,
C (G)
and
T
For each
x,y
in
be elements of
c
Define
F(x,y)
F:G
RB e }
,r] e X
PROOF.
Then
{RB(,)
Horn(E)
consider the mapping
@(x)(y)T
x,y
G
is continuous and has compact support.
an
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
L(r)-(rx,sy)L(s)
(r,s)
of
H
H
Horn(E)
into
Hence, for
v
E
in
H
H
F
element
E
into
L
Since
is strongly continuous, so is this mapping.
the mapping
[L(r)-iF(rx,sy)L(s)] (v)
(r,s)
of
637
is continuous and has compact support.
L(x,y) (v)
E
of
Thus, it defines an
where
ff
FL(x,y)(v)
[L(r)-iF(rx,sy)L(s)] (v)drds
HxH
L
F (x ,y)
The mapping
FL (G
Hom(E))
G
actually have that
Horn(E)
is easily seen to be an element of
F
to verify only that
L
Moreover, since
CL(G
G
Horn(E))
{FL(e,e)
F
e T}
e
Hence,
L
the strong continuity of
Horn(E)
By right translating
see that
{B(x,y)
B
CL(G
G
are uniformly continuous, we
and
Unfortunately, however, we are able
B
g
and
is strongly dense in
It appears from the previous proof that
L
Horn(E))}
G
x
by
and
Horn(E))}
CL(G
The theorem then follows from this fact by the definition of
structure since
Horn(E)
is strongly dense in
{B(e,e)
dense in
in fact,
RB
y
using
is strongly
respectively, we
Horn(E)
for
B
x,y e G
as above.
may not be a complete continuity
In any event, as we have
is not norm-continuous in general.
indicated above, this difficulty is easily circumvented.
LEMMA 4.9.
{B(x,y)
B
(x,y) e G
x G
Let
g
CL(G
’HomL(E
is a continuity structure if and only if
The space
G
Horn(E))}
is dense in
(independent of
G x G
Horn(E)
Horn(E)
denote the norm closure in
{B(rx,sy)
x,y)
Then
G
G
r,s
g
HomL(E
Horn(E)
H
B
for some (equivalently) all
of the span of the set
Horn(E)}
is a saturated subset of
i.e. it is a union of equivalence classes.
HL
(G
G
HomL(E))
Let
638
and
I. E. SCHOCHETMAN
L I
Then
HL
G x G x
(or equivalently, the quotient topology of
HomL(E))
and
a Banach bundle with
i(,) (HL),
HL N H,
m
Similarly, we let
HomL(E,E)
HomL(E)
sponding to
HL &
Consequently,
so that
,
THEOREM 4.10.
PROOF.
,
e
is
x
Hom(E,E)
corre-
BHomL(ED,E)
HL
-topology.
is also equipped with the
The quotient and
(L,X x X,L)
e X
(complete) continuity structure in
is a
HL
-i(,)
denote the closed subspace of
HomL(E,E) (HL),
It is clear that
H
is equipped with the relativized quotient topology of
-topologies on
HL
are the same.
This is proved in the same way as 3.6 now that
has the density
property.
We have thus verified that the continuous cross-sections of the kernel bundle
(HL,X
x
X,L)
HomL(E,E)
correspond exactly to the
Of course, if
has the desired density property, then the
L-subscript may be dropped in 4.10, i.e.
5.
-continuous kernel fields in
C()
CS(X x X,H)
the ideal result.
THE CONTINUOUS KERNELS
We are now in a position to complete our project.
is a continuous mapping of
G x G
k
into
G x G
q)
Horn(E)
Recall (section 2) that
For
in
C (G)
c
I
define
G x G x Horn(E)
by
k (x,y)
Then
k
mapping of
/
(x,y,p(x)-I/qf)(y)l’ql
is continuous and constant on cosets.
X x X
into
G x G x Horn(E)
diagram commutes (modulo isomorphism)
(x,y))
x,y
G
Hence, it defines a continuous
Furthermore, it follows that the following
639
INTEGRAL OPERATORS IN INDUCED BANACH REPRESENTATIONS
k
G xG
Horn(E)
G
G
[2
p.20]
TT x TT
X xX
K
where
K (,)
W
E
E
is given by
K (,)((y,v))
,
for
s X
is a continuous cross-section.
-continuous kernel field.
see that
continuity for
K
(x,y)v)
K
for the operator
i.e.
K
e
RB
K
HOmL(E_,E)
-/q @
B
for
bundle space
by
x
U
)
where
The integrated form of
f
W(x)
W
acts on
X x X
HL
c
H
CS (X,E)
W
of
G
on the
by right translation
is given by
K(,)g()d
e X
X
K
The induced
is isometrically equivalent to a representation
W()g()
where
we
are the same.
LP((E,X,)
x e G
p/qT
so that the two notions of
In conclusion, let us briefly summarize what we have shown.
representation
W()
We had already seen [2, 4.33] that it is an
Finally, since
HL
X x X-
K
-i/qp(y)i/qo(x, I
Hence, the kernel
v s E
y s
x s
pCx)
is a continuous kernel for
g
CS (X,E)
c
W()
C c (G)
Of the four ways we have of realizing the integrated form of the induced
representation
(U,V, together with the vector field and bundle versions of
W)
the last one is the most satisfying and usable.
ACKNOWLEDGEMENT.
Research Fellowship.
The author was partially supported by an Oakland University
640
I. E. SCHOCHETMAN
REFERENCES
i.
FONTENOT, R.A. and SCHOCHETMAN, I.E., Induced representations of groups on
Banach spaces, Rocky Mtn. J. Math. 7(1977), 53-82.
2.
SCHOCHETMAN, I.E., Integral operators in the theory of induced Banach representations, Mem. Amer. Math. Soc., No. 207, 1978.
3.
GODEMENT, R. Sur la
68-124.
4.
SCHOCHETMAN, I.E., Kernels and integral operators for continuous sums of Banach
spaces, Mem. Amer. Math. Soc., No. 202, 1978.
5.
FELL, J.M.G., Induced representations and Banach*-algebraic bundles, Lect. Notes
Math., No. 582, 1977.
6.
FELL, J.M.G., An extension of Mackey’s method to Banach*-algebraic bundles, Mere.
Amer. Math. Soc., No. 90, 1969.
7.
SCHOCHETMAN, I.E., Generalized group algebras and their bundles, submitted.
thorie
des
representations
unitaires,
An____n. Math., 53(1951),