Continuous trace C*-algebras, gauge groups and rationalization

CONTINUOUS TRA CE C ∗ -ALGEBRAS, GA UGE GR OUPS AND RA TIONALIZA TION JOHN R. KLEIN, CLA UDE L. SCHOCHET, A N D SAMUEL B. SMITH Abstract. Let ζ b e an n -dimensional complex matrix bundle ov er a compact metric space X and let A ζ denote the C ∗ -algebra of s ections of this bundle. W e determine the rational homotop y t ype as an H - space of U A ζ , the group of unitaries of A ζ . The answer turns out to b e indep enden t of the bundle ζ and depends only upon n and the rational cohomology of X . W e prov e analogous results for the gauge group and the pro jectiv e gauge group of a principal bundle o ver a compact metric space X . Contents 1. Int ro duction 1 2. Conv en tions 5 3. Section spaces 7 4. Rationalization of top ologica l groups 9 5. Preliminary results: finite complexes 12 6. Limits and function spaces 16 7. Lo calization of function spaces revisited 19 8. Pro of of the main results 20 9. Appendix: on the free loop space 22 References 23 1. In tr oduction W e analyze the rational homoto py theor y of certain top olog ic al groups arising from bundles over a compact metr ic space X . Our res ults are motiv ated by the following situation. Let U n be the unitary gr oup of n × n matrices, and let P U n be the g roup given b y taking the quotient of U n with its ce nter. Let ζ : T → X b e a principal P U n -bundle ov er X , let P U n act on M n = M n ( C ) by conjugation and let T × P U n M n → X be the a sso ciated n -dimensional complex matr ix bundle. Define A ζ to b e the s et of contin uous s ections of the la tter. Thes e sections hav e natural p oint wise addition, m ultiplication, and ∗ -op er ations and give A ζ the s tr ucture o f a unital C ∗ -algebra . The algebra A ζ is called an n - homo g ene ous C ∗ -algebr a and is the most general 2000 Mathematics Subje ct Classific ation. 46J 05, 46L85, 55P62, 54C35, 55P15, 55P45. Key wor ds and phr a ses. cont inuous trace C ∗ -algebra, section space, gauge group, pro jective gauge group, rational H -space, top ological group, lo calization. The first author is partially supp orted b y the National Science F oundation. 1 2 KLEIN, SCHOCHET, AND S M ITH unital c ontinuous tr ac e C ∗ -algebra as studied, for instance, in the bo ok of Raebur n and Williams [1 6]. Let U A ζ denote the top olo g ical g roup of unitaries of A ζ . Our first main result describ es the rational homotopy t yp e of U A ζ . Recall that, from the po int of view of homotopy theory , the simplest g r oups a r e the Eilenberg-Ma c Lane spaces K ( π , n ) with m ultiplication given by K ( π, n ) × K ( π, n ) ≃ K ( π × π, n ) K ( multipl y) − − − − − − − − → K ( π, n ) . Here π is a n ab elian gro up and the space K ( π , n ) satisfies π i ( K ( π , n )) = π for i = n and π i ( K ( π , n )) = 0 for i 6 = n. Only some o f the constructions of a K ( π , n ) y ie ld a bo na fide topolog ical group, but all yield an H -space; that is, a spa ce with co nt in- uous binary o per ation and tw o s ide d unit. How ever, this discr epancy is not har d to rec tify: up to ho motopy all of these H -space struc tur es on E ilenberg-Ma c L ane spaces lift to topo logical gro up structures in the sense that there is a top olog ical group G a nd a homo to py equiv alence to the g iven K ( π , n ) which preserves the m ultiplication up to homotopy . In fact, the H -spa ce structur e o n a g iven Eilenberg -Mac Lane space is unique up to m ultiplicative equiv alence and is homotopy c ommutative . A pro duct Q j ≥ 1 K ( π j , j ) of Eilen b erg- Ma c Lane spaces also has a preferre d H -space structure given b y the pro duct of the structures on the factors. This structure, which w e refer to as the standar d multiplic ation, is also homotopy comm utative. How e ver, in this case this structure may not b e unique (See [4]). Given a simply co nnected CW space X , Sulliv an c onstructed a rationaliza tio n map X → X Q which has the pro p erty that the as so ciated homomorphism on the higher homotopy groups is given b y tensor ing with the rationa l num b ers ([21]; rationaliza tion is a sp e c ial case o f a mor e ge ner al co nstruction, lo ca lization, that can b e made for an y set of primes). Later, this theory was e x tended to include nilpo tent s paces, i.e., spaces with non-tr ivial nilp otent fundamental group π having the prop er ty that the higher homotopy g roups ar e nilpo tent mo dules over π ([9], [2]). It is well-known that top ologica l groups are nilpotent spa c es, so one ca n consider the r ationalizatio n map G → G Q for connected top ologica l gro ups G (whose un- derlying space is a CW complex). Since lo ca lization co mmut es with finite pr o ducts up to homoto py , it follows that G Q has the structure of a n H -spa ce, and further- more, the ratio na lization map is a n H -map, i.e., it preser ves multip lications up to homotopy . This motiv a tes the following: let us ca ll t wo H -spaces X and Y r atio- nal ly H -e quivalent if there is a homo to py equiv a lence X Q → Y Q which is a map of H -spaces. T o state our ca lculation of the ratio na l homotopy groups of U A ζ , we int ro duce some notation. Given Z -graded vector s pa ces V and W , we grade the tensor pr o duct V ⊗ W by declaring that v ⊗ w has degree | v | + | w | . Here | v | deno tes the degre e of the element v ∈ V . Let V e ⊗ W be the e ffect of consider ing only tensors with non-negative gr ading. Given elements x 1 , . . . , x n each of homoge neous degree , write Q ( x 1 , . . . , x n ) for the gra ded vector space with basis x 1 , . . . , x n . Given a to po logical gro up G , wr ite G ◦ for the path comp onent of the iden tity in G . Let ˇ H ∗ ( X ; Q ) denote the ˇ Cech cohomolog y o f a space X with r ational co efficients gr ade d nonp ositively s o that x ∈ ˇ H n ( X ; Q ) has degr ee − n . CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 3 Theorem A. L et ζ b e a pri ncip al P U n bund le over a c omp act metric sp ac e X . L et A ζ b e the asso ciate d C ∗ -algebr a, and let U A ζ b e its gr oup of u n itaries. Then the r ationalization of ( U A ζ ) ◦ is r ational ly H -e quivalent to a pr o duct of r ational Eilenb er g-Mac L ane sp ac es with the standar d multiplic ation, with de gr e es and di- mensions c orr esp onding to an isomorphism of gr ade d ve ctor sp ac es π ∗ (( U A ζ ) ◦ ) ⊗ Q ∼ = ˇ H ∗ ( X ; Q ) e ⊗ Q ( s 1 , . . . , s n ) , wher e the b asis element s i has de gr e e 2 i − 1 . Theorem A is a sp ecial cas e of mor e genera l calculations of the r ational homo topy theory of gauge gro ups whic h we now des crib e. W rite F ( X, Y ) for the (function) space of all co ntin uous maps from X to Y . When G is a top olog ical g roup, the space F ( X, G ) is o ne also with mult iplication of functions taken p oint wise. In this case, the iden tit y comp onent F ( X , G ) ◦ is the space of fr e ely nullhomotopic ma ps. Theorem B. L et X b e a c omp act metric sp ac e and let G b e a c onne cte d top olo gic al gr oup having the homotopy typ e of a finite CW c omplex. Then π ∗ ( F ( X, G ) ◦ ) ⊗ Q ∼ = ˇ H ∗ ( X ; Q ) e ⊗ ( π ∗ ( G ) ⊗ Q ) . F ur t hermor e, F ( X , G ) ◦ is ra tional ly H -e quivalent to a pr o duct of Eilenb er g-Mac L ane sp ac es with the standar d multiplic ation, with de gr e es and dimensions c orr esp onding to the display e d isomorphism. When X is a finite complex, Theorem B is a co nsequence of results of Thom [22] a nd a basic lo calization result for c o mp o nents of F ( X, Y ) due to Hilton, Mislin and Roitb erg [9]. The r e s ult for X finite in this case is descr ib ed in [12, § 4]. Our adv ance here is the extension of this result to the case when X is compact metric. W e deduce Theorem B from an extension o f the Hilton-Mislin-Ro itber g r esult to the case X compact metric (Theorem 7.1). Addendum C. In The or em B, the c alculation of r ational homotopy gr oups holds for any c onn e cte d, gr oup-like H- sp ac e G . F ur t hermor e, if G is r ational ly homotopy c ommutative, then F ( X , G ) is r ational ly H -e quivalent to a pr o duct of Eilenb er g- Mac L ane sp ac es with the standar d multiplic ation. The main results o f this paper concern extending Theorem B to spac e s of s ections of certain bundles. Let G be a top ologica l g roup and let ζ : T → X be a principa l G -bundle. F ollowing [1, § 2], we form the a sso ciated adjoi nt bund le Ad( ζ ) : T × G G ad → X where G ac ts on G ad = G by conjuga tion. The gauge gr ou p G ( ζ ) of ζ is the space of sections of Ad( ζ ), with group structure defined by p oint wise m ultiplicatio n of sections. Alternatively , G ( ζ ) is the gr oup of G -eq uiv ariant bundle automorphis ms of ζ that cover the identit y ma p of X . Theorem D. L et G b e a c onne cte d top olo gic al gr oup having the homotopy typ e of a finite CW c omplex. L et ζ b e a princip al G -bund le over a c omp act metr ic sp ac e X . Then ther e is a r ational H -e quivalenc e G ( ζ ) ◦ ≃ Q F ( X, G ) ◦ . 4 KLEIN, SCHOCHET, AND S M ITH Conse quently, G ( ζ ) ◦ is r ational ly homotopy c ommutative with r ational homotopy gr oups given by the isomorphism app e aring in The or em B . Again, when X is a finite CW complex this result admits a direct proo f. In this case, a result of Go ttlieb gives a multiplicativ e equiv alence G ( ζ ) ≃ Ω h ζ F ( X, B G ) , where the right side denotes the lo op spa c e of F ( X, B G ) based at h ζ : X → B G , the classifying ma p for ζ (see Coro llary 9.2, [8, th. 1] a nd [1, pro p. 2.4]). The equiv alence G ( ζ ) ≃ Q F ( X, G ) then follows from the Hilton-Mislin-Roitb erg lo cal- ization result for function spaces mentioned above and basic rational homotopy theory . (See Theorem 5.6 b elow.) The res ult in this case was recently , indep en- dent ly obtained by F´ elix and O pr ea at the level of rationa l homotopy groups [7, th. 3.1]. Another related result here is due to Cr abb and Sutherla nd, who prove the fibrewise rationalization of the bundle Ad( ζ G ) is fibre homotopically tr ivial, where ζ G is the universal G -bundle [3, pr op. 2.2]. The follo wing shows that the homotopy finiteness assumption on G in Theorem D can sometimes be disp ensed with. Addendum E. Assume G is a top olo gic al gr oup such that B G has the r ational homotopy typ e of a gr oup-like H -sp ac e. Then the c onclusion of The or em D holds for such G . F or example, if G is a connected topolog ical gr oup satisfying rational Bott p eri- o dicity , then B G has the rationa l homotop y t yp e of a group-like H -space. Let P G = G/ Z ( G ) denote the pro jectivization of G ; i.e., the quotien t of G by its center. As the center acts trivially on G ad , one obtains an actio n o f P G on G ad . Given a principal P G -bundle ζ : T → X , form the asso ciated pr oje ctive adjoi nt bundle Pad( ζ ) : T × P G G ad → X . Define P ( ζ ) to b e the to po logical gro up o f sections of the bundle Pad( ζ ) with po int wise multiplication again induced by G ad . W e ca ll P ( ζ ) the pr oje ct ive gauge gr oup of ζ . In Example 3.7 below, we observe that U A ζ ∼ = P ( ζ ) corresp onds to the pro jective adjoint bundle of a principal P U n -bundle. Theorem A is thus a sp ecia l case of the follo wing result. Theorem F. L et G b e a c omp act c onne cte d Lie gr oup. L et ζ b e a princip al P G - bund le over a c omp act met ric sp ac e X . Then t her e is a r ational H - e quivalenc e P ( ζ ) ◦ ≃ Q F ( X, G ) ◦ . Thus P ( ζ ) ◦ is r ational ly ho motopy c ommutative with r ational homotopy gr oups again given by t he isomorphism app e aring in The or em B. Remark 1.1. Suppose that C is a separable C ∗ -algebra . Then its unitary gr oup U C (with the usual mo dification for non-unital C ) has the homotopy type of a countable CW complex. Thus so to o do es U ∞ C = lim − → U n C , a nd the latter is an infinite lo op space, by the Bott Perio dicity Theo rem of R. W o o d [24]. Th us U ∞ C CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 5 satisfies the c onditions on G in Addenda C and E. The same is true for U C itself if C is stable. So our results also apply to C ∗ -algebra s constr ucted simila rly to A ζ but where the initial fibre M n ( C ) is replaced by an appropriate C ∗ -algebra C . W e develop these ideas in a subsequent pap er. The pap er is or g anized a s follows. In Section 2, we establish our basic conv en- tions for spaces, groups a nd bundles. In Section 3, we prove v a rious foundational prop erties of section spaces. In Section 4, we discuss the rationa lization of topo - logical gr oups and the obs truction to homoto py commutativit y . In Sectio n 5, we prov e preliminary versions of the main theorems for X a finite CW complex, a s men tioned ab ov e. By a r esult of Eilenber g and Steenr o d [6], a compact metric s pace X may b e expressed as the inverse limit lim ← − j X j of finite CW complexes. In Section 6 , we use this result and the cla s sical works of Dowker [5] and Spanier [1 9] to identify the homotopy gr oups of the function space F ( X, Y ) in terms of the ho mo topy gro ups of the approximating function spaces F ( X j , Y ). This result is subsequen tly extended to section spaces. As a consequence, in Section 7, we extend the basic lo calization result of Hilton-Mislin-Roitb erg [9, th. I I.3.11 ] for function spaces from the ca s e X finite CW to the case X compact, metric provided the function spa ce compo nent is a nilp otent spac e (Theorem 7.1). In Section 8 we deduce Theor ems A- F by combining the finite complex case with the results of Section 6. Ac knowledgmen ts. W e thank Daniel Isaksen, Gr e gory Lupton, and J. Peter May for many helpful discussions. W e ar e e s pe c ially grateful to N. Christopher Phillips for vita l a ssistance given to us. This pap er is based in many ways upon our joint work [12]. 2. Conventions This pap er brings together results from classical alg ebraic top ology , which is most at home in the c a tegory of CW complexes , a nd functional analysis, which is most at home in the category of compa ct metric spaces. Man y o f o ur technical results deal with extending clas sical alg ebraic top olo gy res ults fro m finite co mplexes to compact metric spaces via limit arguments. W e work in the catego ry of co mpa ctly gener ated Hausdo rff spaces. Whenever basep oints ar e required w e assume that they are non-dege ner ate; that is, we ass ume that the inclusio n of the basep oint into the spac e is a cofibr ation. If the space is a to po logical group then we tak e the identit y of the gr oup to be the basepo int . F ollowing the discussion in [23, pp. 20-21 ], we g ive the function spa ce F ( X , Y ) the top ology obtained b y first taking the compact-op en to po logy and then replacing this with the induced compactly g enerated topolo gy . In particular, b eca use w e are retop olog iz ing pro ducts, by a top olo gic al gr oup w e mean a top ologica l g roup o b ject in the category of compactly generated Hausdorff spaces. Suppo se A ⊂ X is a subspa ce. Fixing a map g : A → Y , we let F ( X, Y ; g ) denote the subspace of those maps f : X → Y such that f coincides with g on A . In particular, when A is a p oint, X and Y obtain the structure of based spaces and F ( X, Y ; g ) in this ca s e is just the spac e o f ba sed maps. If f ∈ F ( X , Y ; g ) is a choice of basepo int , we let F ( X , Y ; g ) ( f ) denote the path compo nent of f . 6 KLEIN, SCHOCHET, AND S M ITH An inclusion A ⊂ X of spaces is a c ofibr ation if it satisfies the ho motopy extensio n prop erty . A Hur ewicz fibr ation is a map p : E → B satisfying the homotopy lifting problem for all (compac tly gene r ated) spaces. A map of fibr ations E → E ′ ov e r B is a ma p of spaces which commut es with pro jection to B . One says that p is fi br e homotopy trivial if ther e is a spac e F and a map q : E → F such that ( p, q ) : E → B × F is a homotopy equiv alence. Given a map f : Y → B we wr ite f ∗ ( p ) : Y × B E → Y fo r the pullbac k fibration (i.e., the fibre pro duct). An H -sp ac e structu r e on a ba sed space X is a map m : X × X → X whose restriction to X × ∗ and ∗ × X is homo topic to the identit y as based maps, where ∗ ∈ X is the bas ep o int. If an H -spa ce structure on X is understo o d, we call X an H -sp ac e. O ne s ays that X is homotopy asso ciative if the ma ps m ◦ ( m × id) and m ◦ (id × m ) are homotopic. A homotopy inverse for X is a map ι : X → X such that the compos ites m ◦ ( ι × id) and m ◦ (id × ι ) are ho motopic to the identit y . If X comes equipp ed with a homotopy asso ciative mu ltiplication and a homotopy inv er s e, then X is said to be gr oup-like. If X is g roup-like then the set of path comp onents π 0 ( X ) acquires a g roup structure. Nilp oten t Spaces. If ( X, ∗ ) is a based space then its higher homoto py groups π n ( X ; ∗ ) co me equipp ed with an actio n of the fundamen tal group π = π 1 ( X, ∗ ). If X is also a connected CW complex, then w e say that X is n ilp otent if π is a nilpo tent group a nd also the action of π on the higher homotopy gro ups is nilpotent. The latter co ndition is equiv alent to the statement that each π n ( X ; ∗ ) p oss esses a finite filtration o f π -mo dules M n ( i ) ⊂ M n ( i + 1) ⊂ · · · such that the action on the ass o ciated gra ded M n ( i + 1) / M n ( i ) is trivial. More gener ally , if X is any based connected space, then we will call X nilp otent if X ha s the homoto py type of a nilpo ten t CW complex. T opo logical gr oups having the homotopy type of a connected CW complex are nilpotent, since the action of π 1 in this case is trivial. Rationalization. A finitely ge ne r ated nilp o tent gro up K admits a rationalization, which is a natural homomorphism K − → K Q ([9, § 2]). The group K Q has the prop erty that the self map x 7→ x n is a bijection for all integers n ≥ 1 (i.e., K Q is uniquely div isible). F urthermor e, K Q is the smallest gro up ha ving this pr op erty in the sense that any homomor phism from K to a group with this prop erty uniquely factors through K Q . When K is ab elian, there is a natural iso morphism K Q ∼ = K ⊗ Q . A co nnected based nilp otent spa ce X admits a r ationalization. This is a nilp o- ten t space X Q with rationa l homotop y (and homolog y) groups, tog ether with a natural map X → X Q and a natural map ℓ X : X → X Q inducing ratio nalization on homotopy groups [9, thms. 3A, 3B]. Again, there is a universal prop erty: if Y is a rationa l space (i.e., a nilp otent space whose homotopy groups a re rational), and f : X → Y is a map, then one has a commut ative diagram X / / f   X Q f Q   Y ≃ / / Y Q , where the b ottom map is a homoto py equiv alence since Y is rational. Consequently , f factors uniquely up to homotopy thr ough X Q , and in particula r the r ationaliza tion of X is uniquely determined up to homotopy equiv alence . More genera lly , we call a CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 7 map f : X → Y a r ationalizatio n if Y is rational a nd the induced map f Q : X Q → Y Q is a homo topy equiv alence. This is equiv alent to demanding f ∗ : [ Y , Z ] → [ X , Z ] be an isomorphism of sets for a ll rational nilpo tent spa c es Z . 3. Section sp aces Suppo se one is given a lifting problem, i.e., a diagra m o f spaces A g / / ∩   E p   X f / / > > B such that A ⊂ X is a cofibration and E → B is a fibr ation. Let us denote this lifting problem b y D . Let Γ( D ) be the space of so lutions to the lifting problem, i.e., the space o f maps X → E making the diagr am commute. When f is the iden tity map a nd A is tr ivial, then one obtains the space of sections of p (it will b e denoted by Γ( p ) in this ins tance). Prop ositi o n 3.1. L et D b e t he lifting pr oblem ab ove. Then one has a fibr ation F ( X , E ; g ) p ∗ − − − − → F ( X , B ; p ◦ g ) whose fibr e at f is given by Γ( D ) . Pr o of. Here p ∗ is given by mapping a function a : X → E to p ◦ a : X → B . The map p ∗ is a fibration b y the exponential la w. The fibre o ver f is clearly Γ( D ).  CW structure. Prop ositi o n 3.2. With r esp e ct to the dia gr am D ab ove, assume that X is a c omp act metric sp ac e and su pp ose E and B have the homotopy typ e of CW c omplexes. Then the se ction sp ac e Γ( D ) has the homotopy typ e of a CW c omplex. Pr o of. Restr iction f 7→ f | A defines a fibration F ( X , E ) → F ( A, E ) in which bo th the domain and co domain have the homotopy type of a CW complex (b y [14, th. 1]). Apply [18, prop. 3] (or [10, lem. 2.4]) to de duce that the fibre F ( X , E ; g ) of this fibration has the homotopy type of a CW complex. Rep eating this argument with the fibration of Pro po sition 3.1 completes the pr o of.  Corollary 3. 3 . L et X b e a c omp act metric sp ac e and G a top olo gic al gr oup havi ng the homotopy typ e of a CW c omplex. Le t ζ b e a princip al G -bund le (r esp e ctively, princip al P G -bund le) over X . Then G ( ζ ) ( r esp e ctively, P ( ζ ) ) has the homotop y typ e of a CW c omplex. Pr o of. W e only prov e the case of the a djoint bundle as the other case is pro ved similarly . The fibr e bundle ζ is clas s ified by a map f : X → B G by pulling back 8 KLEIN, SCHOCHET, AND S M ITH the univ ersal principal G -bundle E G → B G along f . The s pace of so lutions of the lifting problem E G × G G ad   X f / / 9 9 B G coincides with the sectio n space Γ(Ad( ζ )). F urthermore E G × G G ad and B G hav e the ho motopy t yp e of CW complexes beca use G do es. The pr o of is completed by applying Prop ositio n 3 .2.  Example 3.4 . Supp ose that A is a (s e pa rable) unital Banach algebra. Then the group of in vertibles GL ( A ) has the ho mo topy t yp e of a (coun table) CW complex. If A is a (sepa rable) unital C ∗ -algebra then the gro up of unitar ies U A has the homotopy type of a (countable) CW complex . Here is a pro of. As U A is a deforma tion retractio n of GL ( A ), they bo th hav e the same homotopy type. The gro up GL ( A ) is an open subset of a Banach spa ce, and any s uch open set has the homo to py type of a CW complex (cf. [11, co r. IV.5.5]). If A is separable then the op en cov ering in volv ed in the proo f of [11, prop. IV.5.4 ] ma y be taken to b e co un table and then an o bvious mo dification of [11, IV.5.5] implies that the CW complex constructed is coun table. Nilp otence. Prop ositi o n 3.5. With r esp e ct to the hyp otheses of Pr op osition 3.2, assume addi- tional ly that X ha s the homotopy typ e of a CW c omplex and that E is a c onne cte d nilp otent sp ac e. Then e ach c omp onent of Γ( D ) is nilp otent. Pr o of. Co nsider the comm utative diagr am F ( X , E ; g ) / /   F ( X , B ; p ◦ g )   F ( X , E )   / / F ( X , B )   F ( A, E ) / / F ( A, B ) where the horizontal maps ar e all fibr a tions. By [9, th. 2.5], each comp one nt of F ( X , E ) is nilpotent. It follows that ea ch comp onent of F ( X, E ; g ) is nilpo tent b y [9, th. 2.2] applied to the left co lumn o f the diagram. After res tricting the fibration on the top line to connected compo nent s, it follows again by [9, th. 2.2] that each comp onent of Γ( D ) is nilpo tent , s inc e the latter is a fibre by P rop osition 3.1.  Fibrewise Groups. Definition 3.6. A fibration p : E → B is s aid to b e a fibr ewise gr oup if it comes equipp e d with a map m : E × B E → E , a map i : E → E and a sectio n e : B → E , all compatible with pro jection to B , such that • m is ass o ciative, • e is a tw o sided unit for m , • i is a n in verse for m and e CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 9 (In other w ords, ( p, m, e ) defines a group ob ject in the category of spaces o ver B .) If ( p, m, e ) is a fibrewise group, then the space o f sections Γ ( p ) comes equipp ed with the structure of a top ologica l group with m ultiplication defined by po int wise m ultiplication of sections. Here is a recip e for pro ducing fibrewise groups . Suppos e ζ : T → X is a pr incipal G -bundle a nd F is a top olog ical gro up s uch that G a cts on F throug h homomo r - phisms (this mea ns that one ha s a homomor phism G → aut( F ), w her e aut( F ) is the to p o logical group consisting of top ologica l group automorphisms of F ). Then the asso ciated fibre bundle T × G F → X is easily observed to be a fibrewise group. An impo r tant sp ecial ca se o ccur s w he n F is G ad . W e infer that Ad( ζ ) is a fibrewise gro up and s o the ga uge group G ( ζ ) has the structure of a topo logical group. Similar ly , when F is G ad and w e let P G a ct by conjuga tion, w e infer that P ( ζ ) has the structure of a topo logical group. Example 3.7. Retur ning to the C ∗ -algebra setting, we no w show that the group U A ζ in the In tro duction corr esp onds to a pro jective gauge group. Let X b e a compact space and let ζ : T → X be a pr incipal P U n -bundle ov er X with asso ciated C ∗ -algebra A ζ . The n there is an natural isomorphism of topolo gical groups U A ζ ∼ = P ( ζ ) . The pro of is as follows: passing from M n to the subspace U n of unitaries in each fibre of ζ yields a bundle U ζ : T × P U n U n → X . The sections of this bundle ar e exactly U A ζ but it is immedia te tha t the bundle itself is the bundle P ad( ζ ). Remark 3.8. Note that if f : Y → X is co nt inuous then f induces a map of unital C ⋆ -algebra s f ∗ : A ζ → A f ∗ ζ and this restricts to a homomorphism of unitary groups U ( A ζ ) − → U ( A f ∗ ( ζ ) ). The naturality in the result ab ov e is with resp ect to these maps. 4. Ra tionaliza tion of topological groups Now s uppo se that G is a top o logical gro up ha ving the homoto py t yp e of a CW complex. As r ationalizatio n co mmutes with pr o ducts only up to homotopy , the rationaliza tion of the pro duct structure gives a ma p G Q × G Q → G Q which may fa il to b e a gro up structure. It is, how ever, a group-like H -space. The map G → G Q is a homo morphism of H -spaces in the sense that the diag ram G × G / /   G   G Q × G Q / / G Q 10 KLEIN, SCHOCHET, AND S M ITH commutes up to homotopy (and the homomorphism is co mpatible with ho motopy asso ciativity). Homotopy commutativit y and rationalization. Definition 4.1. A gro up-like H -space X is homotop y c ommutative if the co mm u- tator map [ , ] : X × X → X is n ull homotopic. It is said to b e r ational ly homotop y c ommutative if its rationaliza tion X Q is homotopy commutativ e. The comm utator map induces an op er ation on homotopy gr oups ca lled the Samelson pr o duct . After tensoring with the rationals, o ne obtains a graded Lie algebra structure ([23, chap. X.5]). Definition 4. 2. A homomorphism X → Y of connected H -spaces of CW type is said to be r ational H -e quivalenc e if its rationaliza tion X Q → Y Q is a homotopy equiv alence. Prop ositi o n 4.3 ( Sc hee rer [17, cor. 1]) . L et X and Y b e c onne cte d gr oup-like H -sp ac es having the homotopy typ e of a CW c omplex. Then t her e is a r ational H -e quivalenc e X Q ≃ Y Q if and only if ther e is an isomorphism of S amelson Lie algebr as ( π ∗ ( X ) ⊗ Q , [ , ]) ∼ = ( π ∗ ( Y ) ⊗ Q , [ , ]) . W e observe that the ra tionalization of suc h a space X has the ho mo topy type of a generalized Eilenberg Mac La ne space : (1) X Q ≃ Y j ≥ 1 K ( π j ( X ) ⊗ Q , j ) . How ever, a s p ointed out in the intro duction, the multiplication on X Q need not corres p o nd to the standard multiplication. W e may detect when this ident ification is m ultiplicative in several w ays. Prop ositi o n 4 . 4 ( cf. [1 2, th. 4.25]) . L et X b e a homotop y asso ciative H -sp ac e having the homotopy typ e of a c onne cte d CW c omplex. Then the fol lowing ar e e quivalent: (a) Ther e is a homotopy e qu ivalenc e X Q ∼ → ( Y j ≥ 1 K ( π j ( X ) ⊗ Q , j ) which is also an H -map, wher e t he tar get has the standar d multiplic ation. (b) The c ommu tator m ap X Q × X Q → X Q is nu l l homotopi c. (c) The Samelson Lie algebr a ( π ∗ ( X ) ⊗ Q , [ , ]) is ab elian; i.e., [ , ] = 0 . Corollary 4.5. Su pp ose G is a c onn e cte d top olo gic al gr oup such that B G has the r ational homotopy typ e of a lo op sp ac e, i.e. ther e is a b ase d sp ac e Y and a r ational homotopy e quivalenc e B G ≃ Q Ω Y . Then G Q is a homotopy c ommut ative H - sp ac e and is homotopy e quivalent to a pr o duct of Eilenb er g-Mac L ane sp ac es with standar d multiplic ation. CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 11 Example 4.6. A top olog ical group G is said to satisfy r ational Bott p erio dicity if there is rational homotopy equiv alenc e B G ≃ Q Ω j G for some j > 0. An y s uch G satisfies Cor ollary 4.5, and is consequently r ationally homo topy comm utative. In particular, the infinite unitary group U ∞ is ra tionally homotopy commutativ e (cf. Remark 1.1). Example 4.7. Supp os e that the top ologica l group G is a direct limit lim − → n G n , where each G n is ra tionally ho motopy commutativ e. Then G is ratio nally ho- motopy co mm utative. This g ives a second pro of that U ∞ is ra tionally homotopy commutativ e (cf. Remark 1.1). W e now discuss the v ar ious hypothes es on the group G in our main res ults. Recall from Theore m D w e require tha t G b e a topo logical group o f the homotopy t yp e of a finite complex. W e first observe that this clas s includes the co nnected Lie groups. Lemma 4. 8 . Every c onne cte d Lie gr oup G has the homotopy typ e of a finite CW c omplex. Pr o of. By [23, th. A.1.2], G has a max ima l co mpa ct subgroup K , unique up to conjugacy , such that the inclusion K ⊂ G is a homotopy e quiv alence. Then K , being a compact Lie gro up, has the homotop y t y pe of a finite CW complex.  W e will make us e of the follo wing results whose pro ofs are classical. Prop ositi o n 4.9. Su pp ose t hat G is a c onne cte d, top olo gic al gr oup having the homotopy typ e of a finite CW c omplex. Then the fol lowing ar e true: (a) The c ommut ator map G Q × G Q → G Q is nu l l homotopic. (b) π 2 ( G ) is a finite gr oup. (c) The classi fying sp ac e B G has the r ational homotopy typ e of a gener alize d Eilenb er g-Mac L ane sp ac e and in p articular is ra tional ly homotopy e quiva- lent to a lo op sp ac e. Pr o of. The basic res ults o f Milnor a nd Mo o re [15] o n the str ucture of Hopf algebra s of c haracter istic zer o imply H ∗ ( G ; Q ) is a n exterior a lgebra on a finite n umber of o dd degree g e nerators . It follows that H ∗ ( B G ; Q ) is a p o lynomial algebra on a finite n umber of generator s of even degree. Represen t e ach generator b y a map x i : B G → K ( Q , n i ). Then the pro duct ma p f = Q x i : B G → Q i K ( Q , n i ) g ives an isomorphis m on rational homotopy g roups. This prov es (c). Applying the lo op space functor to the map f g ives (a) b y Prop osition 4.4. T o compute π 2 we may pass to universal cov er s and hence assume that the groups are simply connec ted. Then H ∗ ( G ; Q ) = 0 in degrees 1 and 2. This im- plies that H 2 ( G ; Z ) is a finite group. The Hurewicz map π 2 ( G ) → H 2 ( G ; Z ) is an isomorphism, so π 2 ( G ) is a finite group.  Corollary 4. 1 0. If G is a c onne cte d top olo gic al gr oup having the homotopy typ e of a finite CW c omplex, then G Q is a homotopy c ommutative H - sp ac e. In p articular, G Q is homotopy e quivalent as an H -sp ac e t o a pr o duct of Eilenb er g-Mac L ane sp ac es with standar d mu ltiplic ation. Finally , in Theore m F we res trict to the class o f co mpact Lie g roups G . This restriction is chosen to g ov ern the r a tional ho motopy theory of P G as we explain now. First we ha ve the following genera l fact. 12 KLEIN, SCHOCHET, AND S M ITH Lemma 4.11. L et G b e a c onne cte d Lie gr oup. Then (a) P G is a c onne cte d Lie gr oup; (b) t he inclusion Z ( G ) → G induc es a monomorphism π 1 ( Z ( G )) ⊗ Q → π 1 ( G ) ⊗ Q . Pr o of. P G is a Lie group and it is connected since it is a quotient space of G . By Prop ositio n 4 .9, π 2 ( P G ) is a finite group which implies the sec ond statemen t.  The pr eceding re s ults imply in particular that G and P G × Z ( G ) ha ve the s ame homotopy type after ra tionalization. Note, how ever, that ther e is no obvious map in either dir ection. W e need to sharp en this identification for o ur pro of of Theor e m F. The following classica l fact due to E . Car ta n explains o ur restric tio n there to compact Lie groups. Prop ositi o n 4.1 2. L et G b e a c omp act, c onne cte d Lie gr oup. Then ther e is a c omp act, c onne cte d Lie gr oup G 0 such t hat the fol lowing hold. (a) Ther e is a homomorphism q : G 0 → G which is a r ational homotopy e quiv- alenc e. (b) Ther e is a s plitting G 0 ∼ = P ( G 0 ) × Z ( G 0 ) (c) The map q c arries Z ( G 0 ) t o Z ( G ) and induc es an isomorphism P ( G 0 ) ∼ = P G. Pr o of. By [23, th. A.1.1], G has a finitely-sheeted cov ering group q : G 0 → G with G 0 = T ℓ × G ′ , wher e G ′ is a simply co nnected compact Lie gro up with trivial center and T ℓ is the pro duct o f ℓ -copies of S 1 . The results follow directly .  5. P reliminar y r esul ts: finite complexes In this section, we prove Theor ems B, D and F when X is a finite CW complex. F or Theorem B, the result is a dire ct consequence o f classical work of Thom and a lo calization theorem for function s paces due to Hilton, Mislin and Roitb e rg. The pro of of Theorem D makes use of Gottlieb’s iden tity [8, th. 1] for the gaug e group in a ddition to the previous ingr edients. W e deduce Theor em F from Theo rem D and Prop osition 4.12. First, we have the famo us result of H. Hopf, generalized b y Thom [22]: (2) π q ( F ( X , K ( π , p ))) = H p − q ( X ; π ) . Next we have the re s ults [9, th. I I.3.11 , cor . I I.2.6] of Hilton-Mislin- Ro itber g o n the function space F ( X , Y ): Prop ositi o n 5.1 ( Hilton-M isli n - Roitb e rg [9 ]) . L et X b e a finite CW c omplex and Y b e a c onne ct e d nilp otent sp ac e. Then the induc e d map F ( X , Y ) → F ( X , Y Q ) is a r ationalization map on c onn e cte d c omp onents. (More pre cisely , for any map h : X → Y , the ma p F ( X , Y ) ( h ) → F ( X , Y ) ( ℓ Y ◦ h ) is a rationaliza tion map.) Prop ositi o n 5.2 (cf. [12, th. 4.28 ]) . L et G b e a top olo gic al gr oup having the homo- topy typ e of a finite c onne cte d CW c omplex. L et X b e a finite CW c omplex. Then the r ationalization of F ( X , G ) ◦ is a homotopy c ommutative H -sp ac e. CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 13 Pr o of. Reca ll that F ( X , G ) ◦ is the comp onent of F ( X , G ) con taining the constant map. By Cor ollary 4.1 0, G Q is homotopy commutativ e. It follows that F ( X , G Q ) ◦ is also homotopy comm utative. The res ult now follo ws from obser ving that F ( X , G Q ) ◦ is the rationalization of F ( X , G ) ◦ by Pro p o sition 5.1.  Remark 5.3 . The preceding result holds in greater generality: for F ( X , G ) to be rationally homo topy co mmutative, o ne o nly needs to assume that G is a ra tionally homotopy commutativ e H -space (se e [12, th. 4.10]). W e can now prove Theo r em B under the as sumption that X is a finite complex as in [12, th. 4 .28]. Theorem 5.4 ( Preli m inary version of Theorem B ) . L et X b e a fin ite CW c omplex and let G b e a c onne cte d t op olo gic al gr oup having the homotopy typ e of a finite CW c omplex. Then (3) π ∗ ( F ( X , G ) ◦ ) ⊗ Q ∼ = H ∗ ( X ; Q ) e ⊗ ( π ∗ ( G ) ⊗ Q ) . F ur t hermor e, F ( X , G ) ◦ is ra tional ly H -e quivalent to a pr o duct of Eilenb er g-Mac L ane sp ac es with t he standar d lo op multiplic ation, with de gr e es and dimensions c orr e- sp onding to (3). Pr o of. Using Pr o p osition 5.1, w e ha ve π ∗ ( F ( X , G ) ◦ ) ⊗ Q ∼ = π ∗ ( F ( X , G Q ) ◦ ) and we may co mpute the latter using the iden tification (1 ) and Thom’s form ula (2). The identification of the rationa l H -t yp e of F ( X , G ) ◦ is a direct cons e q uence of Prop osition 5.2.  Remark 5.5. Since the iden tification (1) only r equires a g roup-like H -space, the homotopy gro up calculation ho lds when G is group-like. The assumption that G is homotopy finite is used only to conclude that G Q is homotopy commutativ e via P rop osition 5.2. Consequently , the seco nd conclusion of Theor em 5.4 holds when G is a group-like H - space such that G Q is homoto py commutativ e. W e next prov e Theorem D for X a finite CW complex. Theorem 5. 6 ( Preliminary v ersion of Theorem D ) . L et G b e a top olo gic al gr oup having t he homotopy typ e of a finite c onne ct e d CW c omplex. L et ζ b e a princip al G -bund le over a finite CW c omplex X . The n ther e is a r ational H - e quivalenc e G ( ζ ) ◦ ≃ Q F ( X , G ) ◦ . Pr o of. By [8, th. 1] (see also Corollary 9.2 below), there is a homotopy equiv alence of H - s paces G ( ζ ) ≃ Ω h F ( X , B G ) , where the space o n the right is the based loo p space of the function space F ( X , B G ) with ba sep oint given by the c lassifying map h : X → B G for the bundle ζ . Rec a ll that F ( X , B G ) ( h ) denotes the path comp onent of F ( X , B G ) containing h . By Prop ositio n 5 .1 abov e, F ( X , B G ) ( h ) → F ( X , ( B G ) Q ) ( h ′ ) 14 KLEIN, SCHOCHET, AND S M ITH is a rationa lization map, where h ′ is ℓ B G ◦ h , where ℓ B G : B G → ( B G ) Q is the ra - tionalization map. In particular, the display ed map is a rational homoto py equiv a - lence. By Cor ollary 4.10, ( B G ) Q has the homotopy t yp e of a lo o p s pace. I t follows that all the compo nent s of F ( X , ( B G ) Q ) hav e the sa me homotopy t yp e. Cons equently , there is a rational homotopy equiv alence F ( X , B G ) ( h ) ≃ Q F ( X , ( B G ) Q ) ◦ . The result now follows by taking the based lo op space of both sides .  Remark 5. 7. It is clear from o ur pro of that the finiteness as sumption on G was used only to conclude that ( B G ) Q has the homotop y t y p e of a loop space. In fact, one see s that the ab ove a rgument w ork s, without the finitenes s assumption on G , at the expense o f assuming that ( B G ) Q has the structure of a group-like H -space. While the iden tity G ( ζ ) ≃ Ω h F ( X , B G ) do es extend to more g eneral spaces X (see Cor ollary 9.2)), the metho d of pro o f ab ov e is limited b y Pro po sition 5.1. Both the nilp otence result [9, th. 2.5] and the localiza tion r esult Prop osition 5.1 for F ( X , Y ) require X to b e a finite CW complex. In Section 7, we extend the lo calizatio n res ult to X compa ct metric assuming nilp otence. How ever, the nilp otence of the comp onents of F ( X , Y ) is not exp ected to hold for general X . The last goal of this section is to pro ve Theorem F when X is a finite complex. Theorem 5. 8 ( Prelim inary version of Theorem F ) . L et G b e a c omp act c on- ne cte d Lie gr oup, and let ζ : T → X b e a princip al P G -bu n d le over a finite CW c omplex X . Then ther e is a r ational homotopy e quivalenc e of H -sp ac es P ( ζ ) ◦ ≃ Q F ( X , G ) ◦ . Pr o of. Ca se 1. Suppose first that G splits as Z ( G ) × P ( G ). Then o ne has an isomorphism of bundles o ver X with tota l s paces ( T × Z ( G )) × G G ad ∼ = T × P G G ad . The result follows now by Theorem 5.6 applied to the bundle o n the left. Case 2 . This is the ge neral case. By P rop osition 4.12, there is a c ompact Lie gr oup G 0 and a homomo r phism q : G 0 → G which is a rational homotopy equiv alence. F urthermor e, this homomorphism induces an iso morphism P ( G 0 ) ∼ = P ( G ) and o ne also has a splitting G 0 ∼ = Z ( G 0 ) × P ( G 0 ). By Prop o sition 5.1 2 b elow, the eviden t map Q 0 := E P G × P G G ad 0 → E P G × P G G ad := Q of fibrewise groups is a rational ho motopy equiv a lence of nilp otent spaces. Let h : X → B P G cla ssify the bundle with total spa ce T × P G G ad ; then h als o classifies the bundle with total space T × P G G ad 0 . Denote the lifting problem Q   X < < h / / B P G CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 15 by D . Then the space of lifts Γ( D ) is the pro jective gauge group P ( ζ ). Deno te the corres p o nding lifting problem with Q replace d b y Q 0 by D 0 . Then the map Γ( D 0 ) → Γ( D ) induced by the homomorphism q : G 0 → G is both a ra tional ho motopy equiv alence on comp onents and a map of H -spaces (this is by a straightforward induction using the cell structure for X ). Simila rly , q induces a rational homotopy equiv alence o f H -spaces F ( X , G 0 ) → F ( X , G ). By Case 1, we also hav e a rationa l eq uiv alence of H -spaces Γ( D 0 ) ◦ ≃ F ( X , G 0 ) ◦ . Assembling these three equiv a lences completes the pro of.  The r emainder of this section is devoted to proving Pro po sition 5.12 used in the pro of ab ov e . W e nee d some preliminary lemmas. Lemma 5.9. Assume G is a c omp act c onne cte d Lie gr oup. L et E = E G × G G ad and Q = E P G × P G G ad . Then the map E → Q induc es a su rje ction on homotopy gr oups in e ach de gr e e. Pr o of. Ther e is a homo topy fibre sequence B Z ( G ) → E → Q . T aking the lo ng exac t homotopy sequence, we infer that π ∗ ( E ) → π ∗ ( Q ) is an isomorphism when ∗ 6 = 3 (here we ar e using the fact Z ( G ) is a torus). Consequently , we hav e an exact sequence 0 → π 3 ( E ) → π 3 ( Q ) → Z ℓ → π 2 ( E ) → π 2 ( Q ) → 0 where ℓ = rank of Z ( G ). W e can calculate π 3 ( E ) using the lo ng exact sequence of the fibration E → B G ; one s e es (using the fact that π 2 ( G ) = 0 ) that it is is omorphic to π 3 ( G ). Likewise, w e see that π 3 ( Q ) is also isomorphic to π 3 ( G ) and the homomorphism π 3 ( E ) → π 3 ( Q ) is in fact an isomorphism.  Lemma 5.10. If E → B is a fi br ation of c onne cte d sp ac es having the homotopy typ e of a CW c omplex. Assu me π ∗ ( E ) → π ∗ ( B ) is s u rje ctive in every de gr e e and E is n ilp otent. Then B is nilp otent . Pr o of. The quo tient of a nilp otent group is aga in nilpo tent , s o π 1 ( B ) is nilp otent. F urthermor e, when k ≥ 2, we have a short exact sequence of π 1 ( E ) mo dules 0 → π k ( F ) → π k ( E ) → π k ( B ) → 0 where F denotes the fibre at the ba s ep o int. The nilpo tency of the middle mo dule guarantees that π k ( B ) is also a nilp otent π 1 ( E )-mo dule (see [9 , prop. 4.3 ]). This mo dule str uc tur e arises from the ho mo morphism π 1 ( E ) → π 1 ( B ) by restriction. Since this homomo rphism is surjective, it follows that π k ( B ) is a nilpo tent π 1 ( B )- mo dule.  Lemma 5.11. L et G b e a c omp act c onne cte d Lie gr oup. Then the sp ac e Q = E P ( G ) × P ( G ) G ad is nilp otent. 16 KLEIN, SCHOCHET, AND S M ITH Pr o of. Ther e is a homo topy fibre sequence B Z ( G ) → E → Q , where E = E G × G G ad . Then E is homoto p y equiv ale nt to LB G , the free loop space o f B G (cf. Lemma 9.1). W e infer that E is nilp otent by [9, th. 2 .5]. Now apply the preceding lemmas.  Prop ositi o n 5.1 2 . L et G b e a c omp act c onne cte d Lie gr oup and let q : G 0 → G b e as in Pr op osition 4.12. Then the map of fibr ewise gr oups Q 0 := E P ( G ) × P ( G ) G ad 0 → E P ( G ) × P ( G ) G ad =: Q is a r ational homotopy e quivalenc e of n ilp otent sp ac es. Pr o of. Bo th Q and Q 0 are nilpotent by Lemma 5.11. By applying rationalization to the diagram G ad 0 / / q   Q 0 / /   B P ( G ) G ad / / Q / / B P ( G ) whose rows a r e fibre s e quences, and using the fact that rationalization preser ves fibrations ([9, th. 3.12]) w e infer that the map Q 0 → Q is a r ational homotopy equiv alence.  6. Limits and function sp a ces When X is a compact metric space, a classical r e sult of E ilenberg a nd Steenrod [6, th. X.10.1] gives an in verse system o f finite simplicia l (CW) complexes X j and compatible maps h j : X → X j such that the induced map h : X → lim ← − j X j is a ho meo morphism. This result and its generaliza tion are at the core of our metho d for passing from finite complexes to compact metric spaces. In this and subsequent sections, w e consider b oth direc t a nd inv erse limits. Sup- po se { X j , p ij } is an in verse system o f spaces, where p ij : X i → X j are maps, j ≤ i . Given compa tible maps h j : X → X j , o ne has a n induced map h = lim ← − j h j : X → lim ← − j X j . W e recor d the following basic result. Prop ositi o n 6.1 ( Eilenberg-Ste enro d [6, th. X.10 .1, X.11.9 ]) . L et X b e a c om- p act Hausdorff sp ac e. (a) Ther e exists an inverse system of finite CW c omplexes { X j , p ij } and c om- p atible maps h j : X → X j inducing a home omorphism h = lim ← − j h j : X → lim ← − j X j . (b) Given a map f : X → Y in which Y is a CW c omplex, ther e exists an index m and a c el lular map f m : X m → Y such that the c omp osite X h m − − − − → X m f m − − − − → Y is homotopic to f . CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 17 Prop ositi o n 6.2. Under the hyp otheses of Pr op osition 6.1, the map of sets lim − → j [ X j , Y ] → [ X , Y ] is a bije ction. Pr o of. Surjectivity is a direct conseq uenc e o f Prop osition 6 .1 (b). Injectivity is a consequence of Spanier’s metho d of pro o f of [19, th. 13.4 ]. In Spanier’s case, Y = S n and X has Leb esgue cov ering dimension at most 2 n − 2 and his limit is taken in the categor y of ab elian gr oups. How e ver, Spanier r emarks that the dimension condition can be dro ppe d pro vided that the limit is taken in the category of sets [19, p. 228]. F urthermore, an insp ection of his pro of shows tha t it ge ner alizes without change to Y a n arbitrar y finite simplicial co mplex. T he argument is then completed by recalling that any finite CW complex has the homotopy type of a simplicial complex.  W e will need to extend this prop osition to a certain clas s of pair s . Suppose now that ( X , A ) is a pair, where X is a compact Hausdorff s pace and A ⊂ X is a c losed cofibration. W e a s sume that ( X , A ) is e xpressed as an inverse limit of pairs ( X j , A j ) where the latter is a finite CW pair . Suc h a deco mpo sition exis ts by the relative version o f [6, Ch. X, th. 10.1, 1 1.9]. As a b ov e , write p ij : X i → X j for j ≤ i and h j : X → X j for the structur e maps. W e use the s ame notation fo r the r estrictions of these maps to A j and to A , resp ectively . Let Y be a CW complex, and supp ose that one is given a fixed map g m : A m → Y for some m a nd define g j : A j → Y for j > m by g m ◦ p j m . Define g to be the comp os ite g m ◦ h m . Let [ X , Y ; g ] denote the set of homotopy classes of maps X → Y which coincide with g on the subspace A (wher e homotopies are required to b e constant on A ). Similarly , w e hav e [ X j , Y ; g j ] and a map of s ets [ X j , Y ; g j ] → [ X , Y ; g ] (for j ≥ m ) whic h is compatible with the index j . Lemma 6. 3. Assume ther e ar e c omp atible r etr actions r j : X j → A j inducing a r etr action r : X → A . Then the map lim − → j [ X j , Y ; g j ] → [ X , Y ; g ] is a bije ction. Pr o of. Let i : A → X and i j : A j → X j be the inclusions, and let u : [ X, Y ; g ] → [ X , Y ] and u j : [ X j , Y ; g j ] → [ X, Y ] b e the eviden t maps. F or each j , one has a commutativ e diagram of sets [ X j , Y ; g j ] / / u j   [ X , Y ; g ] u   [ X j , Y ] h ∗ j / / i ∗ j   [ X , Y ] i ∗   [ A j , Y ] / / [ A, Y ] 18 KLEIN, SCHOCHET, AND S M ITH where the b ottom terms are po inted sets. F urthermore, if r : X → A is a r etraction, then g ◦ r is a basep oint for [ X , Y ] ma king i ∗ int o a split surjection of ba sed sets. The r ight column is in fact the tail-e nd o f the long exa ct homotopy sequence of the fibration F ( X , Y ) → F ( A, Y ), whic h is also equipped with section. It follo ws from this observ ation that u is one-to-one. Similarly u j is one-to-one. T aking direct limits r esults in a diagram such that middle and bottom maps are isomorphisms. The rest o f the argument follows fro m an elementary diagram chase, using the fact that u j and u ar e one-to -one (we leave the details to the r eader).  Now, let f m : X m → Y be a fixed map and de fine f j : X j → Y for j > m by f m ◦ p j m . Define f to b e the c o mpo site f m ◦ h m . Then the map of function spac e s F ( X j , Y ) → F ( X , Y ) sends f j to f , so we ha ve a map of based spa ces that is compatible with the in verse sys tem. Theorem 6. 4. The inverse system of b ase d sp ac es ab ove induc es an isomorphism of gr oups lim − → j π n ( F ( X j , Y ); f j ) ∼ = π n ( F ( X , Y ); f ) in al l de gr e es. Pr o of. By [1 3, pr o p. IX.2], the limit of a dir e ct system of (ab elian) gr oups coincides with the limit tak en in the category of sets. Case 1. n = 0. This case is just a reformulation of Pr op osition 6.2. Case 2. n > 0. Obser ve that [ X × S n , Y ; f ] = π n ( F ( X , Y ); f ) , where on the left w e ar e taking homoto py classes of maps X × S n → Y whic h coincide with f on X × ∗ = X . Note that each inclusion X j × ∗ ⊂ X j × S n is a retract, and these retractions are compatible. The result then follows from Lemma 6.3 with X × S n in place of X , X × ∗ in place o f A , X j × S n in place of X j and X j × ∗ in place of A j .  Limits and section spaces. Ass ume that ( X , A ) = lim ← − j ( X j , A j ) as ab ov e , wher e each ( X j , A j ) is a finite CW pair. Supp ose that for some index m one is g iven a lifting problem A m g m / / ∩   E p   X m f m / / > > B denoted D m . Here we assume that p : E → B is a fibr a tion in which E and B hav e the homotopy t yp e of CW complexes. Using the maps ( X j , A j ) → ( X m , A m ), we obtain ano ther lifting problem, denoted D j . Then one has maps Γ( D j ) → Γ( D j +1 ) for j ≥ m . Let ˜ f m : X m → E be any lift. Then we obtain basep oints ˜ f j ∈ D m for j ≥ m . Let f : X → B deno te the co mpo site of h m ◦ f m and similar ly , let g : A → E be the co mp os ite h m ◦ g m , wher e h m : ( X , A ) → ( X m , A m ) is the structur e map. Then CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 19 we get a lifting problem D A g / / ∩   E p   X f / / = = B . Let ˜ f : X → E be the ba sep oint of D determined by ˜ f m . Theorem 6.5. The map of b ase d sets lim − → j π n (Γ( D j ); ˜ f j ) → π n (Γ( D ); ˜ f ) is an isomorphism in every de gr e e n ≥ 0 , wher e the dir e ct limit is taken in the c ate gory of sets. Pr o of. F or each n , one has a map of long e xact homotopy seq ue nc e s · · · ∂ / / π n (Γ( D j ); ˜ f j ) a j / / c   π n ( F ( X j , E ); g j ) b j / / d   π n ( F ( X j , B ); f j ) / / e   · · · · · · ∂ / / π n (Γ( D ); ˜ f ) a / / π n ( F ( X , E ); g ) b / / π n ( F ( X j , B ); f ) / / · · · as given by P rop ositio n 3.1. T o prove surjectivity , le t x ∈ π n (Γ( D ); ˜ f ) b e any elemen t. By T he o rem 6.4, a ( x ) = d ( y ) for some y , provided that j is sufficiently large. Then b j ( y ) is trivial provided j is large , again by 6.4. It follows that y = a j ( z ) for so me z . Then a ( c ( z ) − x ) = 0, so x = c ( z ) − ∂ u for some u . If j is la rge, one has u = e ( u ′ ) for some u ′ . Consequent ly , x = c ( z − ∂ u ′ ). This establis hes surjectivity . A similar diagram c hase, which we omit, giv es injectivit y .  7. Lo caliza tion of function sp aces revisited The purp ose of this section is to extend the Hilton-Mislin- Roitb erg lo caliz ation result (Propo sition 5.1) for function spac es F ( X , Y ) to the case X compact metric and Y nilp otent CW provided the pa r ticular function space comp onent is known, a priori, to be nilp otent. Suppo se that X is a compac t metric s pace and X = lim ← − j X j as ab ove, where each X j is a finite CW complex. Let Y b e a nilp otent spa ce. Let ℓ Y : Y → Y Q be the ra tionalization map. Let f : X → Y b e a fixed map and consider the connected comp onent F ( X , Y ) ( f ) of the function space. Theorem 7.1. If F ( X , Y ) ( f ) is nilp otent, then the induc e d map F ( X , Y ) ( f ) → F ( X , Y Q ) ( ℓ Y ◦ f ) is a r ationalization map. Pr o of. By Prop os itio n 6.1, w e can ass ume without loss in gener ality that f factors as X → X m → Y . Let f m : X m → Y denote the factorizing map, a nd define f j : X j → Y for j > m to b e the comp osite p j m ◦ f m , whe r e p j m : X j → X m is the 20 KLEIN, SCHOCHET, AND S M ITH structure map in the in verse s ystem. The a pproximation X ∼ = lim ← − j X j gives rise to a comm utative dia gram lim − → j π n ( F ( X j , Y ); f j ) ∼ = / /   π n ( F ( X , Y ); f )   lim − → j π n ( F ( X j , Y Q ); ℓ Y ◦ f j ) ∼ = / / π n ( F ( X , Y Q ); ℓ Y ◦ f ) where the horizontal maps are bijections b y Theorem 6.4. Apply the rationa lization functor to the diagram and use the fact that r ationalization commutes with dir ect limits. This results in a comm utative diagr am lim − → j π n ( F ( X j , Y ); f j ) Q ∼ = / / ∼ =   π n ( F ( X , Y ); f ) Q   lim − → j π n ( F ( X j , Y Q ); ℓ Y ◦ f j ) ∼ = / / π n ( F ( X , Y Q ); ℓ Y ◦ f ) where the left vertical map is an isomorphis m by Pro p osition 5.1. It follows that the right vertical map is a n isomorphism as w ell.  8. Proof of the main resul ts W e are now in a position to prov e the main theorems in their complete generality . Pro of of Theorem B. Reca ll w e are ass uming X is a compact metric space and G is a co nnected CW top ologic al group having the homotopy t yp e of a finite complex . W e need to establish an isomorphism π ∗ ( F ( X , G ) ◦ ) ⊗ Q ∼ = ˇ H ∗ ( X, Q ) e ⊗  π ∗ ( G ) ⊗ Q  . By Theor em 5.4, the cor resp onding result holds for X j a finite CW complex. W rite X = lim ← − X j as usual for finite complexe s X j . Then for each j we hav e a natural isomorphism π ∗ ( F ( X j , G ) ◦ ) ⊗ Q ∼ = H ∗ ( X j ; Q ) e ⊗  π ∗ ( G ) ⊗ Q  . T ake direct limits o n both sides and use the fact that lim − → ( A j ⊗ B ) ∼ = (lim − → A j ) ⊗ B for ab e lian groups to obtain the isomorphism  lim − → π ∗ ( F ( X j , G ) ◦ )  ⊗ Q ∼ =  lim − → H ∗ ( X j ; Q )  e ⊗  π ∗ ( G ) ⊗ Q  . The con tinuit y prop erty of ˇ Cech co homology [6, th. 12.1] implies that lim − → H ∗ ( X j ; Q ) ∼ = ˇ H ∗ ( X ; Q ) . Then use Theorem 6.4 to iden tify lim − → π ∗ ( F ( X j , G ) ◦ ) ∼ = π ∗ ( F ( X , G ) ◦ ) which gives the result a t the lev el of homotopy gro ups . Finally , use Theorem 7.1 to obtain a homotopy equiv a lence of H -spaces ( F ( X , G ) ◦ ) Q ≃ F ( X , G Q ) ◦ . CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 21 The last part of Theorem B now follo ws from Prop ositions 5.2 and 4.4.  Pro of of Addendum C. As was observed in Remark 5.5, the homotopy gro up calculation of Theorem 5.4 ho lds when G is a group-like H-s pa ce, and the homo- topy commutativit y holds whenever G is rationally homotopy commut ative. The ab ov e pro of of Theorem B, which uses Theorem 5.4, therefore holds in the stated generality .  W e r e fo c us on the case of the adjoint and pr o jective a djoint bundles. As us ua l, let X be a compact metric space, and write X = lim ← − j X j for a n inv erse sys tem of finite complexes X j . Let ζ : T → X be the g iven principal G -bundle, whe r e G is of CW type . Let f : X → B G b e a classifying map for ζ . B y P rop osition 6.2, we can assume without los s in g enerality that f factors as X → X m f m → B G for some index m . F or j > m , define f j : X j → B G by taking the comp osite of f m with the map X j → X m . This defines a principal G -bundle ζ j : T j → X j for each j ≥ m . F or each j we ha ve a lifting problem E G × G G ad Ad( ζ )   X j f j / / 9 9 B G whose space of sections is just the gauge group G ( ζ j ). F urthermore, one has a direct system of topo logical groups G ( ζ m ) → G ( ζ m +1 ) → · · · equipp e d with compatible homomorphisms G ( ζ j ) → G ( ζ ). By Theorem 6.5, the homomorphism lim − → j π n ( G ( ζ j ) ◦ ) → π n ( G ( ζ ) ◦ ) is an isomorphism for n ≥ 0. A similar statement ho lds in the pro jectiv e bundle case. Summarizing, w e obtain the following description of the homoto py groups of the gauge group and of the pr o jective g auge group. Prop ositi o n 8.1. L et X b e a c omp act metric sp ac e and supp ose X = lim ← − j X j for an inverse system of finite c omplexes X j . Then, with n otation as ab ove, π ∗ ( G ( ζ ) ◦ ) ∼ = lim − → π ∗ ( G ( ζ j ) ◦ ) and π ∗ ( P ( ζ ) ◦ ) ∼ = lim − → π ∗ ( P ( ζ j ) ◦ ) . After r ationalization, these b e c ome isomorphisms of ra tional Samelson algebr as. Pr o of. The only thing we need to pr ov e is the last statement. This follo ws b eca us e the map inducing the isomor phism in each case is induced from maps of H -spaces . They th us induce isomor phis ms of rational Samelson Lie algebras.  22 KLEIN, SCHOCHET, AND S M ITH Pro of of Theorem D. Combining P rop ositio n 8 .1 and the preliminary version of Theorem D for finite co mplexes (Theorem 5.6) with Theo r em B one s ees that G ( ζ ) ◦ has ra tional homotopy groups g iven by Theorem B. F urther , since a direct limit of ab elian Lie algebras is abelia n, we conclude G ( ζ ) ◦ has abe lian rational Samelson Lie algebra. This, in turn, implies ther e exists an H -equiv alence G ( ζ ) ◦ ≃ Q F ( X , G ) ◦ by Pro p o sition 4.3.  Pro of of Theorem F. The proo f is similar to the preceding one. In this case, one co m bines Prop osition 8.1 and the pre limina ry version of Theorem F for finite complexes (Theorem 5.8) to get tha t P ( ζ ) ◦ has ra tional homotopy g r oups g iven by Theorem B. The rest of the a r gument is as in the pro of of Theorem D.  Pro of of Addendum E. See Re ma rk 5.7.  Pro of of Theorem A. The pro o f is a dire ct conseq uenc e of Ex a mple 3.7, Theo- rem F for G = U ( n ), and the w ell-known re s ult π ∗ ( U ( n )) ⊗ Q ∼ = Q ( s 1 , . . . , s n ) where | s i | = 2 i − 1.  9. Appendix: on the free l oop sp ace In this section, w e sketc h a pro of of “Gottlieb’s identit y ” for the ga uge group used in the pro of o f Theo r em 5.6. While Gottlieb’s original pro o f requires the base space X of the given principal G -bundle to b e a finite CW co mplex, our pro o f requires only that the bundle be a pullback o f the univ ersa l pr incipal G -bundle. Given a spac e X , let LX = F ( S 1 , X ) be its space of un based lo ops. Ev aluating lo ops at their basep oints gives a fibration LX → X . F or a top olog ical group G of CW t yp e, let ξ : E G → B G be the universal bundle, and let Ad( ξ ) : E G × G G ad → B G b e the ass o ciated adjoin t bundle. Then the follo wing result is folklore. Lemma 9.1. L et G b e any top olo gic al gr oup of CW typ e. Then ther e is a fibr ewise homotopy e quivalenc e L ( B G ) ≃ E G × G G ad of fibr ewise H -sp ac es over B G . Pr o of. Let G × G act on G ad by the rule ( g , h ) · x = g xh − 1 . Then the res triction of this a ction to the image of the diag onal ∆ : G → G × G co incides with the given action of G on G ad . W e hav e a pullback square E ( G × G ) × G G ad E ∆ / /   E ( G × G ) × ( G × G ) G ad   B G = E ( G × G ) /G B ∆ / / B ( G × G ) in which the vertical ma ps are fibrations and the horizo ntal maps a re induced by ∆ . The spa ce E ( G × G ) × ( G × G ) G ad may b e ident ified with B G . T o show this, we first quotient out by the action of the left-hand co py of G in G × G . Since this action is free, we obtain E G . Thus when we take the quotient by the right-hand copy of G w e get E G/G = B G. It follows that E G × G G ad = E ( G × G ) × G G ad is CONTINUOUS TRACE C ∗ -ALGEBRAS AND GA UGE GR OUPS 23 ident ified with the homotopy pullback of the diagonal of B G with itself. But the latter coincides with the actual pullbac k of the diagram ( B G ) I p   B G B ∆ / / B G × B G where ( B G ) I = F ( I , B G ) is the free path spa ce of B G , and p is the fibration which e v aluates a path a t its endp oints. This pullback identically co incides with L ( B G ).  Corollary 9. 2 ( “Gottli eb’s Iden tit y” [8, th. 1]) . L et G b e any top olo gic al gr oup of CW typ e. L et ζ : T → X b e a princip al G -bund le induc e d fr om t he universal princip al G -bund le by a map h ζ : X → B G . Then t her e is a homotopy e quivalenc e of H -sp ac es Γ(Ad( ζ )) ≃ Ω h ζ F ( X , B G ) , wher e the right side denotes the b ase d lo op sp ac e of F ( X , B G ) with lo ops b ase d at h ζ . Pr o of. 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Dep ar tment of Mat hemat ics, W a yne S t a te University, Detroit MI 48 202 E-mail addr ess : klein@math.wayn e.edu Dep ar tment of Mat hemat ics, W a yne S t a te University, Detroit MI 48 202 E-mail addr ess : claude@math.way ne.edu Dep ar tment of Mat hemat ics, S aint Joseph’s University, Philadelphia P A 1 9131 E-mail addr ess : smith@sju.edu