INTEGRAL OPERATORS IN THE THEORY OF INDUCED BANACH REPRESENTATIONS II . THE BUNDLE APPROACH

Let G be a locally compact group, H a closed subgroup and L a Banach rep- resentation of H. Suppose U is a Banach representation of G which is induced by L. 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. Specifically, we show that: (i) the representation space of a Banach bundle; (2) the above oper- ators become integral operators on this space with kernels which are continuous cross- sections of an associated kernel bundle. KEY W@RDS AND PHRASES. Locally compact group, anach reprentation, rnduced repre- senton, integrated form, integral operator, vector field, cross-section, continu- ity structure, anach unle, quotient topology, kernel unle.


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 G/H.Suppose L is a (strongly continuous) representation of H on the Banach space E. Suppose also that U is a representation of G on a certain Banach space F U which is induced by L in the sense of [i, sec.3].The integrated form of U is the representation of LI(G) determined by the bounded operators U() on F U, where for a continuous function on G with compact support, i.e.
e C (G) It is well-known [2] that these operators can be written as integral operators with contin- uous kernels in the following sense: for f in a certain dense subspace of F U we have U()f(x) I I (x,y)f(y)d((y)) x e G x where d((y)) is a quasi-invariant measure on X and I is a continuous mapping of G G into t'he bounded operators Horn(E) on E The existence of the integral is determined by the fact that for each x in G the mapping I (x,.)f(-) is constant on cosets.Thus, the kernel I for U() is defined on G G while the integration is over X Moreover, the mapping I 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 integration are defined over the same space.There are two canonical choices for this space namely G and X In Chapter II of [2] we acomplished our objective over each of these spaces.
In section 3 of [2], we constructed a representation V of G on a Banach function space F V which was isometrically equivalent to U In particular, each operator V() was written in the following form: for each continuous g in a certain dense subspace of F V we have V()g(x) J-J (x,y)g(y)dy x G G where J 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 is a proper closed subspace of a vector-valued LP-space.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 X We constructed a representation W of G on a continuous sum F W of Banach spaces {E : X} which was also isometrically equivalent to U Each operator W() was written as follows: for each "continuous" vector field h in a certain dense subspace of F W we have where K is a "continuous" field of bounded operators in Hom(E ,E) and K(,) g Hom(EA,E)T!This model requires the knowledge of continuity structures and LP-theory for vector fields [3,4], as well as the theory of kernels and integral operators for such Lebesque spaces [4]. (The latter was developed by the author expressly for this purpose.)Since the space F W is a full LP-space, this model of U does not have the shortcoming that V has.In fact, it proved to be quite useful in studying certain compactness properties of the integrated form of U [2,Ch.IV].
However, W 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.Second, although the kernels K do belong to a continuity structure in Hom(E ,E) this structure partially loses a desirable property in the transition from G to X (See pp. 25-29of [2] for a rigorous explanation.) Since the writing of [2], it has been discovered [5] that the theory of contin- uity structures is equivalent to the theory of Banach bundles [6].Moreover, in the bundle context, the appropriate mappings are cross-sections, so that continuity is simply topological continuity.These facts suggest: (i) The representation W can be reconstructed in the setting of Banach bundles.
(2) The kernels K 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 and NHom(E ,E) respectively.Finally, in section 5, we show that the kernels K are continuous cross-sections for the bundle of section 4.

PRELIMINARIES.
The following is a brief summary of sections i and 2 of [2].
Let A G and A H be the modular functions for G and H respectively.The quotient function AH/(AGIH) is a continuous homomorphism of H into the positive reals which we shall denote by 6 Let p be a continuous, non-negative function on G satisfying p(tx) 6(t)p(x) t H,x G Also let d d((x)) be the quasi-invariant measure on X corresponding to p The representation L of H on E is "Banach inducible up to G" if there exists a pair (p,q) such that i <_ p <_ 0 < q <_ and 8(t)i/qllL(t)l <_ b6(t) I/p t e H for some b > i Given such a pair, we can construct [i] an isometric representa- tion of G on a certain Banach function space as follows.Let C (G,L) denote the q linear space of all continuous mappings f: For a suitable LP-norm on C (G,L) [i, sec.3], we find that right translation q f fx by an element x of G is an isometry.Hence, its extension to the comple- tion F U of C (G L) is also an isometry which we denote by U(x) The resulting q mapping U is then an isometric representation of G on F U which we call the induced Banach representation corresponding to (L,p,q) We refer the reader to [i] for a thorough development of such representations.
In this setting, the operators U(), g C (G) may be written as follows: p(y) 8(t) (x-lty)L( dt f(y)d((y)) X for f in C (G L) (where q:p i/q-i/p) Thus, the mapping I referred to q in the introduction is given by I (x,y) AG (x)-I -i f I (x-lty P(Y) 8(t) q: )L(t)dt x,y g G H Before concluding this section, observe that if we let M 6q:PL then M is a bounded representation of H on E with bound b Hence, we may renorm E by defining We thus obtain a Banach space The Banach spaces Horn(E) and Hom() are also equivalent with The following result will be useful in what follows: PROOF.This follows from the fact that L is strongly continuous and locally bounded [1, 3.3].
It will also be convenient to fix in advance a compact neighborhood Z of the identity e in G for use later on. 3.

THE VECTOR FIELD BUNDLE
In order to construct the bundle version of W we begin as in section 4 of [2]   Consider the space G x E with equivalence relation defined as follows: if x,y s G and v,w s E then (x,v) (y,w) if there exists t in H such that y tx and w

L(t)v
Denote the resulting space of equivalence classes (with quotient topology) by E and let o: G x E-E be the canonical projection.Also let : X be the (well-defined) projection given by (o(x,v)) (x) We then have the following composition: The next step is to make (E,X,) into a Banach bundle.As on p. 12 of [2], -i -i define E () X For fixed x in " ()" each element of E is uniquely of the form (x,v) v e E Thus, E becomes a Banach space equivalent to E and under the following (well-defined) operations: llo(x,v)ll p(x)q:Pllvll M v,w E LEMMA 3.4.For each x e G v e E llo(x,v)ll--sup{p(y)q:Pllwll (y,w) (x,v)} PROPOSITION 3.5.The bundle (,X,) is a Banach bundle (as in section i of [6]).
Let S(X,E) denote the linear space of cross-sections from X into E CS(X,) those that are continuous and CS (X,) the continuous cross-sections with C compact support.Analogously, recall [2, sec.4] that there is a continuity structure A in NE given by A {Sf} where Sf((x)) where H (G L) is the space of (L,q)-homogeneous q q functions on G (In [2], A was denoted by Aq.)As in sections 1 and 5 of [4], we then have the space C(A) of A-continuous vector fields and the subspace C (A) C of compactly supported such fields.Note that S(X,) HE T which in turn is in bijective correspondence with H (G,L) via the above mapping f Sf Also, The continuity structure A yields a topology (the A-topology) on making (E,X,) a Banach bundle having the property that the elements of C(A) are the continuous cross-sections relative to the A-topology [6, Prop.i. 6].Actually, the space COl) is also the space of quotient-continuous cross-sections.[7, sec.i] THEOREM 3.6.The quotient and A-topologies are the same.
Before proving this theorem, observe that we then have CS(X,E) C(A) and CS (X,E) --C (A) Furthermore if is the measure d on X then LP(A,) C C [4, sec.6] is the same as the space LP((,X,) ) [6, sec.2], which is the LP-completion of CS (X ) However LP(A,) is isometrically isomorphic to the C representation space F U of U and is the representation space F W of W Hence, the representation space of the bundle version of W will be LP((E,X,) )

H (G,L) C(G,E)
A is an open subset of X and e > 0 Let q 8 o(x,v) be an element of W(h,A,e) so that (e) (x) s A and lle Sh(ie))ll llo(x,v) llo(x,v p(x)-i/qh(x))ll p(x)q:Pllv p(x)-i/qh(x)llM < g The function y _+p(y)q:Pllv p(y)-i/qh(y)ll M is continuous on G Hence, the set N of y in G where this function is less than e/2 is open in G Also, x g N so that x g N N Zx 0 -I(A) which is a neighborhood of x in G Thus, this set contains an open neighborhood B of x For a max{P(y) q:p y g Zx} Since {h(y) h g Cq(G,L)} is dense in E (and hence in ) for each y in G -1/qh [i, 3.8] and h-p is a bijection of C (G,L) there exists h in C (G,L) q q such that llv p(x)-i/qh(x)llM < r/ac By the continuity of h there exists an open neighborhood N of x in G such that N c Zx V and llv p(y)-i/qh(y)ll M < r/ac y g N Let A (N) and g r/c Then W(h,A,e) is a basic A-open neighborhood of o(x,v) 0 in E which is contained in o(B C) Therefore, the two topologies are the same.H the canonical projection.Also, let :H X X be the (well-defined) projection given by ((x,y,T)) ((x),(y)) x,y s G T s Horn(E) We then have the following composition: PROOF.Similar to that of 3.3.
The next step is to make (,X X,) into a Banach bundle.For , s X define H -i(,) For fixed x in y in each element of H is uniquely of the form a(x,y,T) for T in Horn(E) Define: Recall that E, and the E s X are all equivalent Banach spaces.
Thus, the same is true of Horn(E) Hom() and the Hom(E ,E) , s X In particular, if T s Horn(E) then the corresponding element of Hom(E ,E) is the operator T' which is the image of c(x,y,T) under our identification, x s YS Consequently, modulo the isomorphisms, the field {Hom(E ,E):, X} of , x} Banach spaces is the same as the field {, REMARK 4.5.Recall that in 3.4 we verified that llo(x,v)ll sup{o(y)q:Pllwll (x,v) (y,w)} The analogous question here is: does lla(x,y,T)ll sup{(9(x')/p(y'))q:PllT'll ( The answer depends on the question: does IITIIM sup{llM(r)TM(s)-lll r,s g H} T s Horn(E) ?
We believe the answer to both questions is yes; however, we have been able to only partially verify the latter.
As in section 3, we obtain the spaces S(X X,H) and CS(X X,H) of cross- sections and continuous cross-sections respectively.The space CS(X X,) is then the space of continuous kernels.
The vector field analogue of this space requires the existence of a continuity structure in Hom(E ,E) This was essentially accomplished in section 4 of [2] as follows.Let (G G,Hom(E)) denote the linear space of mappings For each such B define RB(,) E E , s X by RB(,)((y,v)) (x,B(x,y)v) x s ,Y s v s E Then RB(,) e Hom(%E) so that RB g NHom(E ,E and we thus have a mapping R:HL(G G,Hom(E)) NHom(E ,E Note that RB(,) B(x,y)' which is the element corresponding to (x,y,B(x,y)) under the bijection between and Hom(E 4) ,E) ( LEMMA 4.7.The mapping R is a bijection. PROOF.The onto property is proved using a (possibly non-measurable) cross- section.
Now consider the linear space The set RB:B e CL(G G,Hom(E)) } was observed to be a, precontinuity structure in Hom(E,E) 2, 4.32].This means that we were unable to verify that {RB(,):RB e } is dense in Hom(E ,E , e X If this is not the case, then the ftopology on is not the same as the quotient topology the latter is Hausdorff while the former is not.Moreover, our proof in 3.6 (which we would like to duplicate here) required this density property.This is not a significant problem since we can bypass it by simply restricting our attention to the appropriate portion of H Before doing so, the following result shows to what extent the density property does hold and (more importantly) suggests why it may not hold in general.
THEOREM 4.8.The subspace {RB(,) RB e } is strongly dense in Hom(Erl, E) ,r] e X PROOF.(After section 3 of [i]).Let  Horn(E) is continuous and has compact support.For each x,y in G consider the mapping and L I Then H L is equipped with the relativized quotient topology of H (or equivalently, the quotient topology of G x G x HomL(E)) and (L,X x X,L) is a Banach bundle with i(,) m (HL), HL N H, H L & -i(,) , e x Similarly, we let HomL(E,E) denote the closed subspace of Hom(E,E) corre- sponding to HomL(E) so that HomL(E,E) (HL), , e X It is clear that is a (complete) continuity structure in BHomL(ED,E) H L Consequently, H L is also equipped with the -topology.
THEOREM 4.10.The quotient and -topologies on H L are the same.
PROOF.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) correspond exactly to the -continuous kernel fields in HomL(E,E) Of course, if has the desired density property, then the L-subscript may be dropped in 4.10, i.e.C() CS(X x X,H) the ideal result. 5.

THE CONTINUOUS KERNELS
We are now in a position to complete our project.Recall (section 2) that I is a continuous mapping of G x G into Horn(E) k (x,y) (x,y,p(x)-I/qf)(y)l'ql / (x,y)) x,y G Then k is continuous and constant on cosets.Hence, it defines a continuous mapping of X x X into G x G x Horn(E) Furthermore, it follows that the following diagram commutes (modulo isomorphism) for , s X x s y s v s E Hence, the kernel K for the operator W() is a continuous cross-section.We had already seen [2, 4.33] that it is an -/q p/qT -continuous kernel field.Finally, since K RB for B @ we see that K X x X-H L i.e.K e HOmL(E_,E) so that the two notions of continuity for K are the same.
In conclusion, let us briefly summarize what we have shown.The induced representation U is isometrically equivalent to a representation W of G on the bundle space LP((E,X,) ) where W(x) acts on CS (X,E) by right translation by x x e G The integrated form of W is given by W()g() f K(,)g()d e X g CS (X,E) X c where K X x X H L c H is a continuous kernel for W() 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.

Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities.This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics.It is possible to make a division in two main categories: astronomy and astrodynamics.By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth.Many important topics of research nowadays are related to those subjects.By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines.The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research.All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
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Manuscript Due
July 1, 2009 First Round of Reviews October 1, 2009 llv-wll < e/2ab} Then C is an open subset of E containing v Thus, o(B C) is a basic quotient-open neighborhood of o(x,v)e which can be shown to be in W(h,A,g) Conversely, let o(B C) be a basic quotient-open subset of E for B open in G and C open in E Let e g o(

3 .
The mapping is continuous and open.PROOF.The openness of follows from the fact that the saturation -i ((A B C)) of a basic open subset A B sB L(r)CL(s) r,s s H} which is open in G x G x Horn(E) LEMMA 4.2.The mapping :-+ X X is continuous and open.The space H is Hausdorff.
o is open, passing to a subnet if neces- sary, we may assume there exists a corresponding net (xi,vi) } in G E such that o(xi,vi) 8 i i 1,2, and (xi,vi)(x,v)in G E Similarly, we may ')} in G E such that assume there exists a corresponding net {(xi,v i o(x.,v[.) 8 i and (x.,v[.)-(x' ,v') Since o(x i,vi o(x.,vi)' there exists The mapping o is continuous and open.PROOF.The openness of o follows from the fact that the saturation -l(o(A x B)) of a basic open subset A x B of G x E is of the form U {(tA) x (L(t)B) o(x',v') Since is Hausdorff.