MAXIMAL IDEALS IN ALGEBRAS OF VECTOR-VALUED FUNCTIONS

Subsuming recent results of the authors [6,7] and J. Arhippainen [l], we investigate 
further the structure and properties of the maximal ideal spaces of algebras of vector-valued functions.


INTRODUCTION
One way to create new topological algebras from old is to look at algebras ,A of functions from a space X which take their values in topological algebras Ax (z E X).If X is itself a topological space (or sometimes even if it is not), these algebras .A can be topologized in various ways.It is natural to ask how the ideal structure of .A is related to the ideal structures of the A The history of this question dates back at least to 1960 and C. Rickart's book [9] and to 1961 and the paper of J M. G Fell [2].Among many other results, this latter paper identified the space of irreducible *-representations of section spaces of bundles of C*-algebras The topological algebras of these sources were commutative Banach algebras with identities and C*-algebras, respectively.Among the more recent studies examining the relationships between the ideal structure of 4 and the ideal structures of the Ax are the papers by J. Arhippainen ], who looked at commutative locally multiplicatively convex A:, and by the authors ([6] and [7]), for whom the A were commutative Banach algebras and arbitrary Banach algebras, respectively The references in these papers provide a guide to some of the record.The purpose of this note is to investigate further the structure and properties of the maximal ideal spaces of algebras of vector-valued functions In it, we subsume results of our own and of J Arhippainen in the works noted above by using the theory of bundles of locally convex topological vector spaces denote by Ct,(X) the space of bounded and continuous complex-valued functions on X Let /A, "x E X) be a family of non-trivial commutative locally multiplicatively convex (lmc) algebras ndexed by X Let A be the disjoint umon O{A, x E X) of algebras (which can, if we like, be thought of as the set ,, x({Xt A, )), and let 7r A X be the natural surjection Assume further that we have on the fibered space A a family of seminorms {,, .9}such that, for each x (where ,' is the restriction of ,, to AT) is a family of submultiplicative seminorms which generates the topology on A, Assume, finally, that we have an algebra .A of selections (--choice functions) (7:X A such that 1) for each x X, ev,(.A) {(7(x) (7 E .A} A (in this case,.A is said to be full).
2) .,4 is a C (X )-module.and 3) for each (7 ,4 and for each .5, the numerical function x u,((7(x)) is upper semicontinuous on X Before going farther, we point out two special cases of this situation If X is compact, and if each A is a commutative Banach algebra (and the set A is a singleton), then we have the situation in [6] On the other hand, if B is a commutative lmc algebra, and if.,4 C(X, B) is the algebra of all continuous B-valued functions on X (so that A B for all x X), then we have the situation described in [1] Returning now to the general situation, we make .A into a commutative lmc algebra First, we select a compact cover of X which is closed under finite unions For each K ' and E .,qwe define a seminorm pr, on .A by pr,((7)= sup v((7(z)) Then the PK., are easily seen to be submultiplicative, so that they generate an lmc topology on ., 4 The sets V(,K,,) {-A: .(--)< } form a subbasic system of neighborhoods of a ,4 as K ,, E J, and every e > 0 vary Note that different choices of covers 6may lead to different topologies on ., 4 In the constant fiber case .A C(X, B), described above, we can let be the family of all compact subsets of X. in which case has the compact-open topology (the topology of uniform convergence on compact subsets of X) If, at the other extreme, we let d'be the family of finite subsets of X, then .At has the topology of pointwise convergence on X In the general case, we note further that since A with the given topology is an lmc algebra, the multiplication on ,4 is (jointly) continuous in the topology given by the seminorms Pg., (see [8]) Moreover, if we endow Cb(X) with the sup norm topology, it is easily seen that the module multiplication (f,a)fa from C,(X) A to A is also jointly continuous, so that A is in fact a topological Cb (X)-module For a subset J c ,A and K let JIK {alK a J}, where crlK denotes the restriction of a to K. Denote the restriction map by restr A PROPOSITION 1. Suppose that J C AlS an 1deal in A which is also a Cb(X)-module of Then J K is an ideal in A IK which is also a C(K)-module.PROOF.Evidently, JIK is an ideal in .AIK Let a J, and let f C(K) We may extend f to f* C(X), see [4, p 90] Then restK(f*a) restK(f*) restK(( 7) f (crlK) .ILK, since f* a E J IZI !-11"! PROPOSITION 2. Suppose thai J C '4 t.s" a Ct,(X)-.submodule and a closed proper Meal.lhen there exlsI.sz E X such thai ev, (J) J, Is" a chzs'edproper Meal m A,.
PROOF.Fix K E 3(, and consider .,41KThis is a space of choice functions over K, whose seminorm functions :c u; (a(x))(a .4,.,0)are then upper semicontinuous over K by restriction, and hence bounded on K By [3, Theorem 5 9, p 49], there is a bundle 7rc AK--* K of lmc topological algebras such that 1-'(TrK) '41K, the topology on AIK is generated by the PK, Suppose now that for each zX, we have J =A,, and let oE'4 We will show that every neighborhood V of contains an element 7-J Since J is closed, this will show that J, contrary to the assumption that J is a proper ideal in '4 We may assume that V is of the form V ( V(,K, ip,), p where the z's are indices in .;From the preceding, JIK is a C(K)-submodule of'4lK F(zrh-) such that ev (JIK) is dense in each A (x E K) Then, using [3, Theorem 4 2, p 39], J[K is dense in By the definition of the topology on '41K, this means that there is a 7-E J such that pr.,;(r 7-) < e for p 1 n But this says precisely that 7-E V E]VIV!PROPOSITION 3. Suppose that H" .At C s a non-trivial contmuous multphcatve homomorphsm; set J ker H. ]hen there exists x X such that J is a proper ideal In Az.
PROOF.It suffices to show that J is a Cb(X)-submodule of'4 If it is not, we may choose a J and f Cb(X) such that f J Since J is in any event an ideal, we have (re)2 (fa) j But H((fa)2) [H(f)] # 0, a contradiction I-il-II-I PROPOSITION 4. Let /x,('4) be the Gelfand space of .At space of non-trivial continuous homomorphsms H" .,4C).If H /x,('4), then there exist x X, h /k(Az) such that H h o ev.
PROOF.Let H E/k(A), set J ker H, and choose x E X such that J is a proper ideal in Thus, A_ :/: 0 Since ev .4 A maps J into J, there is a unique linear map $which makes the diagram commute, where 7r and r: are the natural surjections Since ev .A A is surjective, the induced map --# is also surjective Thus, q5 maps the one-dimensional space surjectively onto the non- @ It follows that is one-dimensional, which means that is a closed regular maximal zero space ideal in A Hence, ker h for some h /k (A).The map h o ev .,4C is clearly a non-trivial algebra homomorphism.If a J, then eva(a) ,/ ker h, so (h o ev)(a) 0 Hence ker H J c ker(h o ev).Because ker H and ker(h o ev) are closed maximal ideals, it follows that ker H ker(h o ev), and hence that H h o ev DE]V] COROLLARY 5.Under the situatton as described, we may tdenttfy ZS('4) as a point set wth the dtsjoint umon of the /X(A). (For bookkeeping purposes, we may also write /(.4) Ux({:} /X(A)).) PROOF.Since ev '4 A is continuous, it follows that, if z X and h /X,(A), then h o ev /x,('4) By using the same method as in the proof of [6, Proposition 6], it may be shown that the map Uex({X } x A(A)) -A(.4), (x,h) h o ev, H .552 s a bljectlon Vll-IVl In all the above, we need to call on the result for lmc algebras which corresponds to that tbr Banach algebras namely, in a commutative Imc algebra/3, there s a one-to-one correspondence between the set of continuous non-trivial homomorphsms from B to C and the set of closed regular maximal ideals in B, see [8, Corollaries 7 l, 7 2, pp 71-72] 3. TOPOLOGICAL CONSIDERATIONS So, under the circumstances described, we have a fibering of &(AI by X For II AIAi, we may write h o ev, for some (unique) x (5 X and h (5/k(A, Let p A(A) X be the obvious projection map, Hh oev, x PROPOSITION 6. /he proleclon map p ." continuous when A(.A PROOF.It suffices to show that whenever {H,}-{h,, o ev, } is a net in /",(.A) such that H,, h, ev, H h ev,, we have f(:r,,) f(:r) for each f (5 C,(X), because when X ts completely regular and Hausdorffis topology is determined by C(X), see [4, p 40] Suppose now that f C,(X) and that cr (5 .,4,with H(cr) h(cr(x)) ?6 0 Snce for (5 A. and since ho o evo h o ev, weak-" in/", (.,4), we have h,([fcr](z,,)) h,,(f(z,,)a(z,)) f(z,,)h(a(z,,)) h([fcr](z)) f(z)h(cr(z)).
Since h,(cr(z,,)) h(cr(:r.))# O, it follows that f(z,) f(z) Since f (5 C(X) was arbitrary, we ave the desired result I-I1-11--i On the other hand, we can look at how A(A embeds into PROPOSITION 7. Gtve A(A) tts weak topology and, fi)r each x X, gve A(A.)tts weak-" topology.Then A(A embeds homeomorphcally into A(A).PROOF.Fix z X Evidently, the map % A(A.) X(A), h h o ev., is one-to-one if ht h,2, then we may choose a (5 A such that h(a) h2(a), and use the fullness of ..4 to choose cr (5 .,4such that or(z) a It is then clear that (h oev)(cr) # (h2 oev)(cr) Now, suppose that we have a net h, C A(A such that h, h ZX(A) when A(A) is given its weak-" topology Let o-(5 ., 4 We then have (ho oev)(cr) ho(cr(z)) h(cr(:r)) (h oev)(cr), ie %(h,) hoev in A(.A).It is likewise easy to show that if {h, oev} is a net in %(A(A:)) which converges weak-" to h o ev (5 %(A(A:)), then ho h weak-" in A(A) I-'11"-I1"-!Previous work of the authors [6] has provided examples which demonstrate that the projection map need not be closed, even when each fiber A is a Banach algebra with identity Moreover, the projection need not be open, even when each fiber A is a Banach algebra with identity and .,4satisfies the even stronger condition that it contain the identity selection Both of these examples use the weak-" topologies Suppose now that we re-examine the situation when each A: is a commutative Banach algebra and X is compact Under these special conditions, .,4 is the space of sections of a bundle of Banach algebras n-" A--, X We may look at the Seda topology on .M I,.J:x({:r} x A(A))= [3exA (A) Recall from the Banach bundle case that the Seda topology is the weak topology on (R) Uex({Z} x B((A.)')) (where B(Z) denotes the closed unit ball of a Banach space Z) which is generated by the conditions (:c,,F,) (z,F) (5 .Ad iff z z (5 X and Fo(cr(z,)) F(cr(:r)) for each cr (5 .,4It is shown elsewhere that (R) is compact in the Seda topology (See [10] and [5] for more information about this topology PROPOSITION $.Let X be a compact Hausdorff space, and suppose that ,4 ['(rc) ts the space of sections of the bundle of commutatn,e Banach algebras 7r" A X. Then the weak-" topology on A(A) and the (relative) Seda topology on .M are homeomorphc.PROOF.As above, for H E/k(.A), write H h o ev, for some x E X and h /k(A, The map H (x,h s a bjecton If H,, h, oev,.H h oev, weak-" in/A(A),thissayspreciselythat Ha(o) h.(cr(x.))H(a): h((x)) for each cr A, above we have shown that x, x Thus, (x,, h.) (x, h) in the Seda topology The other direction is clear l-li-II-I We may also consider the continuity of the projection map and the embeddings when /k(.A) and /k(A, are endowed with their hull-kernel topologies PROPOSITION 9. l/nder the given general circumstances, suppose lhal /(.,4 1s gtven its hull- kernel topology, attd that each /(A, )(x X) ts given its hull-kernel topology, lhen the prolectton map p /(A X and the embeddmgs of the/(Ar) into/(A) are continuous.
PROOF.To show that the natural projection p'/(.A)-X is continuous in the hull-kernel topology, let H, hoo evo be a net in/(A) with ho o eVo h o ev H /(.A) in the hull- kernel topology We claim that :r, x If not, we may then choose an open neighborhood N of x and a subnet {x,, of {x, such that :r,), N Choose a A such that h(a) O, and choose a' .,4such that a'(x) ev(a') a Since X is completely regular, we may choose a function f Cb(X) with f(X) C [0, 1] and with f(x) 1 and f(X\N)=O Set or=for' Since h,, oeV:o,--h oev, we have P= ["lo, ker(ho, oev,)cker(h oev) Since cr(zo,)-0 for all a', we have cr E P C ker(hoev:) But this is a contradiction, since (h o ev)(a) h(cr(x)) h(a) Now, fix x X For the second part, it suffices to show that for a set W C/(A), and for h /(A), we have h in the hull-kernel closure of W iff H h o ev is in the hull-kernel closure of -(w) {h'o .h' e W} Suppose, then, that h is in the hull-kernel closure of W in/k(A) Then {kerh" h' c= W} c kerh, we claim that ["]ker {h' oev h' E W} C ker(h oev) So, let a ,A be such that a ker(h' oev) for each h'W Then h'(a(x))=O for each h'W, e a(x)kerh' for all h'EW, so that a(x) ker h Hence, a ker(h o ev) A proof of the reverse inclusion, which uses the fullness of A, is equally straightforward We note that these are essentially the proofs used in [7, Propositions 17, 18] ill-IV] Recall (see [8, p 332]) that a topological algebra B is said to be regular provided that any weak-" closed subset W of/k(B) and point of/k(B) disjoint from it may be separated by an element of B It happens that B is regular iff the weak-* and hull-kernel topologies coincide on/X,(B) PROPOSITION 10.Suppose that we are given the general data on .A, as above.If .A ts a regular algebra, then so s each A.
PROOF.Choose x X We know that /(.A) contains a homeomorphic copy of/'x(A) in the weak-* topology, in particular, {x} W p-(W) is weak*-closed in A(A) whenever W is a weak-* closed in /(A), where p'/(dt) /(A) is the continuous projection map.Hence, if h /(Ax)\W, then (x,h) /(.A)\p-l(W), and so there exists a .A which separates (x,h) and p-l(W) Then it is evident that a(x) A separates h and W in/(A) I'11"-I1-1 Now, if x E X, and if I c A is an ideal, set A(x,I) {a .,4a(x) I} It is easy to see that .A(x, I) is always a closed proper ideal in ,4 whenever I is a closed proper ideal of A (In fact, .A(x, I) is also a closed Cb(X)-submodule of.A when I is closed PROPOSITION 11.Let J c .,4 be a closed ideal which ts also a Cb(X)-submodule of .A. Then J NxX(:, ), PROOF.Clearly, S C Nx.a(, To show the reverse inclusion, we use a partition of unity argument similar to that of Theorem 8 of [1 Let cr J' To show that cr J, it suffices to show that for K Off, 3, and > 0 there is 7-J such that PK,, (or 7-) Fix K, t, and t, and let x E K be arbitrary Then o(x) E J-, and so there exists a' E J such that u,' (o(x) o'(x)) < since the seminorm functions x' u," (o(x') '(x')) is upper semicontinuous, there is a neighborhood U, ofx such that when x' E U, we have u[' (a(x') o'(x')) < Since K is compact, we may choose a cover Us U., of K, with corresponding o',..., ap J such that u;'(o(x') (x')) < whenever :r' C U,r(r-,p) Now, {UT]K "r= p} is an open cover of the compact Hausdorff space K, and so there is a partition of umty {f, "r-1,...,p} cC(K) subordinate to {u,rlK} In particular, 0_ f,(x)_ l(xCK,.P supp(f, C Ur]K for r 1 p, and fr(x) 1 for x E K As in Proposition l, we may extend f, to f Cb(X) Then "r fo'r J, and it is easy to check that p-.,(o "r) < l-Ii-ll-I COROLLARY 12. Suppose that A has an tdentity e, and let J C ,4 be a closed ideal."lhen N, .-(,).
PROOF.It suffices to note that J is a Cb(X)-submodule of A Let f Cb(X) and a E J Then f o f(eo) (fe)o J [-lOi-I COROLLARY 13.Let J C 4 be a closed proper ideal, and let (J) denote the closed C(X)- .bmodulem 4 generated by J. Then (J) x,A(x,-ff-).
PROOF.This follows immediately from the method of proof in Proposition 11 I'-!i-11-1 We point out in closing the crucial role which the assumptions on the space X play Complete rgularity of X allows us to extend the functions appearing in the proofs of Propositions and 11, and provides sufficiently many continuous functions to demonstrate the continuity of the projection map p-A(4) X in Propositions 6 and 9.That X is Hausdorff means that each K f is a compact Hausdorff space, and allows us to use the full power of the cited theorems from [3] in the proof of Proposition 2