GENERALIZED GROUP ALGEBRAS AND THEIR BUNDLES

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.

Neither of the previous approaches is more general than the other.However, they were designed for the same purpose.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.This is the main shortcoming of the Fell approach via bundles.On the other hand, since Fell's approach allows for non-lsomorphlc fibers in the bundles (i.e.non-trivial bundles), this should also be true of the bundles in the general theory.This is the main shortcoming of the Leptin and Busby-Smlth constructions.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.This extension will then include the Leptln and Busby-Smith constructions.However, the method by which we realize the measurable *-algebra structure in a (Banach) bundIe will be quite different from that of Fell.In fact, it will be more consistent with the other two approaches in that it ill be done by means of operator fields.This is our secondary objective.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 com- putations 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 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 ' bundle structure arises in specific examples.Thus, in short, our aim is to 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.Replace A by a field {Ax:XeG } of Banach spaces indexed by G In Section i, we review the notion of continuity structure A in and the x our 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.If S is any set, then E S will denote the identity mapping on S and YS its characteristic function.If A and B are Banach spaces, then Hom(A,B) will denote the Banach space of bounded linear operators from A into B If X is a locally compact Hausdorff space and is a positive Radon measure on X then C(X,A) (resp.C (X,A)) will denote the linear space of continuous (resp.compactly supported) A-valued functions on X and M(X,A,) the linear space of (equivalence classes of) such -measurable functions.The phrase "for -almost all x in G" will be abbreviated by "-a.a.x in G." I. CONTINUITY STRUCTURES.
Let X be a locally compact Hausdorff space and {Ax:XeX a field of Banach spaces over X Let A denote the disjoint union of the A xeX Define :,%/X x by -l(x) Ax,XeX Note that A is a linear space.An element of A will x x be called a vector field.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.This was first accomplished by Godement in [7] (see also [8]) by means of a "continuity structure."Actually, Godement's original terminology was "fundamental family of continuous vector fields."The term "continuity structure" is taken from Fell [9].
DEFINITION I.i.A continuity structure A in A is a subset satisfying: x (i) A is a subspace of KA x (ii) For each h in A the scalar function x-li h(x)II is continuous on X (iii) For each x in X the subspace {h(x):heA} of A X is dense.
Given a continuity structure, we define continuity for a vector field as follows DEFINITION 1.2 Let x eDX and feDAx Then f is A-continuous at x o o if, given >o there exists a neighborhood N of x in X and h in A o such that llf(x) h(x)ll<e for x in NQD The field f (defined on D) is A-continuous if it is so at each point of D Denote the space of such fields by C(D,A) if D X we write C(A) for C(X,A) Clearly, A _c C(A) in general.
Hence, in this context, to say that f is a continuous vector field means that f is an element of the linear space C(A) (See sections 1 and 5 of [8].
and only if the element h in A can be chosen from F Some time after Godement, Fell introduced the notion of continuity into our context in a very different way--by axiomatically topologizing A[I] For this purpose, it will be convenient to speak of the elements of I[A as cross-sections x and denote the space of them by S(X,A) DEFINITION 1.5. (A,) is a Banach bundle over X if .A is a Hausdorff space, (iv) If xEX and {a i} is a net in such that flail 0 and w(a i) x in X then a. 0 (the zero of A in A 1 x x Let CS(X,A) denote the subset of S(X,A) consisting of continuous cross-sections.
Observe that the relative topology of A on each fiber A is precisely the x norm topology [i, p.10].More importantly, we have the following: LEMMA 1.6.The space CSX,is a continuity structure in with equality holding in part (iii) of i.I.
PROOF.Part (i) of i.i follows from [i, p. ii].Part (ii) is clear.Part (iii) is a very recent development and follows from the fact that (A,) has enough continuous cross-sections [i, Thin.ii].
Letting A--CS(X,A) we obtain that C(A) A [8, p. 13].Therefore, start- with a Banach bundle, we obtain a continuity structure in A which is the largest such structure in its equivalence class [8, Prop. 1.23]. (Recall that AI A2 if C(AI) C(A2) .)Conversely, it has been known for some time that this process is reversible (up to equivalence).
LEMMA 1.7.Let A be a continuity structure in A Then there exists a x _unique topology on A making (A,) a Banach bundle.Furthermore, A .c CS(X,A); in fact, C(A) CS(X,A) LEMMA 1.8.Let x oeD c.X and feAx so that f:D-Then the cross- section f is A-continuous at x if and only if the vector field f is A-cono tinuous at x Hence, C(D,A) CS(D,A) o Thus, starting with a continuity structure A we obtain a Banah bundle (A,v) for which A~CS(X,A) This shows that the two methods for obtaining con- tinuous vector fields in A are equivalent, i.e.C(A) CS(X,A) For the re- x mainder of this paper, we will let A be a continuity structure in KA and (A,) x the unique Banach bundle guaranteed by 1.7.In particular, if X is discrete then HA itself is the essentially unique continuity structure [8,1.22] in A Next we turn to the notion of measurability.Let be a positive Radon measure on X First we consider the vector field context.DEFINITION 1.9.Let D be a locally compact subspace of X and f an element of DAx Then f is (A,)-measurable if, given compact K c_ X and e > o there exists compact KE c_ K such that (K-Ke)< and f is A-continuous on K e Denote such f by M(D,A,) Analogously, we wish to introduce measurability in S(X,A) (as in [i0]).
DEFINITION 1.10.Let D and f be as in 1.9 Then f is (A,B)measurable if, given compact K _c D and >o there exists compact Kc K such that (K-K)< and f:K is continuous for the A-topology of A Denote such f by MS(D,A) aEMARK I.Ii.On p. 22 of [1], Fell defines the notion of measurable cross- section in a different way.However, as can be verified by the results on p. 23 of [i], the two definitions are equivalent.
LEMMA 1.12.If D is a locally compact subset of X then
REMARK 1.13.Since continuity and -measurability depend only on the equlva- lence class of A [8,1,3], we may replace A by C(A) CS(X,A) without loss of generality.(Of course, this is false if we find it necessary to consider equl- continuous families of vector fields [8 2]).Thus we may (and wILl1) assume A C() CS(X,) Actually, this is advantageous because of the last equality, as well as the fact that {h(x) hC()} is equal to A x for x in X In what follows, C(A) or A will be used according to which is appropriate when they are no__.assumed to be equal.
The remainder of this section is devoted to separability considerations.
LEMMA 1.15.The bundle (A,) is strongly separable if and only if it is second countable.
Suppose for the moment that A Is arbitrary again, i.e.A c C(A) in general.
Recall [8, p.10] also that is separable If C(A) is countably dense and A is locally separable if, for each compact K X, the restricted continuity structure AIK [8, p.9] is separable.In this context, we have: A strongly separable implies A separable.The converse is false in general.However, if we again assume --C(A) then: A is separable if and only if A is countably dense.
Therefore, under this assumption, we have the following separability summary: LEMMA I. 16.The following are equivalent (i) A is second countable, i.e. strongly separable.
(ii) is countable dense and X is second countable.
(lii) is separable and X is second countable.EXAMPLE i. 17.Let be an exact sequence of locally compact groups.If H is second countable, then so is G and the group algebra LI(N) is separable.In Section 9, we will construct a Banach bundle over G 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.

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} is a field of Banach spaces over the locally compact Hausdorff space Y with A a continuity structure in B Let (B,T) denote the Y corresponding bundle.Suppose also that :X/Y is a continuous mapping from X into Y The induced bundle [ii] (B,T) over X is then given by: If is one-to-one, then B can be identified with r-l((X)) xX} The cross-sections S(X,B) are in one-to-one correspondence with HB(x) {f:X-f(x)eB(x),XeX} Suppose B is equipped with the relativized topology of X B LEMMA 2.1.Let A {ExXk keA} Then A is a continuity structure in B (x) LEMMA 2.2.Let feHDB(x) for D c X and XoeD Then the following are equivalent (1) The vector field (Ex, f is A-contlnuous at Xo (li) The cross-sectlon (Ex, f) in S(D,B) is continuous at Xo From 2:2, it follows that and cs(x,m) c(a) {(Ex, f):fgC(X,),f(x)E(x) xgX} MS(X,,U) M(a,U) EXAMPLE 2.3.If X c_ y and @ is the injection mapping, then h is simply the restriction AIX of A to X [8, p.9].In this case, B@ is homeomorphic to i.e. (Ex,h)+-h heC(A) EXAMPLE 2.5.If :YX+Y is the left projection % then A % is the lifted continuity structure obtained from A [8 4]   The corresponding bundle ( can be identified with (Xx, ExXT (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 bundles.Nevertheless, in going through this section, it would be worthwhile for the reader to give special attention to the case where the domain is a single bundle- -and not a product.
Let Xl,...,Xn,Y be locally compact Hausdorff spaces, {Ax1:XlX 1} {A :x X {By:yY} fields of Banach spaces over these base spaces, AI,...,%, x n n n the bundle spaces and l:Al/Xl,...,n:An/Xn T:Y the projections.Let denote the product bundle over X XI... X n given by A i....wAn and -1 ... (Recall [11].)n his context, an (abstract) bundle morphism n (,') from the product (A,) ino (,T) will be a pair of mappings satisfying T-Thus, such is fiber preserving in the sense that (Ax c BCx xeX where x-(x I ,x n) and A A x...xA Note that we X X 1 X n are not requiring that or be continuous.In fact, our objective here is to develop the notions of continuity and measurability for (,) Clearly, a bundle morphism (,) may be viewed as a pair (,) conslating of a mapping :X-Y together with a family of mappings # { :xEX} satisfying ed n-linear mappings.)Thus, may be viewed as a vector field in the product rxm (x' B,(x)) Now let Di _c X i fieDiAxi l<i<_n with D-DI...Dn and f'(fl'"''fn Then defines a mapping #f:D given by (x) x((x))-x.,...,xn(.(x.)'''" ,n(xn)) (fl(Xl)'''''fn(Xn) xED Clearly, Cf .f Moreover, since fi(xi)eAxl l<_i_<n we have f(x)zA x so that f(x)eB(x) xED Thus, #fEEDB(x) i.e.
:DAx / DB(x) Next suppose that AI,...,A n A are continuity structures in the spaces AXl ,A x ,B If is aontlnuous, then it induces a continuity structure n Y A in B(x) as in section 2. On the other side, we have the product A AIX" xA of the continuity structures AI,.. ,A in [L% ...xIL% ExAx n n x I x n DEFINITION 3.2.Let (,) be as in 3.1 with aeA Then (,) is con-.
tlnuous at a (relative to A,A) if is continuous at a and is continuous at (a).
DEFINITION 3.3.The operator field is continous at x in X (relative to A,A) if, for each h in A the mapping Oh=.h is continuous at x The field $ is locally bounded at x in X if, for each compact subset K of X (2) If is continuous on X then is continuous at x in X if and only is A@-continuous at x if, for each h in A the vector field h in HB(t) x in X the zero element @ __(@i of A satisfies: (2) This follows from 2.2. X.
PROOF.The proof of 9.2 of [8] can be adapted to this proposition.
COROLLARY 3.6.If is continuous at x in X then it is locally bounded at x--assuming A i C(A i) l_<i_<n PROOF.By hypothesis, f:X/B is continuous at x for each feiC (A i) C Hence, the scalar functions Cf (')If are also continuous.This corollary then follows from 3.5, together with the fact that the supremum of continuous functions is lower semi-continuous and hence, locally bounded.REMARK 3.7.If we were not assuming A i C(A i),l<_iin then the two versions of continuity for in 3.3 would be equivalent for locally bounded # [7, p.84].
This is reasonable in view of 3.6.

THEOREM 3 8. Let xeX
Then (0,) is continuous at each a in A if and X only if is continuous at x (as in [i, p.32]).
COROLLARY 39 Let x.eD.cX l<i<n Suppose (,) is continuous at then for f (fl ,fn the mapping 1 1 ...XD +B #f:DlX n is continuous at x for the A-topology of B Next we develop measurability for ($,) and # Let H i be a positive is a positive Radon measure on Radon measure on X i lii<n Then H HI Q'''Hn x [10].
DEFINITION 3.10.The Banach bundle morphism (,) (resp.operator field $) is measurable (relative to A, A, H) if, for each h in A the mappin h (resp.h) of X into is measurable.Thus, (,) is measurable if and only if $ is measurable (recall 3.8).
(2) If is continuous, then $ is measurable if and only if, for each h in A the vector field Sh in B(x) is (A,)-measurable (recall 2.2).
There is a stronger notion of measurability for $ and (,) (as in [8,10])   which is useful for our needs.
DEFINITION 3.12.The Banach bundle morphism (,) (resp.operator field is ultra-measurable if, for each compact subset K of X and >0 there exists a compact subset Ke of K such that H(K-Ke)<e and (,') (resp.$) is continu- ous on K Thus, (,) is ultra-measurable if and only if is.
THEOREM 3.13.If is ultra-measurable, then it is measurable.The converse is true if, in addition, is locally bounded and each A i is countably dense, lin PROOF.This is proved as in Prop. 20of [7].
THEOREM 3.14.Suppose $ is measurable.If f=(f!,...,fn) is an element of HiM(Ai,i) then fgM(X,,) REMARK 3.15.Once again, if we were not assuming A i C(AI) l<i<n then there would be two versions of measurability for $ given in 3.10.However, in view of 3.14, these would be equivalent.is of special interest.Specifically, for this case, this section contains the de- finitions and properties of continuous, measurable, and ultra-measurable Banach bundle morphisms.

PRODUCT FIELDS.
Having established the foundations for our analytical needs, we turn to our algebraic needs--namely multiplication (i.e.convolution) and involution.In this section we develop multiplication in A by means of a product field of operators.
Suppose G is a locally compact group and (A :xG} is a field of Banach X G with A as before.Then Hom2(AxXAy,Axy) is the Banach spaces over space (A3) For convenience, let P HHom2(AxXAy,Axy) Note that an element of P is no.__Since the mapping :GGG given by group multiplication satisfies (x) p it follows that (p,) is a Banach bundle morphism of (A,'w) into (A,) We will also require that product fields be measurable relative to A and (rit) Haar measure B on G as in Section 3. It will be instructive to summarize he results of Section 3 for (p,P,) THEOREM 4.2.Let P be an element of P with p as above.at (Xo,Yo) If fEDIA x gED2Ay are A-continuous at Xo,Y respectively, then the mapping P(f,g):DIXD2-A is continuous at (Xo,Yo) (resp.A'-continuous at (Xo,Yo)).
(ii) For each h,k in A the mapping P(h,k):GG/A is measurable.
For each h,k in A the vector field (EGG,P(h,k)) in IiAxy is (A', @)-measurable.
(iv) For each h,k in A the cross-section (EGG,P(h,k)) is measurable A' relative to and Q PROOF.Combine 3.10, 3.11, and 1.12.
THEOREM 4.5.Let PeP Then the following are equivalent: (i) P is ultra-measurable.
(ii) For each compact subset K of GG and >0 there exists a compact subset Ke of K such that Q(K-Ke)<e and P is continuous on Kg If P is a product field and A is countably dense, then the previous are equivalent to: (iii) P is measurable.

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 A For each x in G let A denote the Banach space conjugate to A X X i.e.A and A are identical except for scalar multiplication which is given in X X A by: X a'a aa eK aeA X Consider the fields {A :xeG} {Ax_l:XeG} of Banach spaces over G The spaces X {Hom(A,Ax_l)} form a field of Banach spaces over G and Hom(Ax,Ax_ I) is a linear space of operator fields.Note that the linear mappings from Ax into Ax_I are precisely the conjugate linear mappings from A into Ax_I Thus if I is X an element of NHom(Ax,Ax_I) then, for each x in G the mapping is bounded and linear (resp.conjugate linear) with range contained in Ax_I For convenience, let S :A A A A x,y x y y x denote the switching mapping (a,b) (b,a) aEAx, bEAy x ygG DEFINITION 5.1.Let I be an element of HHom(Ax,Ax_l) Then I is an involution field if lllxll i Ix_I Ix-I and IxyPx,y Py-l,x-l(lyIx)Sx,y x,ygG for each product field P in P For convenience, let I HHom(Ax,Ax_ I) Note that the elements of are not necessarily involution fields.
For the purposes of this section, let @:G+G be the inversion homeomorphism

(x) x -I xeG
If f is a mapping defined on G it is customary to write fv for f@ Consequently, the induced continuity struc6ure A @ in A(x) flax_1 will be denoted by A v i.e.
Av {hV:heA} Also, denote by Athe continuity structure A viewed as being in KA HA X X Hence, for oSK and h-sAwe have (oh)(x) oh(x) xEG Now let i denote the mapping of A into itself given by i(a) l(a)(a) aeA Then (i,@) is a bundle morphism of (A-,n-) into (Av,v) where: (i) A-= A DA-(disjoint) with conjugate scalar multiplication in the X fibers.
(ii) A v A UA-i (disjoint) with A-I the fiber over x in G X X (iii) (a) (a) x for aeA A xeG As in the previous section, an involution field will be required to be measurable relative to A and Here also, it will be instructive to summarize Section 3 for (i,l,) THEOREM 5.2.Let leI with i as above.Then the following are equivalent (ii) For each h in A the mapping l(h-) G-A is measurable.
(iii) For each h in A the vector field l(h-) in A -i is (Av,)-measurable.X (iv) For each h in A the cross-section Ih-:C--A v is measurable realtive to (v,) PROOF.Recall 1.12, 3.10, and 3.11.(i) I is ultra-measurable.
(ii) For each compact subset K of G and g>0 there exists a compact subset Kg of K such that (K-Ke)<g and I is continuous on KE (resp.i is -i -i continuous on (7-) (K) (K)).
If I is an involution field and A is countably dense, then the previous are equivalent to (iii) I is measurable.
We are now ready to construct a measurable analogue of Fell's Banach *-algebraic bundle [i].Let {A :xgG} once again be a field of Banach spaces over x the locally compact group G with A,A,, as above.Suppose also that P is a product field in P with p the corresponding product bundle mapping.One of our goals is to generalize the above to the case where the convolution is just measurable--and not necessarily continuous.In this regard, we have: PROPOSITION 6.3.The product in A is measurable if and only if P is measurable.It is ultra-measurable if and only if P is ultra-measurable.If A is countably dense, then these are equivalent to measurability for P (4.5).
To introduce involution into A suppose further that I is an involution field in with i as before.dense, then these are equivalent to the measurability of I DEFINITION 6.7.By a (measurable) Fell bundle (A,; P,I) over G we will mean a Banach bundle (A,) over G together with measurable product and in- volution fields P and I (The underlying continuity structure A will be understood to be CS(G,A) as before.)If P and I are continuous, we will say that (A,;P,I) is a continuous Fell bundle over G (equivalently, a Banach *-algebraic bundle over G ).
In Section 3 of [i], Fell defines a continuous Fell bundle (A,:-,*) over G to be a Banach bundle (A,) over G with product and involution * given axiomatically.Clearly, the corresponding product and involution fields P and I are then determined as follows: Of course, these fields are continuous relative to A CS(G,A) since and are continuous operations on A (6.2, 6.5).To extend his construction to that of a measurable Fell bundle, the appropriate measurability requirements on and * would be given by 4. Before leaving this section, we wish to point out one of the main consequences of replacing a continuous Fell bundle by a measurable one.DEFINITION  Hence, the converse implication is questionable for measurable Fell bundles since its proof depends on the continuity of the product in A This also affects Propositions 11.4 and 11.5 of [2] in the same way.
The purpose of this section is to extend (and rephrase) Fell's study of approxi- mate identities (units) in [1,2] to the context of a measurable Fell bundle.
Let (A,;P,I) be such a bundle.It is clear then that the fiber A over the e identity e in G is a Banach *-algebra DEFINITION 7.1 [i,p.34].A (bounded) approximate identity in A is a net {u.} in A satisfying: e (i) llujll-< 8 all j for some 8>0 (ii) Iluj'a all 0 aeA (iii) Ila'uj all o aeA In particular, {uj } is an approximate identity in Ae in the usual sense.DEFINITION  identity if it is an approximate identity and (ii) and (iii) of 7.1 hold uniformly on compact subsets of A In Prop.ii.I of [2], Fell showed that these two notions agree in the case of continuous Fell bundles.This appears to be false in the measurable case.To determine what is true, we proceed as follows.
Let D-CG feDAx and agAe The left and right translates of f by a are defined by f(x) a-f(x) P The following is our measurable Fell bundle version of Prop.

MULTIPLIERS.
There are two additional significant distinctions between continuous and measurable Fell bundles which involve multipliers.We will briefly discuss here the DEFINITION 8.1.If x is an element of G then a multiplier m of order x for the measurable Fell bundle (A,;P,I) is a pair (ml,m2) satisfying the following (i) ml,m 2 are continuous mappings of A into itself which are bounded in the sense that Ilmill sup{lira i(a)ll :aeA} is finite for i 1,2 (ii) For each yeG ml(res p. m2) is a linear mapping of A into A y xy For convenience, as usual, we will write ma for ml(a) and am for m2(a) aeA Let M (A) denote the set of multipliers of order x and M(A) UGMx(A) x Also, let z:M(A)+G be the canonical projection.
Each M (A) is a Banach space under the canonical linear operations and norm x given by llmll max(llmlll,llm211), mgMx(A) xeG Thus, (M(A),z) is algebraically a Banach bundle.Moreover, there is a product and The operations have the following properties: (i) If meMx(A),m'eMy(A) then mm'eMxy(A) (2) The product is bilinear on Mx(A)XMy(A) (3) The product is associative.
(4) If meM (A) then m*gM -I(A) x x (5) The involution is conjugate linear on M (A) x (6) The involution is anti-multiplicative.(7) The involution is self-invertible.
and llm*ll ilmll (9) The left and right identity mappings of A fon the identity of M(A) in M (A) e We are now ready to describe one of the distinctions between continuous and measurable Fell bundles referred to at the beginning of this section.
If the Fell bundle (A,;P,I) is continuous, then A can be mapped into by right and left multiplication: tuba ba am b ab a,bgA However, 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.x LEMMA 8.5.If P is continuous and A has an approximate unit as well as enough unitary multipliers, then A is saturated (6.10 and [2,prop.ll.5]).
REMARK 8.6.In general, if A is saturated, then it may not have enough unitary multipliers even if P is continuous and A has an identity [2,p.130].
The implication in 8.5 is questionable in the measurable case because its proof, namely Prop.11.4 of [2], is questionable for non-continuous P (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, M(A) can be equipped with a topology called the strong tppology [1,5].In this topology, a net {M.} in M(A) converges to m in M(A) if m.a ma am.
am agA 3 3 For this topology, involution is continuous and the product is separately continuous.
The mapping :M(A) A is continuous relative to this topology but possibly not open.Consequently, .vIU(A) is a continuous homomorphism of U(A) into G (onto if there exist enough unitary multipliers).Note that multiplication in U(A) is separately continuous but possible not jointly continuous.

Let MI(A)
{meM(A) :llmllo-<l} The following will be useful in Section i0.
PROPOSITION 8.7.If A has a strong approximate identity and P is ultra measurable, then the mappings (m,a)+ma and (m,a)/am of MI(A)A "into A are measurable in the sense of 3.10.In particular, if P (i.e., p) is continuous, then these mappings are continuous (compare with [l,Prop.5.1]).
To construct a Banach *-algebra from a measurable Fell bundle (A,;P,I) over G let LI(A,) denote the Banach space of (null equivalence classes) of (A,)- measurable vector fields f which are -integrable, i.e. for which Ilfll 1 ./*Gill (x) lld (x)< (See [7,8] for the details.)The subspace Cc(A) of C(A) consisting of vector fields with compact support is well-known to be dense in LI(A,) Also, in view of the results in Section i, LI(A,) is the vector field version of the space LI(A,) of -measurable integrable cross-sections [1,2].Let be the (right) modular function for G Before proceeding further, let us record an important result for future use.Once again, let %:GXG-G be the left projection (recall 2.5).
LEMMA 9.1.Let fELI(A%,) Then for -a.a.x in G the integral fGf (x,y)d (Y)   belongs to A and the resulting (-a.e.defined) vector field x x fGf (x,y)d (y) belongs to LI(A,) PROOF.This is the vector field analogue of Prop.2.11 of [i].
LEMMA 9.3.The vector field of 9.2 belongs to LI(A ) and its L I -norm is at most llfll l'llgll I PROOF.This is a straightforward application of the scalar Fubini Theorem.
As a result of the previous discussion, for f,g in LI(A,) we may define a vector field f'g in LI(A,) by We thus obtain a binary operation (multiplication or convolution) on LI(A,) which satisfies <_ llflll-Ilgll I f,ggLI(A,) REMARK 9.5.Before going any further, observe that Fell defines convolution first in C (A) [i,8] and then extends it to all of LI(A,) Note that in his C case (that of a continuous Fell bundle), C (A) is closed under multiplication as a C consequence of the continuity of multiplication in A However, this is not true for a measurable Fell bundle.Hence, for us, there is less advantage in first multiplying elements of C (A) since the products may not be in C (A) 6(x)-if(x-l)* 6(x)-ll(f-) (x) 6(x)-II (f(x)) xeG X LEMMA 9.7.For each f in LI(A,;P) the field f* also belongs to LI(A,;P) in fact, llf*ll I llfll I Hence, we obtain a mapping * (involution) from LI(A,;P) into itself.PROPOSITION 9.8.Under involution LI(A,;P) is a Banach *-algebra which we denote by LI(A,;P,I) LEMMA 9.9.If the conditions of 1.16 hold, then LI(A,;P,I) is separable.
PROOF.The underlying Banach space LI(A,) is separable by the Corollary to Proposition 2.2 of [l,p.20].
Next in this section, we turn to a study of identities in LI(A,;P,I) Suppose A has a strong approximate identity.In the proof of Prop. 8.2 of [i], Fell shows how to construct an approximate identity for LI(A,;P,I) Also observe that this proof does not use the A-continuity of P or I Consequently, we have LEMMA 9.10.If A has a strong approximate identity (recall 7.5), then LI(A,;P,I) contains an approximate identity (with the same bound).
PROPOSITION 9.11.The Banach *-algebra LI(A,;P,I) contains an identity if and only if G is discrete and A has an identity.
PROOF.The proof of the corresponding result for classical group algebras [12,310] can be adapted to the vector field context.In particular, the identity u of A and the identity f of.LI(A,:P,I) are related by f(e) 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.
Suppose (B,T:Q,J) is another Fell bundle over G and LI(A,:Q,J) is the corresponding generalized group algebra.Let :(A,)+(B,T) be a Banach bundle morphism as in section 3 (with =EG) Then we have a field { :xeG} g GHOm(Ax,Bx ..
xgG If f is a vector field in HAx then it follows that f is a vector field in B Furthermore, the correspondence f f is linear x between the underlying linear spaces of vector fields; notationally, we have HA +liB x x Now suppose that is measurable (3.10).Then by 3.14, we have a linear mapping M(A,) M(A,) which is constant on null equivalence classes.If is continuous, then (3.9) For convenience, let I111 G denote sup {llxl :xgG} <m Then we have: We will say that is bounded if ..llII G PROPOSITION 9.12.If (i.e.) is measurable and is bounded, then (modulo nullity) is a bounded, linear mapping of Ll(A,) into Ll(A,) with norm at most IIII G In particular, if each Cx'XgG is an isometry, then is an isometry.
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) is a Fell bundle morphism if : (A,n)-(B,y) is a measurable Banach bundle morphism and (i) Px, i.e. # P y Qx,y xy x,y Qx, y (xXy) (ii) I x Jx i.e.I J X X X X for x,y in G THEOREM 9.14.If (as in 9.13) is a bounded Fell bundle morphism, then LI(A,p;P,I)/LI(A,p;Q,J) is a Banach *-algebra homorphism.
Conversely, suppose -I B+A exists and is a Banach bundle morphism, so that 9.12 and 9.13 apply.

PROPOSITION 9.15
If ,-i are bounded Fell bundle morphisms for (A,;P,I) and (B,T;Q,J) as above, then LI(A,;P,I) and LI(A,;Q,J) are isomorphic Banach *-algebras which are equivalent as Banach spaces.In particular, if each is an x invertible isometry, xgG then these algebras are isometrically isomorphic.

I0. 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.
Recall that the unitary multipliers U(A) for the Fell bundle A form a group and a topological space with the relativized strong topology.DEFINITION i0.i.The Fell bundle A is (measurably) homogeneous if: (i) A has enough unitary multipliers, i.e. (U(A)) G (ii) The mappings (re,a) ma and (m,a) am of U(A)xA into A are measurable (as in 3.10).
REMARK 10.2.For each x,y in G and m in U (A) the mapping a ma of x into A is a linear isoetry.Therefore, if o is onto (in particular, if xy is homogeneous), then the fibers {Ax:XgG are all isometrically .somorphlc.
LEMMA 10.3.If A has a strong approximate identity and P is ultra-measur- able, then A is homogeneous if and only if A has enough unitary multipliers.
(Compare with Remark 3 of [i,p.49].) PROOF.It follows from 8.5 that (ii) of i0.i is automatically satisfied under the given hypotheses.
REMARK 10.4.In particular, if A is a continuous Fell bundle with approxi- mate identity, then A 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 suf- ficiently 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 shows that all continuously homogeneous Fell bundles can be constructed (up to isomorphism) from a given set of "ingredlent's."We will next extend this construction to the measurable case.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.This field is constructed in the same way as is the field in Section 4 of [13].However, the contexts in which these constructions take place are different in point of view.Consequently, we will also adopt the viewpoint of Section 14 of [13J--namely, that of group repre- sentations.This will allow the results of Section 13 to follow immediately from [13] and this section.
Let A be a Banach *-algebra with (bounded) approximate identity.Let N be a subgroup of the topological group U(A) of unitary multipliers on A Suppose also that H is a topological group extension of N with q:H-H/N the canonical epimorphism on the space of right cosets.Assume also that G=H/N is locally compact in the usual quotient topology.
REMARK 10.5.The local compactness of G is not assumed by Fell in [i].This is essential for us here, since G will play the role of a base space X i.e. all of Section I must apply to G However, this is not a severe additional assump- tion, since Fell assumes G 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 G by defining an equivalence relation N in the space HA Consistent with our stated point of view, observe that the topologicl group N is represented on A by: The projection mapping :A-G is then given by (x,a)~) q(x) agA, xEH For each in G let AS -I().Then A is the disjoint union of the non-empty) fibers A, EG.In this way, we obtain a field {A:eG} of Banach spaces over G The continuity structure underlying the bundle (A, This is obtained from C(H,A) in the following way.For D a "saturated" subset of H i.e.D a union of N-cosets, define CN(D,A) {fEC(D,A) f(tx) Rt(f(x)) xgH tgN} Then CN(D,A) is a linear subspace of C(D,A) For f in CN(D,A) define f~(x) (x,f(x))~J (f(x)) xeH x The mapping f~is constant on cosets and hence, defines a'mapping f~:G+A which is easily seen to be a cross-section.Since f~is also the composition of continuous mappings, it is also continuous, i.e. f-f~is a mapping of CN(D,A) into CS(q(D) ,A).In particular, {f~:feCN(H,A)} c C(A) CS(G,A) LMMA 10.6.The mapping f+f~of CN(H,A) into A is a linear bijection.
Furthermore, if hgA then h=f for the unique f in CN(H,A) given by f(x) J _l(h(q(x))) xeH x Next suppose we are given a mapping of H into the topological group AUtl(A) of isometric *-automorphisms of A having the following "admissibility" properties (i) T is a group homomorphism.
-i (ii) Yt(a) tat R t(Rt(a*))*) aeA teN (iii) Tx(t xtx -I xgH teN where T' is the unique extension of T to x x M(A) defined by: T' (m)rx(S) Tx(ma) -, aeA meM(A) x The *-algebra structure of A is defined by means of T In terms of operator fields, this structure is given by: In view of 10.9, we propose the following: DEFINITION i0.i0.A mapping :H AUtl(A) is measurable if, for each a in A the mapping x-W[ (a) is measurable in the following sense: For each compact X subset K of G and >0 there exists a compact subset Kg of K such that -I (K-K a < e and the mapping x I x(a) is continuous on q (K ,a There is also a stronger notion of measurability as in the case of bundle morphisms.DEFINITION i0.ii.A mapping :H Aut I(A) is ultra-measurable if, for each compact subset K of G and g > 0 there exists a comDact subset Kg of K such -i that (K-Kg) < and T is continuous on q (Kg) to Aut I(A) for the topology of pointwise convergence, i.e.
-I T:q (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.Suppose I is an admissible mapping of H into AUtl(A) Let P and I be as above.
(i) lf is measurable, then the oerator fields P and I are measurable.
(2) If is ,itr-measurable, then P and I are ultra-measurable.
(3) If T is continuous, then P and I are continuous (relative to A ).
It follows from the above that the ingredients (A,N,H,) yield a Fell bundle A when T is admissible and measurable.The obvious next question is whether or not A is homogeneous.For each y in H define: M (x,a) The previous lemma shows that (U(A)) G for the bundle A Consequently, A will be measurably homogeneous if (ii) of i0.I holds.In particular, this will be the case if (i) T is ultra-measurable, and (2) A has a strong approximate identity (10.3). (Note that if {u.} is an approximate identity in A then {(e,uj) ~} is an approximate identity in A .)These appear to be false in general-- unless is continuous (7.5).Hence, it seems unlikely that A is homogeneous in the techncial sense of i0.i.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) with continuous.In particular, the group extension H is chosen to be U(A) This is not possible in the measurable setting, since it is not clear that U(A) is a topological group.Perhaps there is another way of obtaining a measurable generalization of his Theorem 9.1.For example, it may be possible to replace U(A) by its image in U(LI(A,:P,I)) [2,pp.137-139].We refer the reader to Section V of [5] for further information regarding this possibility.
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 similar to each other, but very different from that of Fell.Although their objects 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.
Let G be a locally compact group and A a Banach *-algebra with approximate identity.Let M(A) U(A) and AUtl(A) be as above.The ingredients for the Leptin construction [5] are G,A together with the following: DEFINITION ii.i.A unit factor system (T,W) for (G,A) is a pair of strongly measurable mappings T:G/Aut I (A) W:GG-U(A) satisfying: (i) Wxy,z.T -l(Wx,y) Wx,yz.Wy,z x,y,z in G (multiplication in M(A)).(EA,EA) T e EA, x,y in G, a in A e,x x,e REMARK 11.2.As in the case of T there is a notion of measurability for W,T betwaen strong measurability and strong continuity--namely Bourbaki measur- ability [i0,p.169].This is what we will call ultra-measurability for W,T for obvious reasons.As in i0.12, we have: LEMA 11.3.For W(resp.T) strongly measurable and A separable, W(resp.T) is ultra-measurable.
Let (A,) be the trivial bundle GA over G (product topology)with z(x,a) x xEG aA Let A be the Banach space {x}XA canonically isomor- x phic to A, xSG For each f in C(G,A) let fG be the mapping EGf GG GA Then {fG fgC(G,A)} is a continuity structure A in Ax for which A C(A) CS(G,A) The algebraic structure in A is defined as follows in terms of operator fields: x X PROPOSITION 11.4.(i) The field P (resp.l) is a measurable product (resp.involution) field.
(2) If W,T are ultra-measurable, then so are P,I (3) If W,T are (strongly) continuous, then so are P,I (relative to % ).
The resulting Fell bundle (A,;P,I) will be called a Leptin bundle and will be denoted by (G,A:T,W) PROPOSITON 11.5.Every Lepti bundle has enough unitary multipliers.PROOF.As on p. 329 of [5], for z in G define: .Iz(X,a (zx,Wz,xa) (x'a)lZ (xz'Wx,zT -l(a)) (x,a) e GXA z 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 {(e,uj)} is 3 an approximate identity in A where: (e,uj)(x,a) (x,T _l(uj)a) x and (x,a)(e,uj) (x,auj) (x,a) e A PROPOSITION 11.6.If T is strongly continuous, then {(e,u])} is a stron approximate identity for A PROOF.This can be proved directly or obtained as a consequence (by 7.5) of the following result.PROPOSITION ]1.7.Let (G,A;T,W) be a Leptin bundle.Then: (I) C(A) is locally a right A-module.
(2) If T is strongly continuous, then C(A) is locally a (two-sided) A-module.
COROLLARY 11.8.If A is separable and T is strongly continuous, then the Leptin bundle (G,A;T,W) is homogeneous in the sense of i0.i.PROOF.This follows from i0.3 together with ii.3, ii.4,ii.5, and II.6.
PROPOSITION 11.9.Every Leptin bundle is saturated.PROOF.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) corresponding to the underlying Leptin L 1 bundle is the Banach space (G,A,) with convolution and involution ifor ight for f and g in LI(G,A,) This Banach *-algebra is isometrically isomorphic to our generalized group algebra LI(I\,;P,I) of Section 9 by the mapping fLl (G,A;T,W) f-fG Moreover, by 11.6 and 9.11, LI(G,A'T,W) will have an approximate identity if T 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.Let (A,;P,I) denote the corresponding Fell bundle, where A HA/ and (x,a) Then it is easy to verify that is an invertible, bi-measurable Fell bundle isomorphism.Hence, the algebra isomorphism follows from 9.15.
Let G and A be as in Section ii.The Busby-Smith approach [4] to con- structing a generalized group algebra from LI(G,A,la) requires the following: DEFINITION 12.1.A twisting (S,V) for (G,A) is a pair of strongly x,e e,x (EA' ,y,zgG aeA LEMMA 12.2.For V(resp.S) strongly measurable and A separable, V(resp.S) is ultra-measurable, i.e.Bourbaki measurable (recall 11.2).(2) since the mapping x S (h(x)) is continuous at x for h in C(G,A) if x o and only if h is continuous at x o COROLLARY 12.7.The Fell bundles (G,A;T,W) and (G,A;S,V) are homomor- phically isomorphic if T (equivalently S) is strongly continuous.(Note that the 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.
The twisted group algebra LI(G,A;S,V) corresponding to the underlying Busby- Smith bundle is the Banach space LI(G,A,) with product and involution (for right and hence, to our algebra LI(A,;P,I) by the same mapping.Moreover, by 9.11, 11.6, and 12.5, LI(G,A;S,V) will have an approximate identity if S is strongly continuous.
EXAMPLE 12.9 [5,p.330].Let A be LI() and G the circle group identified with [0,i] under addition modulo i. Define Sx Tx E A xgG and V,W:GxG U(A) by and (S,V) is a twisting pair (equivalently, (T,W) is x,y x,y a unitary factor system) for (G,A) The mapping V is not strongly continuous at any point (x,y) in GxG where x+y 1 However, V is ultra-measurable ll.2).
In the next section (13.2), we will see that LI(G,A;S,V) is isometrically isomor- phic to the group algebra LlR) 13. GROUP EXTENSIONS.
Suppose N is a closed normal subgroup of the locally compact group H with G H/N (right cosets) and q H-G the quotient mapping.Let 0, be right Haar measures for N,H respectively.Let be the right Haar measure on G defined [15] by IHf(X)dg(x) fG /N f(tx)dO(t)d(q(t)) for f in Cc(G) Let 6H G 6N be the respective modular functions for H,G,N Of course, 6HIN 6N Fell then verifies directly that the cross-section algebra LI(A,;', *) is isometrically isomorphic to the group algebra LI(H,)) In this section, we will accomplish these two tasks in very different ways.We will show that the mapping arises naturally from a certain induction of Banach space representations ( [16,3] and [13,4]).As a consequence of this approach, it will then be automatic that the previous Ll-spaces are isometrically isomorphic.

L I L I *)
Transferring the *-algebra structure from (H,) to (A,;', Let G,H,N be as above with E {e} and iE(e) i Then i E is both the L I identity representation of E and the regular representation of E on (E) The is a continuous, open surjection and: (i) The function a a is continuous on A (ii) The operation + is continuous from {(a,b)eAxA (a).= (b)} into A(iii) For each 8 in the mapping on A given by a 8a is continuous.
b) x (x,b) e B We then have the following commutative diagram: mapping is the projection (x,b) b (Also see [2, p.101] in this regard.)The set B is (roughly) the disjoint union of the field {B(x) to T-l(x) c_ and T IB EXAMPLE 2.4.If :X/Y is a homemorphlsm, then B B and r essentially.Of course, C(A) -+ C(A)

(
1.A Banach bundle morphism from (A,w) into (B,T) is a bundle morphism (See the Appendix for the definition and required properties of such spaces of bound- xi)II:xigXi} l<i_<n PROPOSITION 3.5.For (,) as in 32, we have: necessarily a product field.Moreover, let p denote the mapping of AxAA defined by p(a,b) Pv(a),w(b)(a'b) a,bEA

THEOREM 4 .
4. Let PEPThen the following are equivalent: v (a) (a) -I x for aeA,v. (x A -i xEG x

THEOREM 5 . 4 .
The pping i is continuous at eac point a ine A o PROOF.Combine 1.8 3.2, 3.3 3.4,nd 3.8.PROPOSITION 5.3.Let DG xeD Suppose I in [ is continuous at x If fEHA is A-continuous at x then the mapping l(f-) :D+A is continuous at Y x (resp.AV-continuous at x PROOF.Recall 2.2 and 3.9.Let le[ Then the following are euqivalent (i) I is measurable.
) a* aA x bAy,x,yEG

4 .
Then af fa belong to DAx which is a linear space, i.e.DAx is a (two- sided) A-module in general.Now consider C(D A) c DAx ePROPOSITION 7.3.Let F be a total subset of C(A)Then the following are equivalent (i) C(D,A) is a left (resp.right) A-module.e (ii) For each h in F We say that c(A) is locally an A "-module if C(K,A) is an e A -module, for each compact subset K of G e an involution in M(A) given by (m-ml)a m(m'a) a(m-m') (am)re' in M(A)The unitary multipliers U(A) in M(A) form a group under multiplication.Also, if Ux(A) u(A)OMx( aEA tEN Such R is a bounded, strongly continuous representation of N on A which we call the right regular r.epresentation.If (x,a) and (y,b) are elements of the space (HxA)/õ f equivalence classes (x,a)" by A equipped with the quotient topology from HA Observe that A can be viewed as the orbit space of HXA under the right topological transformation group (HXA)N HXA where ((x,a),t) (tx,Rt(a)) agA, tEN xgH well-defined) bijection, since x Jtx(a) Jx(Rt(a)) tgN agA xgH Therefore, we may transfer the Banach space structure of A to A by such J Specifically, (x,a) + (x,b) (x,a + b) .(x,a)(x,a) ll(x,a)~ll llall ,a,bgA xgH where P N((x a) (y,b) ~) (xy a'T (b)) I((x,a) ~) (x-l,r -l(a)*) , a beA xe y] x LFLMA 10.7.The fields P and I are product and involution fields respec- t ive ly.EXAMPLE 10.8.(One-dimensional fibers [i,p.75])If A is the complex numbers then N U(A) is the circle group.Also, T must be trivial, since AUtl(E) is trivial.Thus, for H as above, we have: CN(H,A) {fgC(H) f(tx) f(x) tN xgH} P,n((x,a) ~,(y,b) ~) (xy,ab) l((x,a) ~) (x-l,) 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.Let T:H Aut I(A) The following are equivalent for x in H (i) T is strongly continuous at x o (ii) The_ mapping of HxA into A given by (x,a) T (a) is continuous on x {x }A o (iii) For each f in C(H,A) the mapping of HxH into A given by (x,y) Tx(f(y) is continuous on {x }H o (iv) For each a in A the mapping of H into A given by x (X,lx(a))~o is continuous at x o (v) The mapping of HA into A given by (x a) the above conditions, for each f,g in C(H,A) the mapping of HH into A given bv (x y) f(x)'Y .g(y) is continuous on {x }H X O PROOF.All implications are straightforward with the exception of "(iv) implies (i)" which follows from Lemma 9.1 of [l,p.70].
10.14.For each y in H M is a unitary multiplier of A i.e.Y M U(A) Y PROOF.See p. 71 of [i].
is locally compact (as before) and that there exists a a Leptin bundle (G,A;T,W)[5,p.333].Leinert also shows that the group algebras LI(A,;P,I) LI(A,;P,I) and LI(G,A;T,W) are isometrically isomorphic.This can also be concluded from the following.
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 ))* xgG f*(x)for f and g in LI(G,A,) The algebra LI(G,A;S,V) is isomorphic to the algebra LI(G,A;T,W) (9.13) by the mapping f [x S-l(f(x))] fELl(G,A;S,V) On p.77 of [i], Fell shows how to construct ingredients (A,N,H,T) for a homogeneous Banach *-algebraic bundle (A,n;-,*) called the (H N) -group extension bundle.Specifically, A LI(N,D) N is identified with a subgroup of U() d0(xtx-l)/dO(t) xeH teN aeA (i.e. to LI(A,;P,I)) then completes the picture.
To obtain continuity for the product mapping p:AxA/A recall 4. i PROPOSITION 6.2.The product in A is continuous if and only if P is continuous.
The involution in A is continuous if and only if I is cont inuous.PROPOSITION 6.6.The involution in A is measurable if and only if I is measurable.It is ultra-measurable if and only if I is.If A is countably 7.2 [i,p.34].The net {u.} in A is a strong approximate