INTEGRAL OPERATORS ON THE SECTION SPACE OF A BANACH BUNDLE 449

Let π:E→X and ρ:F→X be bundles of Banach spaces, where X is a compact 
Hausdorff space, and let V be a Banach space. Let Γ(π) denote the space of sections of the 
bundle π. We obtain two representations of integral operators T:Γ(π)→V in terms of 
measures. The first generalizes a recent result of P. Saab, the second generalizes a theorem of 
Grothendieck. We also study integral operators T:Γ(π)→Γ(ρ) which are C(X)-linear.


INTRODUCTION.
A bounded operator T V W between Banach spaces is called an integral operator (in the sense of Grothendieck) if there is a bounded linear functional on the inductive tensor product VW* such that d(z(R)y*) y*(T(r)) for all re V and y*e W*.The integral operator T" V--.
W carries a norm IITII int _<IITII which is the norm of the functional r T e (V (R) W*)* where rT(r(R)y* y*(T(z)).In this paper, we study integral operators T r() W, where E X is a Banach bundle and the base space X is compact Hausdorff.In Section 3, we obtain two representations of such operators by means of measures.The first of these is described in Theorem 3, which generalizes a result of P. Saab  [1] for integral operators on C(X, V), the space of continuous functions from X to a Banach space V. (Of course, C(X, V) r(x), when :r is the trivial bundle whose fiber space is X x V with its product topology.)The second representation occurs as a corollary to Theorem 4, which itself generalizes a theorem of Grothendieck [2].Finally, in Section 4, we study integral operators T r() r() which are C(X)-linear, where and p are Banach bundles with base space X.
In this paper, the base space X of a Banach bundle :E-X is always assumed to be compact and Hausdorff.The reader is referred to Kitchen and Robbins [3] for details about the canonical bundle x E X of a Banach module M over C(X) and the Gelfand representation of M as a space of sections in Our two representations of integral operators on r(r) involve a compact Hausdorff space which we shall call the carder space of the Banach bundle E X.The space g was introduced by A. Seda in [4], and is described in Section 2.
Throughout the paper, we shall use B(V) to denote the closed unit ball of the Banach space V; that is, B(V) {r V: ,r, _< 1}.In particular, we shall use this notation in describing the carrier space g of a Banach bundle r E X.If l0 X, then we denote by El0 the fiber above that is, E, t-({p}).As a point set the space can be described as the disjoint union of the unit balls of the dual space of the fibers.Thus 13 B((Ep)*)x {p} {(,)'X, (E*,IIII -< There are two important maps associated with : the obvious coordinate projection -X, defined by ,((f,p)) p, and a map o" g B(r(r)*).The map assigns to the point (I, p)in the function FI, p r() C defined by FI,I,() =/((p)) for all r().It is easily verified that FI,p is a bounded linear functional on r() whose norm is the same as the norm of I.As a result, maps $ into B(v(,)*) as we already noted.We topologize $ by giving it the weakest topology which makes the maps : $ X and #: * B(r(,)*)) continuous, where B(r(,)*) is given its compact weak-, topology.It follows that a net {(fa, Pa)} in g converges to (f,p) if and only if lim Pa P and lim fa(tr(pa)) f(a(p)) for all t, in FOr).
When g is topologized in this way, the projection g X is not only continuous, but open, an,d for each p X, the topology which B((Ep)*) inherits from is its compact weak-, topology as a subset of (Ep)*. (See Seda [4] or Kitchen and Robbins [5] for a further description of the topology on g.)The space C(g) can be viewed as a C(X)-module in a natural way.Given a C(X) and g C(g), we define ag to be the pointwise product (a r)g; thus, for all (f, p) (a)(0',p)) a(p) ((I,)).
Finally, there is an important embedding F(,)-C(8).Given a I'(), I(a) is the function g C defined by ((I, )) FI, () I(())" From our definition of the topology on g, it is clear that Y is continuous.The most important facts concerning are contained in PROPOSITION 1.Let E X be a Banach bundle.The carrier space g is compact Hausdorff and the map I I'(=) C(*) is a C(X)-linear isometry.
PROOF.The compactness of follows rather easily from the compactness of X and of B(r(,)*)).(Seda [4] proves that , is locally compact under the assumption that X is locally compact.)Let {(fir, Ptr)} be a net in .Since X is compact, there exisis a subnet {p} of {p} such that ,B p in X.Now, consider the subnet {(I, 1of)} of {(la, pa)} As before, define 7 ,p_(a).Then B is a net in B(r(.)*) which converges weak-, to some g B(r(,)*).But noer tat for a r(,) and a C(X), we have () =,i 7() i () 7() a() (); i.e. (ag)() a(p) g(a).It follows from e.g.Gierz [6] that g for some I B((Ep)*) and it is clear that (I, pf) (I, p) in the topology on .
The first part of the statement is obvious from the definition of the module operation on C($).
The next result is known.We include its statement here for easy reference.PROPOSITION 2. Suppose that e:Z V and q Y W are bounded linear maps between Banach spaces.Then the tensor product map O (R) q Z (R) Y V (R) W extends uniquely to a bounded linear map e(R): ZY-VW such that 110 (R) _< O # If O and are isometries, then O (R) is also an isometry.
PROOF.The proof of the first assertion can be found in Diestel and Uhl [7, p. 228].
Suppose now that O and are isometries.Then O(R) (idz(R) )o(O(R)idw) where id Z (R) , Z Y Z W and O (R) id W Z W V W, and id Z and id W are the identity maps on Z and W, respectively.By Diestel and Uhl [7, p. 225], both idz(R) and O(R)id W are isometries, and hence so is O (R) .niX] 3. REPRESENTATION OF INTEGRAL OPERATORS BY MEASURES.
The first result of this section is a generalization of a result of P. Saab to operators on the section space of a Banach bundle.THEOREM 3. Let E X be a Banach bundle.Let V be a Banach space, and let T I'() V be a bounded linear map.Then T is an integral operator iff there exists a regular V**- valued Borel measure # on g, the carrier space for , such that for all r(=), where J V V** is the natural embedding.
PROOF.Suppose that there is a measure E M(S, V**) the space of regular, V**-valued Borel measures on , for which (,) holds.We will show that T is an integral operator.
Since the space M(g, V**) of all regular Borel V**-valued measures on which are of bounded variation is isometrically isomorphic to the space of integral operators from C(S) to V** ', it follows that if # M(, V**)is such that (,) holds, then the operator " C($) V** defined by T(g)= g g dp,forgC(g), is an integral operator.Thus, if r() C(g) denotes the natural embedding described earlier, that is {I(.)}(j',p)f(,(p)) for all (],p)G , then (,) states that (J T)() ( I)() for all G r(x), that is, the diagram commutes.r() T ) V J } V** c() Since I is integral, it follows that J T is integral, and hence that T is integral, by Diestel and Uhl [7, p. 233].Now, suppose conversely that T is an integral operator.Then there exists a bounded linear functional O on FOr) V* such that e(.(R) v*) v* (T(.)) for all e r() and v* V*.Since r()--.C(g) and id V*-V* are isometries, the map I (R) id: r()V* C($)V* is an isometry, whose range, W, is a closed subspace on which the linear functional O (I (R) id)-1 is bounded.By the Hahn-Banach theorem, this functional can be extended without increase in norm to a bounded linear functional on C($)V*.Then for all r() and v* V* (I(a Since belongs to the dual space of C()V* _C(,V*), there is a unique measure eM(, V**) such that for all g e C() and v* e V*, (g(R)v*) f$ g d(v*,t,), where (,,*,#) is the C-valued Borel measure such that (v*,)(B) {u(B)}(v*) J"(*){u(B)} (J"(*) )(B) for every Borel subset of g (where J" is the natural embedding of V* into V***).We must now show that (J T)(a) f , f(a(p)) du(f, p) for all r(r).Now, for all v* V* Thus, J T)(r) f I(a) dv f, /(a(p)) dg(L p).
In the special case when F(r) C(X, V) for a Banach space V, g is thc product space X x B(V*), and we recover Saab's result.The effect is to reduce the space on which the measures lives to be as small as possible, and the result may find its relevance in the fact (see e.g.Behrends [8]) that every Banach space is isometrically isomorphic to a space of sections over its center.
Our next result generalizes a result of Grothendieck [2] to Banach bundles; our proof is similar to that of Diestel and Uhl [7] for the special case of continuous bilinear maps from the iductive tensor V W of Banach spaces V and W to C. THEOREM 4. Let r E X be a Banach bundle, and let be its compact carrier space.
Let V be a Banach space, and give B(V*) its compact weak-, topology.A bilinear functional on FOr) x V defines a member of (F(t)V)* iff there exists a regular Borel measure v on I x B(V*) such that ('Y) ]*B(V*) f(a(p))y*(v)dv((/,p), y*) for all F(=) and y V.In this case the norm of as a member of (F(x) V)* is the variation lal(* B(V*)) of .
for all a e r0r), v V, (f, p) e ,, and v* B(V*).If we let W be the range of R, then W is a closed subspace of C(gxB(V*)) and R: r(t)V-, W is a bijective linear isometry.Thus, the bounded linear functional 0o R-1 on W can be extended without increase in norm to a bounded linear functional on C(* x B(V*)).Moreover,

R-1[]
By the Riesz representation theorem, there exists a Borel measure u on *xB(V*) such that (g)= f g du, for all g e C (g x B(V*)).Thus, for all a e I'(r) and v V, w have Moreover, the norm of as an element of (r(,)v)* is equal to Ildll=llttll the variation of on B(V*).Suppose conversely that there is a Borel measure u on g B(V*) such that (' Y) f B(V*) R( (R) u) du for all aer(r) and yeV.If we let be the bounded linear functional on C(gxB(V*)) which corresponds to u, then according to the above equation, ( (J(a (R)   for all ae I'(,r) and yeV, which is to say that oR is a bounded linear functional on I'(t)V which extends the bilinear functional .
COROLLARY 5. Let a E X and V be as above.If T r(:r) V is an integral operator.
then there exists a Borel measure p on g B(V**) such that y*(T(a)) / x B(V**) l(a(p)) it(y*) du((/, p), ,) for all a e r(,) and y* e V*.PROOF.Because T I'() V is an integral operator, the continuous bilinear functional on r()x V* defined by (a, y*) y*(T(a)) determines a bounded linear functional on F(,r)V*, so the theorem guarantees the existence of a Borel measure u on g x B(V**) with the required properties. 13130  4. THE SPACE OF INTEGRAL C(X)-MODULE OPERATOP AS A C(X)-MODULE.
We turn now to the study of the space of integral operators T between sections spaces of Banach bundles which also have the property that T is a C(X)-module homomorphism.That is if E X and p F X are bundles, then T r() r(p) will satisfy the equation (aT)(a) a. W(a) T(aa) for each a e C(X) and.e r(,).. LEMMA ft.Let = E X be a Banach bundle, and let V be a Banach space.The space M can be made into a C(X)-module which is C(X)-locally convex.If " G X is the canonica!bundle for M, then for each p e X, the stalk Gp -l(p) can be identified with Moreover, if evp r(t) -Ep is evaluation at p and id V V is the identity map, then evp (R) ld r(t)V EpV is a quotient map whose kernel is IpM={ am" me M, ae C(X), and a(p) 0}.
PROOF.Given a e C(X), we let ua: r()--, r() be pointwise multiplication by a.The tensor product of ga with id V V then yields a bounded linear operator ua(R)id F()$V F()V on M, and IIua(R)id[[ < IluallJJidJl Ilall.We make M into a C(X)-module by defining am (I.t a (R) id)(m) for all a e C(X) and m e M. Then ,,,, _< IluaidlI,,,, _< ,, ,, and we have a(a (R) I) (ta (R) id)( (R) I) Ua(a) (R) id(I) (aa) (R)   for all a e C(X), a e r(,r) and e V.It is then straightforward to complete the verification that in this way M becomes a C(X)-module.
We next apply Theorem 4.2 of Kitchen and Robbins [9].We can view V as the section space F(1V) where 1V denotes the Banach bundle whose base space is the singleton set {1} and whose one stalk is V.Then, according to the theorem cited, M r(r)V F(,r)F(1V) is isometrically isomorphic to rot IV).The bundle ,r 1V has X x as its base space, and, for p:X, the stalk above (p,1) is EpV.The isometric isomorphism O M F(,rIv) is characterized by the equation O(a (R) y) a (R) v for all a r(,) and v V, where (ry)(p,l) a(p)(R) (1) (In other words, we identify an element y V with the section in F(1V) whose (one and only) value is v.) Now, r(= 1V) is a Banach module over C(Xx {1}).By identifying Xx {1} with X in the obvious way, we can view r(t 1V) as a C(X)-module.
We will now show that our isometric isomorphism O:M r(= 1V) is C(X)-linear.It holds for all a C(X), a r(), and !
, o(,, (R) f).Hence, M and r(.g 1V) are isomorphic as C(X)-modules.Since r(.1V) is C(X)-locally convex (because of identification of C(X) with C(X x {1}) ), M is C(X)-locally convex.It follows that the canonical bundle G X is bundle isomorphic to V Thus, for t, X, the stalk Ot, can be identified with Et, V, in which case the Gelfand representation of an element m M is given by ) and f V, where evt, I'(x)-Et, is evaluation at t, and id V V is the identity operator.By lineaxity and continuity it follows that r(p) (evp (R) id)(m) for all m M.
Of course, ,(p) is actually IIp(m), where lip" M Gp is the natural surjection.Thus, we have a commuting diagram M lip evt,(R)id where $ is an isometric isomorphism.Thus, ker (evt,(R)id) ker lit, It,M and since lit, is a quotient map, evt, (R) id M Et, V is a quotient map also.lXX3 TtIEOREM 7. Let =" E X and p" F X be Banach bundles.Let T r(=)-F(p) be a bounded C(X)-homomorphism, and for each t, X, let Tt, Et, Ft, be the induced fiber map which results in the commutative diagram r(,) T r( evp evp Ep Fp Tp where evp denotes the evaluation of sections at p.If T FOr) r(p) is an integral operator, then, for each p X, Tp Ep Ft, is an integral operator.
According to Lemma 0, M r(,)$(F)* can be made into a C(X)-module in such a way that .(tr(R)f) (ao)(R)f hold for all a C(X), o I'(=), and f (Ft,)*.Also, according to Lemma 6, by taking the tensor product of evt, F(r) Ep and the identity map id (Ft,)*--, (Ft,)* we get a quotient map evt," M Et, (Ft,)* whose kernel is It,M.
Because elements of the form (R) !span a dense subspace of I'(r)(Ft,)* M, and because rp is a bounded linear functional on M, it follows that rp(am) a(p) rp(m) holds for all a C(X) and all m M. It follows that the kernel of rt, contains It,M.evp(R)id E(F)* Now, let, Ep and f (Ft,)*, and choose a section tr l'(r) such that (p) e.Then (e (R) f) " (o(p) (R) f) 7 ((evt, (R) id Since is bounded, this identity shows that Tp is an integral operator. [3[3[3  Let X be a compact Hausdorff space, let C c_ X be closed, and let C {a q C(X) a(C) 0}.Let c be the.system of open neighborhoods of C.An approximate identity for C is a collection of continuous functions {i V X [0,1] v *'} such that iv(C 0 and iv(X \v) for each V v.If F(r) is the section space of the Banach bundle E X, then lim V a a for =, /ll ll > I1 -Pp()I1+11 Ppl} / <-PP()' (1 -iv) rl) or all v ," with V _c V 0. Since lim (-Pp(#), (1-iv)a1) lim ((1-iv)(,-Pp(b), al) (Pp(-Pp()), Ul) (Pp() Pp(), rl) O, it follows that ='+IIII>IIPp()II+II -P()II" om (Actually, more is true than this string of lemmas would indicate.It can be shown that we actually have (1-iv) PC(C) in the norm topology of r(r)* whenever C c_X is closed and {iv} is an approximate identity for C _c C(X).However, our proof uses the preceding lemmas, and we do not need this more general result in what follows.) We may regard the space M Intx(r(= ), r(p)) of integral operators from r() to r(p) which are C(X)-module homomorphisms as a Banach space under the integral norm (Diestel and Uhl [7]) that M is a C(X)-module follows because, for T M, a C(X), and a r() we have (aT)(a) a. T(a)= (#a o T)(a), where #a is the operator r ar on r(p).Our final result involves the fibers of the canonical bundle of M as a C(X)-module.THEOREM 11.Let =: E X and p: F X be Banach bundles, and M Intx(r(r), r(p)).
If TM, then ]]T+IpMII int IITpl] int inf {liT +aT'l] int aIp, T'M) where T p Ep F p is the induced map described in the statement of Theorem 7. Thus, the map T Tp is a quotient map, and if G X is the canonical bundle for M, then for each p X there is a isometry ap from Gp the fiber over p, to Int(Ep, Fp), the space of integral maps from Ep to Fl0, with ap(T + IpM) Tp.
PROOF.In this proof, all norms of operators in M are assumed to be the integral norms, and norms of cosets are arrived at by taking infima of integral norms.
Let T, T' M, a e Ip, and let > 0^b e given.We may choose an element E %(R)y r(.)r(p)* such that ll -I and such that lit + T'II < E (v:, (T + aT')(%) )I +.

ITp+3
Since was arbitrary, we have lit + IpMII _< IITpII.In much the same fashion as above, it may be shown that if V is a Banach space and E X is a Banach bundle, then the space M Int(V, r()) of integral operators from V to rt) is a C(X)-module.The fiber over p of the canonical bundle for M then turns out to be e' , T. This is analogous to the result in Gierz [11], regarding compact maps from V to r(x).ACKNOWLEDGMENT.The authors wish to thank the referee for several valuable sugges, .ons,including an improvement in the proof of Theorem 3.

RFERENCES
[I]o Since evt, (R) id M Et, (Ft,)* is a quotient map whose kernel is It, M, there