Banach categories with conjugation.

BANACH CATEGORIES WITH CONJUGATION

LAZARO by

RECHT

Licenciado en Matematicas, Universidad de Buenos Aires

(1963)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY September,

Signature

Certified

Accepted

Autho

r

Signature Redacted

DepartmI of _{Mathiatics, August 25,1969}

by

Signature Redacted

T>6sis Supervisor

by

Signature Redacted

Chairman, Departamental Comitee on Graduate Students

Archives

MASS. INST. Cc

OCT

₂

₁₉₆₉

k1'FRA RIE59 1969 of

2

BANIACH CATEGORIES WVITH CONJUGATION. Ldzaro Recht

Submitted to the Department of Mathematics on August 25, 1969, in partial fulfuillment of the requirement for the degree of Doctor of Philosophy.

ABSTRACT.

This work is devided into two chapters.

Chapter 1 is mainly concerned with the following problems:

i) Given a Banach category 4, to construct represen-tations for the functors .4(X)l and K(X;4) , where 14(X)}

is the monoid of isomorphism classes of bundles over X with fiber in 4 and K(X;4) is the Grothendie& group of the

category 4(X) of bundles over X with fibte in 4. These functors qre defined on the category of compact spaces.

ii) Given a linear continuous functor T: ---- + e'

where 4 and 4' are Banach categories, to construct a re-presentation for the functor K(X;T) on the category of compact spaces.

The notions of a Banach category, the category _6(x), the group K(X;k?) were introduced by M.KAROUBI (Theses, Universite de Paris, 1967, not yet published).

Problems i) and ii) are solved and from the cons-truction of such representations an exact sequence

(*) . . +1(XY;4 ) -+Kn(X, Y; I):-- Kn(X,Y ;4)--K~~~~ (0 XY -+...-- Ko (X, Y; k?) -+Ko (X, Y; 4) ---p Ko (X, Y;4'

is derived when the functor If is quasi-surjective, which is natural with respect to the compact pair (XY).

In Chapter 2 generalized cohomology theories

(.,.;4) and K (.,.;4) are constructed on the category of compact pairs, based on the notion of a Banach category with conjugation. Such a category 4 is a graded Banach

category together with an involutive functor F :4---+v 4 and a natural equivalenece :id4----e T of degree one

These theories are "periodic" of period 8 . Finally, in concrete instances 6f the category 4 long exact sequences are bonstructed relating the groups **(pt;4) with KR4(X), K₇*(X), KO*(X) , where X is a Z2-space, KR*(X) is the

2

"real" K-theory of X (cf.M.F.ATIYAH, K-theory and reality) Kz*(X) is the real equivariat K-theory of X (cf. M.F.

2

ATIYAH and G.B.SEGAL: Equivariant K-theory (Lecture notes) Oxford, 1965) and KO*(X) is the usual real K-theory of X. These sequences are found as aplicationz of a fundamental sequence which is derived from (*).

Thesis Supervisor: Donald W. Anderson Title: Associate Professor of Mathematics

3 TABLE OF CONTENTS page ACKNOWLEDGE.JENTS... INTRODUCTION .. ... ... CHAPTER 1. BUNDLES ... 1.1.BASIC DEFINITIONS ... ... 1.2. BANACH ALGEBRAS ... 1.3. THE CATEGORY

'(4)

. ... 1.4. NETS ... 1.5. THE SPACE BGL(4) ...

1.6. THE CLASSIFICATION THEOREM ...

1.7. THE GROUPS K(X;oC), K(X;e) ... 1.8. THE SPACE

1.9. 1.10.

B(4)

.*.* .* *

THE CLASSIFICATION THEOREM FOR K THE PRINCIPAL BUNDLE

GL(4) - P(4) -- +BGL (4)

...

1.11. A CLASSIFICATION THEOREM FOR THE FUNCTOR K(*;I) .*. .. .

1.12. THE RELATIVE CASE

14

5

63 125 70 2.2

CHAPTER 2. BANACH CATEGORIES WITH CONJUGATION ... 9 2.1. BASIC DEFINITIONS ... ...

2.2. THE CATEGORIES 4 AND e ... V8

pvq

piq

2.3. THE GROUPS KPC'(X,Y;4) AND

q

(X,Y;4) ..

(o

2.4. THE COHOMOLOGY THEORIES P AND Cn q?

BIBLIOGRAPHY ... 2 BIOGRAPHICAL NOTE ... .. .... 103

I

11 .. .. ,

0

.. . .. . . ...

I - _________

--11

ACKNOWLEDGEMENTS

I am deeply indebted to R. Ricabarra who taught me Topology and to D.W.Anderson for his help, suggestions and encouragement during the writing of this work.

I want to thank my wife who took care of the typing and made it easier for me to live these last three years

in a foreign country.

Finally, I wish to thank E. Santos Discepolo, E. Rivero and Fred I. Perticani,

S

INTRODUCTION

Recently TL.Karoubi [31 gave a unified description of the various types of K-theories (KO, KU, the equiva-riant KG , Atiyah's KR , etc. )based on the use of Banach categories and Clifford algebras.

In the first chapter of this work we represent se-veral of the functors that appear in

(3},

notably

.'(.)f

,

K(.;P) and K(.;i) (see 1.6.,.1.9., 1.11.) , Moreover,

from the representation of K(.;Y) we obtain the sequence in 1.12.2. when T is a quasi surjectivefunctor.

In chapter 2 , Banach categories with conjugation

are introduced (2.1.). The cohomolohy theories 0.;)

and K(.,.;G) are constructed (2.4.) where 4 is a Banach category with conjugation, using Clifford algebras. The fundamental theorem 2.4.1. can be proved exactly as in

[3), using methods of Wood [5Jp and periodicity follows, Finally, concrete instances of Banach categories with

con-jugation t are studied and exact sequences qre found

e2.4.5., 2.4.6.)relating the groups 2*(pt;) with KO*(X)

Atiyah's KR*(X) [1] and the real equivariant KZ*(X) [21, 2

where X is a Z₂-space. These sequences are found as an aplication of the general sequence in 2.4.4. &

El

--6

CHAPTER I. BUNDLES

1.1. BASIC DEFITITIONS. (See

(3])

Let k denote one of the fields F or M.

l.l.l.Definition. A pre-Banach category 4 over k is an additive category (we assume the existence of a zero-object and of finite direct sums) together with the following

adi-tional data:

i) for each pair (A,B) of objects of 4 it is given a k-vector space structure on Hom6(AB) such that the

compo-sition maps are k-bilinear;

ii) for each pair (A,B) of objects of 6 it is given a topo-logical vector space structure over k on Hom6(A,B) such that HomnA,B) becomes a normable complete space and the composition maps are continuous.

1.1.2.Remark. Observe that condition ii) says that Hom4 (A,B)

is a complete topological vector space which has a bounded neighbourhood of 0.

Observe that for each non-zero object A of 4 1.-1.-3. Remark.

7

Home(AA) = End6(A) becomes a topological algebra over k with identity 1 J 0 , which admits a Banach algebra

struc-ture.

1.1.4.Definition. Let 4 and 4 be pre-Banach categories

and Y: &-- ' a functor (we assume T covariant). We say

that

f

is a lc-functor (linear continuous functor) if for each pair (A,B) of objects of 1 the map Y:Hom4(AB)

--Hom , (fA,fB) is k-linear and continuous.

Let q: 4 - 4' be a lc-functor. We want to define an abelian group K(T) associated with the functor I.

1.1.5.Construction of K(T). Consider the class I((9) of all triples of the form (A,B,o() where AB 60b(6) and

om:fA---B is a 4'-isomorphism. Two elements (A₁,B₁, ) ,

(A₂,B₂' ₂ ) in I(Y) will be called isomorphic if there exist isomorphisms

(

_,

y

_{in 4 , such that 9}

_{( )o}

and de ₂()

both make sense and are equal.

An arce in I(9) will be a family (A,B, 1t)t6[O,1 of objects in I(T) such that the map t .0 ( from L0,1] to

Iso(fA,?B) is continuous. The "end objects" ( (ABos ₀), (A,B,N)) of an arc in I((?) will be called homotopic elements.

We define a relation in I(f) by saying that two ele-ments (A,B,a) and (A',B', cK) are equivalent if they are iso-morphic to two homotopic elements. We call L(f) the quotient

of I(k) by this relation.

WIe define an operation in I(k) by the rule (A₁ ,B₁ ,d₁ ) + (A₂,B₂,f(₂ ) = _{(A oA2,BloB2' l}

It is clear that this operation also defines an operation in L(f). Moreover, L(Y) becomes a commutative monoid with zero.

1.1.6.Lemma. The sum of the elements (A,B,) and (B,A, ) is equivalent to (AeB,AeB,id) in I(P).

Proof: Observe that

(A,B,c) + B,Ayo) =(A*BB*A,&%0~ )

0 et-1

is isomorphic to (A.B,AB,) , where = ~ l) Next consider the homotopy

cos t (-sin t)%-(sin t)'( cos t

-1

It shows that is homotopic to )(O t Th/2).

Finally, as (A*B,A*B,

rr/

r

₂

is isomorphic to (AeB,A*B,id), the lemma is proven.

Now it is clear that if one wants to obtain a group out of the monoid L(f) one should "divide" by the classes of the "elementary triples" (AA,id) or, what is the same, by the triples homotopic to one of the form (A,B,T(N)) , for o :A - B a 4-isomorphism. More precisely, let us con-8ider the subclass LT(T) of L(') consisting of the equiva!-lent classes of the ele-mentarytriples (A,A,ic). Divide L(%?) by M(t), i.e., declare x,y EL(t?) equivalent if there exist m,n eM(f) such that x + m = y +- n. Observe that M('f) is

additively closed and consists exactly of the classes of those triples (A,Bo) such that = '(an isomorphism in

4).

Given (A,B,c) in I(T) call (A,B,k) its class in L(T) ; then, for every (A,B,O() in I(f), 1.1.6.Lemma shows that (AB,k) + (B,A,&~ ) 6M(t?). Therefore, the quotient L(T)/i(T) is an abelian group. We call this group K(f), and given (A,B,0) in I(T) we call d(A,B,N) its image in K(f).

El -4

Let 'O, OC ,t2, 2 be pre-Banach categories and

consider a diagram of the form ~fl

)0k

j~2

2T

where i ,2' ' 2 are lc-functors and Vlil =

We want to construct an abelian group associated with this diagram, called K(f).

1.1.7.Constraction of M(). First, we define the auxiliary category P(V) as follows:

i.- an object in

F

() is a triple (A₁,A₂,d) where A 6Ob(e()

and c(: pA T2A2 is an isomorphism;

ii.- a morphism ,:(A₁ ,A₂,d) (A ,A?, ') in () is a pair of morphisms :A A! in ki (i = 1,2)

such that

Yl~~lhr 2 04 1) ?,A

TA

YA

2 2 2 2

commutes. We topologize Hom )((AlAo),(Aj,A ,o.9)) as a subspace of Hom (A,A{) x Hom (A₂,A ) . This subsapce is clearly closed and therefore we get a pre-Banach category.

Now we define a lc-functor W(V): ' r- (ID) as follows: i.- ( = ( ₁ (), ₂ (E),id) , for E 4b O ;

ii.- ( = (? 1(f), 2(f)) , for f (.or '

El

-I0

We define K(i) = K(L(9)

WNe describe the &lements in K(W(V)): Such an element is of the form d(A₁,A₂, k), where A₁,A₂ 6Ob 4 and

V:((A ),2 (A,),id) ? (f ₁(A2 2(A ₂ ),id) is an iso-morphism in P('i), i.e., d= (d ,(2) where a :i (A,)

-i(A2) (i=1,2) are isomorphisms in O% ,and such that IP= 02( 2): P1O(Al) ,P2 YP₂(A₂)*

1.1.8.Bundles with fiber in f. Here we want to define the

notion of a bundle over a compact space X with fiber in a pre-Banach category e (see [3)). We want such bundles to form a Barach category (i.e., a pre-Banach category such that all idempotent morphisms-projectors admit a kernel). We start with a general procedure.

1.1.9.Definition. An additive category is called pseudo-abelian if all projectors have kernels.

Given an additive category 4 we want to solve the following problem: to find a pseudo-abelian category ' and

an additive functor X: 4- V 4with the following universal property:

For every pseudo-abelian category VD and every addi-tive functor T: -- ,

P

factorizes through ' by means

of X..

The construction is straightforward. We take as ob-jects in ? the pairs (E,p) where E EOb'd and p:E - P E is a projection operator. HomZ((E,p),(E',p')) is defined as a quotient of the subgroup of Hom,(E,E') of those f such that fp = p'f . We want two such maps f and g be equival lent if fp = gp. The composition of maps in 4 follows from the composition in 4. The functor X is defined by

= B)= f , for E Ob4 , f&Jor f.

11

Ye show that

4

is pseudo-abelian: Let q be a pro-jector in (E,p) ( q = class of q ). Then qp = pg and

(q2-q)p = 0. 17e claim that a kernel for q is given by

-:(E,(1-q)p) - (Ep) . Indeed, if 7:(Fr) 4 (E,p) is such that 77 = 0, then qfr = 09 so f also defines a

map T:(Fr) - (E,(1-q)p), because fr = pf = (1-q)pf

(as qpf = 0). Clearly, O = (1-q)f' because fr = (1-q)fr. The uniueness of

r

is also clear. The universal property of (1-,4) is easily checked.

In the case where 4 is a pre-Banach category, 4 be-comes a Banach category.

Let X be a compact space and 4 a pre-Banach catego-ry.

1.1.10.Definition. The category T(X,4) of trivial bundles over X' _{with fibre in 4 is described as follows:}

i.- an object in T(X,4) is simply an object in

4;

ii.- a morphism f:E - F in T(Xe) is a continuous map f:X _{P Homi(E, F).}

If f:E - F and g:F - H are morphisms in T(X,4),

the composition gf is the map gf(x) = g(x) o f(x) The pre-banach structure of T(X,d) is clear.

P1..1l.Definition. The category ?(X) of bundles on X with fibre in 4 is the pseudo-abelian category '(X,) associated with the pre-Banach category T(X,4). It is a Banach category.

12

1. 2. BANACH ALGEBRAS

In this section we are going to prove soma facts about Banach algebras which will be used later on.

Let A denote a Banach algebra with identity 1 (not necessarily commutative). Vie denote by

II i

the norm in A and we assume:

i.- l(I = 1;

ii.- 11XyQ U 4 y1

Let PcA be the set of idempotents of A; we denote by GL(A) the set of invertible elements of A and by SL(A) the conn-ected component of the identity in GL(A).

1.2.1. Lemma.

Proof: (p,p') E(PxP)

Let (Px) ₁ be the subset of PxP consisting of the pairs (p,p') such that lip-pIll (/1 and let I denote the unit interval [0,13. Then there exists an uniformly continuous function ft:(PxP)1 xi - p SL(A) such that

i) p = f₁(pp')p'(f₁(p,p'))71 _{, (p,p') E(PxP)}

1

ii) f (p,p') = 1 , (p,p') C(PxP),

Put (p,p') = 1 + t(2pp'-p-p') , t eI

1 WVe claim that ft(pp) EGL(A) and there-fore ft(ppt) ESL(A). Indeed,

ft(p,p') . ft(p',p) = 1 + t(t-2)(p-p')2

-_

*)

so that, as

It(t-2)(p-p')

2

_1<l,

_{it follows f}

t(p,p') EGL(A). The property ii) is clear. We prove i) Vie have to show that f 1(p,p')p'a = pp'a is equal to p, where a =

(fl(p,pl))~ . Multiplying the relation (1+ 2pp' -(p+p'))a = 1 by p on the left, we get (p + 2pp' - p - pp') a =

15

7e say that two idempotents p,p' are equivalent if there exists a ESL(A) such that p =

apa~.-1.2.2.Corollar. i) The relation of equivalence in P devi-des P into its arc-connected components. The distance between two different compo-nents is t 1.

ii) P is closed and the arc-connected com-/4 -p are

ponentslopen-closed subsets of P. iii) P is locally contractible.

Let paP be fixed. We want to define a principal bun-dle on the arc-connected component of p , fl. Let Cp be ' the set of all a eSL(A) such that ap = pa, and let SL(p)

cC be the set of all a such that a(l-p) = 1-p . Si-p

milarly, let SL(l-p) be the set of all a such that ap =

=

p

.

1.2.3.Lemma. i)

P

is homeomorphic to the left coset space p

SL(A)/Cp

ii) Cp is isomorphic as a topological group with SL(p) x SL(l-p)

Proof: We use the map TT:SL(A) -- + p defined by

TT(a)

= apa~ for a e-SL(A). This map is clearly continuous and onto because of 1.2.1.. It is also open and defines a principal fibration with group C. Indeed, let q d

P

and

P p

let us define a section over the open unit ball B around q in rp . Let p = p0, p, -.., pn=q _{be elements in rp}

such that ltpi+l - p 11<1 , _{i=O,...,n-l. Then given q'E Bq,}

define Gr(q') = [f(po,p). ... .f(pn-l'nn)'f(Pnq) The map G:B - SL(A) is a section of TY, (q ) p

*(q')~' = q1 .This ,hows that

Tr

is open.

I1t

image of an orbit is constant. Clearly, the fibers of

Tt

are the left cosets of C in SL(A); therefore, the induced map

1%.0 p

TT:SL(A)/C - p is a homeomorphism. This proves i).

p p

In order to prove ii), define g:SL(p)xSL(1-p) -- C

by g(a,b) = ab. This defines a homomorphism because if a e.SL(p) and b eSL(l-p), then ab = ba . Conversely, given c eC put h(c) = (1-p+cp,c(l-p)+p) ; then h is a

homomor-p

phism h:C - SL(p) x SL(l-p). Clearly, h is the inverse

p of g,

1.2.4.Corollary. The map Tr defines a principal fibration SL(A) R P p over P with group

p p

SL(p) x SL(l-p) .

For the rest of this paragraph we introduce the foll-owing notation:

W will be a fixed projector in A. WVe will say that p EP is dominated by Tr if Thp = pTi = p (observe that in any algebra this relation of domination defines a partial order in the set of idempotents of the algebra with greatest element 1 and smallest element o ).

Wde define S CP as the set of all p rP dominated by 1.

We want to study the inclusion S - P .

1.2.5. Proposition, Let UCP be the set of all p EP such that the distance from p to S is less than \\l-TrIiV1 (U=P when IT=1). Then there is a retraction r:U -.--- S.

Moreover, is U'6U is the set of all those pEP whose distance to S is less

than (2 HIR I 2 + l l-Th 1 )- , then for every p CU' , [p-r(p)(1

<1.

15

Proof. Let p EU and a = (1-p)W' . 7e show first that the sequence an , n=l,2,..., converges. Moreover, con-sidering the an as functions from U to A we show that the convergence is unoform on any Vc U consisting of all p EP whose distance to S is 4 6 for any fixed 6<11l-Tr-1.

Indeed, write p = q+e, where q eS and lieV & E . Then, using the fact that p is a projctor and that q eS we get

a T-q-eir

a2 _{= a + (1-e)(1-') e'I}

an = a + (1-e)(1-r) 11 + e(l-) +...+ (e(1-)_n-2 Ie T, for n>2. The usual argument about geometric series shows that an

converges uniformly on such V's. Then it is clear that the limit lim an is defined and continuous on U. We observe

n --&O that

lim an = T - q - IN [1 - e (1-T') e 'IT n - oo

Next we define r:U -- S by r(p) = II - lim an

n-- oo

Onserve that r(p) is in S because r(p) = q+'T1-e(l1T)]e l and therefore

it

dominates r(p). Also observe that r is a retraction, because if p ES we can take q=p, e=O , and therefore r(p) = p.

It remains to prove the last assertion. Let p 4EU' and write p = q+e , where hiell ((211'1111 2 + 1ll -T 1L) and q

ES

. Then

I\r(p) - pU= IT [1-e(1-) -e T - e(1 T 1= 2 Ie + Lie (*) 1- It e it Ill-T Ii

But the condition on e implies Itell1Y112 _ < and 1 - I ell lll-TTll

10

if Tr / 0 then the same condition implis 11 ell <-. On the other hand, if 17 0 the inequality (*) shows that

11 r(p) - p ( 4 (1 e (1 (observe that the norm of a projector is either 0 or = 1). The proposition is proved.

1.2.6. Corollary. The closed set S is a strong deforma-tion retract of a neighbourhood in P. Proof: Consider the homotopy ht:U' -- P

(U' as in 1.2,5.) given by ht(p) = ft(r(p),p)pft(r(p),p)F toEI , where f is as in 1.2.1.. The properties of f show that h₀ = idU,, hl = r and that during the homotopy the points in S remain fixed. This proves the corollary.

We now want to introduce certain special norms on certain Banach algebras. ie need some notation.

Let M be a topological vector space. We say that M is a Banach-like space if there exists a norm I1 [tan U which induces its topology and which makes it into a Banach space. We call any such norm a compatible norm on M.

Let A be a topological algebra. We say that A is a Banach-like algebra if there exists a norm 11 g on A which induces its topology and which makes it into a Banach alge-bra. 7e call any such norm a compatible norm on A.

Let A be a Banach-like algebra and M a Banach-liie space. We call M a left A-module if there is a map &:AxM IrM _{which is continuous and which makes M a left}

A-module in the algebraic sense. Similar definition for right A-nodule.

1.2.7.Lemma. Let (A,.1 1) be a Banach algebra and M a Banach-like space which is a left A-module. We denote the map AxM --- M by (a,m)

17

I I on M such that I a-rn _{6 | al Urmi for every} a 6A, m eM.

Proof: Let I , be any compatible norm on U . We put, f or m M, I m I = sup jam I' / a ; a EA,aO I . Then, clearly,

I I _{is a norm on M; also, Imi ( m (". But AxM --- M being} a bilinear continuous m-ap, there is a constant K such that

lami' ' K Ita11 Imi' for a eA, m e . So,

Imi = sup amlI/lad; _{a fA,a/0j L sup KIaU1(m'/a1I;aaA,a/Oj}

=

k

Im

Then, It and I I' are equivalent.

Finally, let a EA, a/'O , then Imi = sup (xm I'/Ktx t ; x E A, x jd 0

k sup ixam I '/ xa V; xa/0 , x ( A A sup Si xam I /it a I lix ; xa/O , xr.A

= sup \lxam I'/ta 11 Uxu ; x/0 , xcA

=

1amt/1iall

, for mM.

This proves the Lemma.

We now consider pairs (A,(pi)i,₁ ) , where A is a

Banach-like algebra, I is a finite set, pi is an idempotent of A for ieI and pipj = 0 for igj , p = 1.

iEI

Given a pair (A, (p ) e₁ ) we _{say that} a norm 0 lion A is admisible if there exist compatible norms I 1. on A.. =

= p.Ap. such that if we write jai _{= sup} j[a ; ijrI J, for aeA, where a.. _j = p _{p ap} ,ap, then

mu

= sup IaxI/jx( ; x#O, xcA for every aeA.

Given a pair (A, (pi) ieI) and a subset KcI , K $ ,

we write p- pi . Then PK is an idempotent. 16K

1.2,8.Lemma. For every K c I, IpK Il 6 1.

Proof: Indeed, _{1 PK 1= sup} pKx t/ (xi ; x CA, x40J But IpKxI= sup Ix .. ; i6K, jeI

x\ = sup xijlij;

i,j6

I where x = pixp .

Co, for xeA, x/O , pKxI/ixL 6 1 . The Lemma is proved.

1..29.Proposition. There exists an admisible norm Il III on A such that the natural inclusion

(AK, 1 II) C(A, IlIl II ) is an isometry. Proof: We define norms I

1.

. on all A. . for

13 13

i,j61 as follows:

a) if i,jeK, we put the norm

I

I. that we already had on A. b) if iLK, j#K , we put on A.. a compatible norm I lI. such

that abi = UaiIL (b lij for a A, b. .A. which is possible because of 1.2.7. (Observe that A. . with

the restriction of the norm

(I I

is a Banch algebra by

1.2.8. and that A. . is a left A. .-module).

13 il

c) For the rest of the pairs (i,j) we put on A any compa-tible l3anach space norm.

Then, we put Jal = sup ja; i, EIJ for any a = ea A , a= piap , ijeI , and

i,13 aI

III

a III = sup I axI/IxI ; X, _{A, xO I} Then clearly Il (it is a compatible norm on (A, (pi). ₎ . We have to show that li a ii =

= III a ill _{for every a6AK. Clearly,} 11 a 1[6 1(a I for a6_A.

consider x6A, x = J ax I i, j 4K

x.,

r~~ 13j a . = sup Ilaijxjl(il ; i, j6C,

= max (sup Ia ixj ₁ ( ili _{IjK,liK ,} max (sup

II

all

Ila

il lxi

1

IX jK1 1 j1 ;RlOK ,

So, sup

laxI/1x[;

x/O,xEAI =

_lII

_a1il

_{a il.}

YJe used the fact a .11= 11p ap f 1.3. THE CATEGORY 4) Througho The catego i) An object ii) A morphism 4-morphism

ut this section we fix a pre-Banach category

ry 08(6) is defined as follows: in 6 (6) is just an object in 16 .

s

(:E - F9

i:L -E-- F,

iii) if O = (i, j):E - F an

in CC() is a pair (i, j) of

j:F - _{E such that}

_ji

_{= idE'}

.d ot' = (i1,

j'):F

---- H are

13 x .eA.. . Then ; now

Xkl I

_'I

a. .X. 13 31 leI

1=

lE]

|t all

lx

)

that 11 4 Ila

l

and _b).

20

morphisms in

(4),

we define Wo'( = (i'ijj"):E -PH. iv) The set Hom (EF) CHom₄(z,) x Hom6(F,E) carries

the induced topology and clearly the compostion maps in

?(4)

are continuous in these topologies.

1. 3.1. Remark. Observe that if 4 is a Banach category a morphism ot:E ---- F in (6(4) gives a direct sum decomposi-tion of F, F F 0 F₁ with E F0 (if 0(= (i,j) , define

pO = ij, p₁ = 1-p0 , and take F = Ker(p1 ) and F1 = Ker(p )).

We now define a functor

)-: e(e)

v Top, where Top is the category of topological spaces with continuous maps. 1.3.2.Definition of the functor 5 :d(4) ---- Top.

i) If E 6.Obt() , we put ) (E) = p6end(E) ; 2

ii) If 0(= (ij):EJ o F is a morphism in d(P), we put (a) (P) = ipj for p in

V(E)

.

13.3.. Remark. If 0(=(i,j):E -- F is a morphism in

e(4)

observe that if we call ''r = ij

,

then t(ao)(4(E)) is the set of all projections q 6Endg,(F) which are dominated by ' .

Let EFH be objects in q(4) and let o'=(ij):E

---+ F and (=(kg):F - H be morphisms in q (4). We call

p

=(m,n):E -PH the composite morphism V=pO( . We put IT i j C End4 (F) and = mn C. End4 (H) Finally, let

U = p (F) ; distance (p f (ot )(()) )((217 ++1l-tLI) and V = \q6.(H); distance (q, t ()( i(E)) ) /(2 tlI+ \A-Observe that vie have defined homotopies h t:U P f(F) and

hf:V --- + (H) according to 1.2.6. (taI). These homoto-pies begin at idU and idV respectively and end at the re-tractions r:U-yg (o) (E)) C- - U and

r':V C

( )(,g(E)) -- V repectively. The following

1P 4, Lemma.. The mnp-s t( )

U

n

(

₎

h t _{and h4 (P) agree on}

Proof: First we show that the compositions agree on U

ntf(

) (V). In

Call a = (1-p)Tt. Observe

='Y -lim an and hence

n

--

koo (()r(p) = k ( Y-lim But k'(Tg= mn= = (1-kpg) f = (1 n --i oo

?

and - ( (P)

an) g = k'Ig - lim n- co

(kag)n

kag k(l-p)gkTrg = (kg -kpg) to =

) f so that

r(p) = f - lim ((1- f(t)()) )n= r'( )(p). (observe that kg

y=

y).

Now, according to the definitions, we have ( ) ht(p) = k ft(r(p),p) p ft(r(p),p)~' g = k f (r(p),p)g t()(p)k ft(r(p),p)~' (*) g But k f (r(p),p)g = k(l+t(2r(p)p-r(p)-p)) = k g + t(2r (k()(p)) ( (p) k g ,

where f and fl are the homotopies described in 1 2.1. f the Banach algebras End₄(F) and End.(H), respectively and where we used the fact that for any

= kg ()(a) = (b)(a) we have . Similarly, atEnd,(F), j()(a)kg calling a= t(2-t)(r(p)-p k f (r(p),p)~ g = k ((l+t(2pr(p) - r(p) - p)) = (kg + t(2kpgr'(kpg)- r'(kpg) .;( (t(2-t)(r'(kpg) - kpg 0=1 kg ((l+t(2kpgr'(kpg) - r 00 . (

:2:_

(t(2-t)(r'(kpg) 0=0 a

₎

_g - kpg) ) , )2) (kpg) -

kpg)

+ kg) - kpg) ). 2 ) D ZI and r' U g() ~ ( ). fact, that let r(p)

t( )r

p

6 = I (r,( ( )(p)) ( )p )

I

2z

= k g f (r'((W)(p)), ( )(p))~

where we used the formula (*) of 1,2.l.Lemma to obtain the inverse of ft(r(p),p) and the above mentioned commutation relation with kg. Finally, loking at formula (*) we get

( )

ht(p) = k ft(r(p),p) g t()(p) k ft(r(p),p)' g

= f (r'((()(p)), ( )(p)) k g ( (p) kg

= h' T( )(p).

1.4. NETS

Let T be a partially ordered set, the order relation being denoted by 6. Vie may then consider % as a category with one morphism [a,b] from a to b whenever a i b(a,b cl' ).

The objects in this category are the elements in % . We shall speak indistinctly of an ordered set or the category determin-ed by it.

Throughout this section 4 will denote a pre-Banach category.

1.4.1.Definition. _{Let 'P be a partially ordered directed set.} A net in 6 is a pair ( where

S

is a covariant functor

23

(9,b) is called a subnet of ($', ') if there exists a pair

(,f),

where 6 is a covariant functor C:D--+' and

- is a natural transfoxTiation %U:

S

- ' & .

1.4.3.Definition. The net (T,S) in e6 is called a cofinal net _{if for every object} E e d(4P) there exists ael and a morphism f:E - S(a) in e (4).

1.4.4.Definition. The net (VS) in 'o is called a total net if for every aeC and every finite set of morphisms f.

S(a) 3pci , i=l,...,n in d(6), there is an element b C- , b - a and a set of morphisms gi:Ci -+ S(b) , i=l, ... ,n in (4) such that gif. and b([a,b3) are homotopic in Hom 6(a),6(b)) for i = 1,... ,n (i. e., there are arcs joining gifi and ([a,b]) in Hom ) ( 6(a), cS(b)) ).

Next we _{introduce the category 9(e)'O .}

i) An object in 3(Q0" _{is a sequence A =} (,A ,... ) of objects in & such that there is an integer n such that Ar is a zero-object for r A n.

ii) A morphism f:A -- B in d(G()" is a sequence of mor-phisms fi:Ai - Bi , i=ol,... in (4) (here

A = (A ,A,... ) and B = (B ,B ,...)

iii) if f = (f _)i= :A ) B and g = (g )i=₀ :B _- C

are morphisms in 6(4)' we define gf = (g f.). :

A C1

We define the functor S:&6(4)W - (4) as follows:

0D

i) If A = (A 0 ,A,...) we put S(A) = ~. A . i=O

'00 00

ii) If f = (f ) :A - B, we define S(f) = f.

I I ~

Observe that we cnuld similarly define the category towhich is easily seen to be a pre-Banach category and then e( VU3) 9(4)0 ,

1.4.5.Proposition. If (Z,) is a cofinal net in

4k,

then (1, SS) is a total cofinal net in

4

. Proof: _{Clearly, (5,3S) is a cofinal net in G,} .7e show that it is total.

Let aci, and fi:S$(a) OD Ci , i=l,...,n , be mor-phisms in @(6). Let b(a) = (A ) and let p a 0 be such

000

that A is a zero-object if j>p . We call C = (Dk 1_)ko

EOb R' , where Dk is a zero-object in 19 if k d p+i and where D C. , _{for i=l,...,n. We choose an index}

b 6 , b Aa, such that there are morphisms g!:CY - (b) 1 1

in 0(4) . (This can be done because (D,$) is cofinal in 'w). We shall prove that S(g )f is homotopic to S(.a,b})

1 1

in Hom (4)(S6(a),Sb(b)). 00 p+i-l Suppose

_b(b)

_{= (BM)m=0 . We call}

21=

E

_Bm m=o

O0

E2 = B , E3 =

E

B. Then clearly sS(b) is iso-morphic to E. loreover, using this decomposition of

SS(b) , the map S(g!)fi can be represented by 0

S(g )f u (0,v,0) ₎

for some u:Sb(a) E , v: ₂ S- S8(a) in 4 such that vu = id5 E(.)

r

Similarly, _{SS([a~bJ) =(0 ,(s,0,0) ) for some} (0

r:3S(a) 7 E ,

s:E

E S>(a) in C> such that sr =

= idSS(a)

A homotopy ht is given by the formula cos 9. r

ht =

(

sin

e.

u , (cos9.s , sine.v ,

)

)

;

0

clearly h 6.Mor(o()) _{and ho = SS([a,b1) , h T/}

2 = S(g!)f

This concludes the proof.

Again we fix a pre-Banach category 6.

1.4.6.Definition. A family of generators (A ). for the category 6 is a family (A )i of objects of o such that for every object Ee. _{there is a map f:E ---- A} *A ...

1 2 * .eA in <(4) for somne i, .. , in eI

in

'Oe shall fix a family of generators (A ). in 4

and we shall fix a well-ordering & in I. W/e add to I a last element o ; we choose A0 0 = some zero-object in

4,

we call 100 the well-ordered set 100 = I U lool _{and we}

are interested in the family of generators (A.). =

02 We want to consider two sets, I ₀₀ and I

00

~LO

1.4.7. Definition of I . An element i of 10 is a sequence i = ( I, i, , where io,1₁, ... e.I 00 with the following conditions:

i) io _{ii 1 i2'}

ii) there is an integer n such that i= o if k A n . We introduce a prtial ordering 6 in I : if i=(i

z (

and (j0 ,j 1 ,...) we say that I -= if there exists an injective increasing function N:N - N (N = 10,1,2,..j)

such that i k = j(k), keN . Observe that among such functions d there is one which is the smallest, i.e., c(k) has the smallest possible value for each keN. It can be defined re-cursively by saying that o'(k+l) is the least n>o((k) such that ik+l = J4(k+l). Given i,A! E I₀ , 6 A, we call this

smallest function d4:N -- N , o(.. . The following

tran-sitivity property holds: if i A l and

a

L k then

k 0(. =i C%. ,

Now we show that the relation defined in IO is a 00

partial order. It is clear that i 4 i for every ieI . Let

OD 00 OD

r

r=o

'

Z

~ r 0 be elements in I and suppose

. Let n be the smallest integer k such that

i ;

k _then oc. .(m) = A(..(m) = m for m<n .But if

d..(n)> n , then

j

ni and also as (..(n) n, in j

n A n n1n

which is a contradiction. So, c<. .(n) must equal n ; but then we get the contradiction in = in & Then necessarily

= . The transitivity of the relation 4 in I is obvious. Moreover, we claim that I is directed by

-00

Indeed, let i = (i

)=%

and = (jr'O , where ir =0

for r> n₁ ,

jr

= co for r> n₂. We need to construct an ele-ment k = (kr) 0 following both i,A.

Let J = ir ; r nU jr ; r n_{2 0} oCI

let f:tO,l,.,p --- J be a bijective increasing function; let _{s =} Card _{Ir 4} n;ir=f(s) ,+ Card [r & n₂ jr=f()j I for s elo,,...,p . Now we define

27

k = f(1) for +

-0 ,

k = f(p) for 0+... ÷ & r+ - 1

kr = o for r + +

Clearly, by construction k EJ and obviously i m k, j i k. We call co EI the sequence ( o, o,...).

1.4.8.Definition of I O An element = in Icc is a sequence (1,,1 , .0.) of elements in I such that

(io Y 1100

there is an integer n such that ik -co whenever k a n.

tie introduce in I the product order obtained from the order in I which we still denote by & . It is clear that

(I ,

)

is a partially ordered directed set.

We now construct the fundamental net

(,

>) in i) We put 9 = I

W

ii) If i = (1k) k0 we put -(i) = (S(A. ))k= Ob(6LL,

where if i ~~ ) 0 ,we denoted by A. = (A ) 0

1k 'kr r=-O .1k i kr -o

6Ob(~-k k,r

GOb(4n .

iii) _{Now if} i in I 'z i =(Lk)= ' = '

00 k=)M=

kC r= 'r imr_ s=O , we need a morphism

(

): (j)

-(

()

in ig (a') . As for every

k4N we have i, 2k , we get a function o( :1k --- N as described in 1.4.7. We now define

I.

-29

S([i,])k:S(A.

) )S(A. ) as being the pair

k k ik ( )

(

q

)

, where qk _ (q

)

' _ rs r,s=O k _: rs _{ks Akr} Prs:A ₃ - A ks 1kr k 'p _=~rs (pk )

_)r,

_s=O0 k qrs k Prs id A = lks 0 id A k 10 if r = 0. . (s) otherwise if s = (r) otherwise. ;

This clearly defines a net in Vu because of the tran-sitivity properties of the oc's. On the other hand, this net is clearly cofinal in 16o because eC is a family of genera-tors of

C

. So, according to 1.4.5., ('*,S>) is a total co-final net in

G

.

1.5.THE SPACE BGL(e).

In this paragraph we indicate by ' a pre-Banach

ca-tegory and by (4, >) a total cofinal net in W7e introduce the following notation:

29

i) If

&L

is any category we denote by JAJ the collection

of isomorphism classes of objects in

-k.

If

T*:*.

---

1)

4'

is a functor, we get a map J :

-

1

3

.

Observe that t. behaves as a covarianit functor defined on the category of categories with values in the cate-gary Ens of sets.

ii) Suppose F:-k k Ens is a contravarint functor and

H:ekA *' is some covariant functor for given

ca-tegories

-,-k';

we say that F is H-representable if there is some object A in k' and a natural eqnlivalence

X:

F - Hom ,(H.,A)

as contravariant functors A - Ens. In case F is

H-representable. the pair (A,X.) is called an H-repre-sentation of F.

iii) 71e denote by CTop the category of compact spaces with continuous maps. The natural functor 1:CTop -- Top gives a functor

jtL3:

CTop] -f [Top]. (Where

bra-ckets indicate that morphisms are homotopy classes of continuous functions).

Let (4,S) be a fundamental net in 6' for some fa-mily of generators a-of 4 and consider the net (4,SS) in

4 (see 1 4.)

1.5.l.Definition of BGL(V). We call BGL(e) the space lim

S,

the limit being taken in the category Top. For a 6.T) we call i the natural map ia:PSS(a) > BGL(4). We now define a natural transformation

X:

(4()} Hom ITop(t.,BGL(G)) , where both functors are

contra-variant from [CTo4p to Ens. This is clear for

HomTopJ (&.,BGL(e)) and will be shown later for t6(.)J. Let X be a compact space and (E,p) a bundle over

30

X with fiber in d , i.e. EOb(&) and p:X E End,(e) is continuous and p(x)2 = p(x) for x6X. To define Y(

j(E,p)j)

where j(E,p)j denotes the isomorphism class of (Ep) in

<(X), we do the following:

i) choose a cb such that there is a map f:E S S(a) in J(6) ;

ii) consider the map i -- BGL(f) ;

D(f)p:X

iii) put

%(

t(Ep)j ) = [iak(f)pj , where [.1 stands for

thehomotopy class of the map we put inside. We are gc

i.e. that X ( but before we si

[CTopi

---- Ens.

ing to show that (E,p)] ) does not ow that

IQ(

₎

these rules define X depend on the choices is a functor _{C(. )i}

1. 5. 2. Lemma. If we call

q

: CTop ICTop3 the natural functor, then there is a functor still called

b?(.)1:[CTopA o-7--Ens such that the foll-owing diagrr.m is commutative

CTop PEns

[CTop )

Proof: _{Let X be a compact space. We define}

{o(.

)): [CTopI ) Ens as follows:

1o(X)l = the collection of isomorphism classes of bundles over X with fibers in eq

if [f :1 0- Y is a map in [CTop) and (E,p) is a bundle over Y we put j46(f)j(((E,p) ) = j(E,pf)j

The only thing to prove in order to conclude the Lemma is that if f ₀ --f -:X Y , then \(E,pf₀ )) = (E,pf)f ,

,

made;

El

3'

or, in other words, that (E,pf ) is isomorphic to (E,pf,).

₀

Consider the Banach algebra End,(E)X of all continuous functions X End4(E). Observe that p is a functionE

p:Y - Ende(E) such that p(Y) cjD(E), so that if we call h:XxI - Y the homotopy between f₀ and f 1(h Xx{01 = fo; hIXxjIA = f₁ ), thenthe map

r

:I - Enda(E)L given by

S(t)(x) = ph(x, t) , xeX, tel, defines an arc in Endg(E) begining at pf and ending at pf . We consider the local-ly trivial bundle SL(End4(E)X) W described in

1.2.4.; we lift the arc in pf to the are in

SL(End((E)X) with starting point 1 . Then for each t we have P(t) = f(t)

r(

0 ) (t)~ , or in other words, for each xeX and tE6, ph(x,t) = (t)(x)pf (x) (t)(x)1 and in particular pf(x) = ?(l)(x)pf (x)I"(l)(x)3' ; so that

the map a = (1):X --- EndC(E) satisfies (pfl)a = a(pfo) and so defines a morphism of bundles (E,pf0) (E,pf ) over X with inverse a . The lemma is proved.

1.5.3.Proposition. X. is well defined and is a natural trans-formation of functors.

Proof: Let (E,p) and (E',p') be two isomorphic bundles over X. We have maps f:X > Hom4(EE' ) and

g:X Homa(E',E) such that for each xeX the following relations hold:

f(x) p(x) = p4(x) f(x) , p(x) g(x) = g(x) p'(x)

g(x) f(x) p(x) = p(x) , f(x) g(x) p'(x) = p'(x)

Nlext consider a QS such that there is a map h :E -- SS(a) in j(4) and a' 6.9 such that there is a map hi:E'- SS(a')

Let ia, if(hi)p' are homotopic maps X -- BGL(4).

(ilj 1 ) , h' = (ifji)

Homr(SS(a)SS(a') ,SS 1 -

i

p(x)j i-f(x)p(x)

_j

₁

; consider the map r:X

-a)*S6(a')) given by i g(x)p'(X)jj 1 - itp'(x)j Then r(x)2 = 1 and, (ilp(x)i₁ r(x) . 0 for every x.X ,

0

0

J

0

0

- (r(x))~

We now define the maps h 2:3 (a) -- y ( S (a) *S6(a I) ) h' :6(a)- ₂ (SS(a)*SS(a')) *(S(a)*SS(a)) S(SSka)*Sb(a ))

in 9(5o)

given by h2 = (i 2 j 2) 0 0 1 0

)

F(0 0 1 0)) 1 h = (ifj?) Call F = r(x) =0 s(x) 0 r(x) 0 0 0 1

)

(0 0 0 1)). and define for x6X . (ht) (hl)pl(x) Then we have

for xeX. Moreover, the map s:X - 7 End,(F*F) is homotopic to the identity by the homotopy _{s t:X -- End(F*F) given}

r(x) 0 PI p(x)j : F*F -- : F*F s (x) [ _{(h 2))hiYp(x)l S(X)~}

33 by S (x)

r(x)

0 O cos 1 )(sin t t where 0 & t diT/2 (s = S,

-sin t cos t 1 0 STr/2 = 1).

Cost

-sin t 0) r(x)) sin t Cos t Therefore, the maps and t(h )tf(hj)

_{p' (x)}

are homotopic maps X -.- (YF) .

Now choose _some be'), b t a, b h a' , and a map h3:F --- SS(b)

30

h 3h 2"-S b

h₃h' .,SS({a',bl) This can be done

in A (G) such that

in Homj(,) ( S S(a), S(b)) in Hom (S5( a'),SS(b))

because the net (9, SS) is a total cofinal net in Co. ia Thus we have (h 1) p = ib4(S 6( a,bl)) 4(hl) i ib (h 3) t(h 2) +(hl) Finally, iately that the proof.

x(

= ia, (h') the fact that is a natural

is well defined shows immed-transformation. This concludes

show that

X

is a natural equivalence or, in other words, that the functor

table and that the pair

149(, )1 is '.-represen-(BGL((),X) is a 'L.-representation . Before doing this, however, we shall construct a spacial open covering of BGL(4).

steps. We proceed in several (h2)f(hl) p(x) p p Our aim is to of

I(,

)

~ ib (h3) (h') (h') p' s b fSS([a',b]) +(h') p, pf

i) if i = (ik) 0 & I we call n.

'k k=O (M $ 1 the only integer 0 such that i /oo and im =o for m>n . We call n. the order of i . We agree that 00 has order -1

ii) If i

= ( k .1₀₀ we put n. for the only

integer 3 0 such that i 4 o and im o for m

>

n

We agree that 00 = (op, _o,...) F.I has n = -1 . We call

00 O

n the order of i

iii) Let i C , i= ( 0k = (i -) 00

- - ~k kj j=0 for

k = 0,1,... . Then SS1) is cannonically isomorphic to ni n.

o ( * -k A

)

, where we use the convention that an k=0 j=0 kj

empty sum is to be replaced by a zero-object.

iv) Given i 6'% we want to consider the set of pairs I= \(kj) ; o 6 k 6 ni , 0 4

j

4 n . Thvn we just

re-;-k marked that SS(i) = e _A

a ae I. a Now for aeIi, we call jector defined by the matrix

in a canonical way. i

pa:3S(1) --- SS(i) the

pro-i ( a bc id _A 0a if b=c=a otherwise where b,c eI.

v) Given iC50 we consider the pair (B(i),(p )aEI. in the sense of 1.2.8., where B(i) is the Banach-like alge-bre End (S&(i)). Observe that p = ids(i) 1 B(i).

a

35

vi) Let 6,k6, k; let 1\ 11 be an admisible norm k

on (B(k),(p~) ) . Let aak

U(01 1111, , ) e t SO(k) where distance means Then, according to 1.2.5.

; distance _{Ix(p,4S'S(Ukl) (tSS( )))<1/311}

the distance in the norm U Iin B(k). a retraction

r(lI

1,,k):U(I

1,1,k)

--

-(-4

,_k)(tSS(

))

is defined and also a homotpy h (I[ 11t, ,s):U(111 ,2 ,_)tS8(_k

begining at idU(l (,_) ending at r(tI IIA, ) ing js6(9,k})(()) fixed (observe that

1~

.1 and 11-

_p

11=

aeI. CI a

vii) Now we put U(,k) =

k

admisible norm on (BQk),(p-)

e)

1 because of 1.2.8.).

Ui

,

,_k) ;

i

11i s an -*k in V . Observe that we still have the retraction r(A,k):U(A,k)

-fSf>(,_))(tSS(,)) anf the homotopy ht(U,k):U(Uk)

--4

SS(k) with the same properties as in vi) (because obvious-ly, for example, h (& lt,,k) and ht(I lI',Ak) for two admi-sible norms 11 H,11

I['

agree on U(II 1(,j,k)C\U(I( 11',A,_k) ).

viii) Suppose j L k i m in9 . Then because of 1.2.9. we have tS$>([k,gr])(U(A,)) cU(j,L) , and because of 1.3.4.

we have VS6([km\) ht(I,k) = h

(!

,

([L_,L])

on

U(,k),

leav-pa and

NO

ix)Let I E9. We put U(j) = U ik(U(Ik)) . Then

be-cause of remarks in viii) , U(j) is an open neighbourhood of'

i

(VIS(i)) = Z .C BGL(PD) . For

Ii)

we call Z . the A-th skeleton of BGL(f). Because of the compatibility of the ht(A,k) indicated in viii), and the fact that the natu-ral map lim (+S(A)xI) 7-BGL(G) x I is a homeomorphism, we get homotopies ht(j) for I L

,t(j):U(j) ---BGL(4)

which begin at the identities of U(A), end in retractions r(j):U(j) --- Z . and leave Z . fixed.

x)Finally, because of 1.2.8. and 1.2. 9., we have that if A k in 5 , then U(A) C.U(k).

1.6. THE CLASSIFICATION THEOREM.

Let X be a compact space and [f} a map in

Hom (TopX tX,BGL(P6)). We define a map ':Hom[Tl(X,BGL(P)) *7jI(X)j by the following rules:

i) Choose A.0 such that f(X) CU(A) (this can be done be-cause the family (U(j))A₆9 is an open cover of BGL(4o), and f(X) being compact is covered by e finite union of U(U)'S and then because of x) of 1,5. there is a 1E6 such that U(A) contains all these finite U(A)'s ).

37

ii) Let p = i.~ hi(-) f:X - SS(A) (observe that i :S (j) : BGL) is a homeomorphism i : SS()

iii) We put [f = ( )

1.6.1.Lemma. Let E,F be objects in ' and g = (i,j):E - F

be a map in q(P). Let X be a compact space and (E,p) a bundle on X with fiber in 4,

(p:X - End&(E), p(x)2 = p(x) for xeX). Then (E,p)e(F,ipj) as bundles over X with fibres in C.

Proof: The maps i and j "are" bundle paps i:(E,p)---(Fipj) ,

j:(Fipj')

-p (E,p) in 4(X) and moreover

ji

= id(Ep), ijipj = ipj = id(,ipj) so ij is equivalent to the identity map of (F,ipj) in 4(X). Therefore, i and

j are inverse of one another in 4(X). The Lemma is proved.

1.6.2.Lemma. Let X be a compact space and (E,po), (E,pl) bundles over X with fiber in

&.

Then, if

POPpl, then (L,p0) r (E,p 1 ).

Proof: Let h:XxI ) , (E) be a homotopy h(x,O) =

=p₀ (x), h(x,l) = pl(x) for xeX. We consider the bundle (E,h) over XxI. Call i0:X - XxI the map i (x) = (x,O) for xeX

and i 1:X - XxI the map i 1(x) = (x,1) for xeX. Then,

(Epo) = (Ehi0 ) and (Epl) = (Ehi ) . The maps i0 and i being homotopic, the bundles (E,p ) and (E,p1) become

isomorphic because of 1.5.2.

1.6.3 Propc sition. The map X' is well defined and is a natural transformation of functors. Proof: To see that X' is well defined, let

be a compa.ct space and let [f) = f'1 :X - BGL(ec)

[Top) .Suppose Call

f(X) C U() anf

f'(:) CU( ')

for some

f

>r

ft - )

We have to show that (SS( ),p) and (SS(I'),p phic bundles over X with fiber in 6.

Let h:XxI -> BGL(6) be a homotopy f', h(x,O) = f(x), h(x,l) = f'(x) for xeX. C

are

isomor-between f and onsider the map ven by

f(x)

h(x,3(t-1/3))

h3(t-2/3)(A')

Then H is continuous and H(x,l)

f(x) for xeX. Now let

if xCX, 06t&l/3

if xEX, 1/3 6 t - 2/3 if xEX, 2/3 ! t A 1

= h

(A')

f? (x) , H(x, 0)

k 6i be such that ku j,k j', This k clearly exists because XxI is co-pact. Now consider the map G:XxI -p BGL(Z)

= h₁(h) H(x,t) fo a) G(xO) = H(x,o) G(x,1) = H(x,l) r xeX, = h( = h( a retraction of U(k) b) G(x,t) CZ k given by te.I. We have ) f( ,

') f'(x) for xeX because onto Zk and Z C Z , _{Zi, C Z k} for xcX,

We call q = i have tS6(A,kl)(p(x)) so that the bundles (SS

hi () f ; q' =- ~ h_( ) f

= q(x) , SS([',k j)(p '(x))

( ),p)

and (SS(k),q) are isomorphic and (S6Q),q') are isomorphic because hl (1) H(x, t) = hl () H(XxI) CU(k). G(x,:t) is 0 I _{. We} =, W'x, in p =i H: XI- 'I BGL(fo) gi h 1-3t(i)

U ~

59

of 1.6.1..

So, to conclude, we have only to show that (SS(h),Q) (L(),q'). But i G(x,t) , CX5, teI, is a homotopy

between q and q', so the desired isomorphism is a consequence of 1.6.2.

The naturality of XI follows from the fact that it is well defined by rules i), ii), iii).

1.6.4. Theorem. The functor 14(.)j is t-representable on CTop3 and (;t,BGL(()) is a 7-represen-tation.

Proof: One easily checks that

X'

is the inverse

of

X.

1. 6. 5. Remark on the monoid structures on 14(, )1 i)Let i = (ir )- 0 I . We define i2 in

r r_- 0 00 follows: 2 = (kr ) ₀ in in and BGL(6'), I as 00 where if r = 2n, n a 0 if r = 2n+1 , n A 0

We define two increasing injective functions e,o:N - N

by e(r) = 2r, o(r) = 2r+1 , r A 0.

ii) Let i eI"3. Vie define two maps e ,o :S(A ) ->_S(Ai2), where A = (A. ) 00 E(PVand similarly A 2 ,

r

in ( The map e is to be the pair (q,p) , where q = (q _rs) s 00

r, s=O p = (p rs rs=O)00 are matrices defined by ~idA. rs 0 if e(s) = r otherwise kr

idA.

0

if e(r) = s

otherwise

where if A.2 _- = (Ak _r )0' , then _t q:A. _{s 1} A _r=O

k and

Prs:Ak₅ -A--Ar . The map o is defined in exactly the same way using the function o instead of e .

iii) The set i being a subset of the product I x

200

xI x ... admits a squaring operation i i defined coordenate-wise using the squaring operation in I 0. Let

0r 0 ,

ei

=

-r

1 -

oi

= (qj _Ppi ) _{- We define} -r -r -r -r -r --r map E,

O

in q?(,f) ,

E :S()

---- 3( 2_{) ,}

_O.:SS(i)

--- S3(1S 2_{) , by} 00 00 00 E #( q _{. = p. ),} = ( q!r = r=0O ir r=0 -Ir = -=0 Ir 0O r=0O

ir

iv) Let iE6* ; we define a map m :S(i)x4Ss(i)

-- 'S,>(i2) : ve _{put m (p,p') =4EQ)(p) + _(O )pf),}

where the sum is taken in End₆(SS(i2)) ; note that this sum is an idempotent because jE i)(p) and

4.

0 )(p') are ortho-gonal projections.

Let i,j& I, i & ; then the following diagram commutes:

) x

S>(i)

S([, 1'1})x ( ,9 ))

> )

x

( )

SS(i2) SJ(12) AO0

41

Therefore, we get a map -m': -6()x+S5(i)

lim , Si 2₎

Now consider the canonical map

L?:lim vS6( ) xtgss(u) BGL(<P) x BGL(PS)

which is a homeomorphism, also the diagonal inclusion

I-...: Sxz gives a map

T: 1i S(U) x +SS( ) --- lim t ( )xsE( )

which is a homeomorphism; finally, the map i -- 2 for

6.% gives a map

i

s6(i

2 _{) -- BGL(4). We define the}

map yn:BGL(()xBGL(k) --- BGL(4) by m = Tm" .r v) The map m is easily seen to be homotopically associative and commutative. Moreover, given a compact space

X we get a commutative monoid structure in HomcTopf.(X),BGL(4)) and the natural equivalence . is an isomorphism of commuta-tive monoids.

1. 6. 6.BRemark on the "naturality" of BGL(4) with respect to 'e.

Let

4

, fo' be pre-Banach categories and T: - go' a linear continuous covariant functor; then i defines a map t.l: 4

.y g9

X

. If X is a compact space, the functor

9

de-fines in an obvious way a linear continuous functor P(X): O(X) -- '(X) and therefore a map k(X)1:

)(X)

-Let o, , be as above; let (A.) be a family of generators of C>, where I has a well ordering . We consider the family (A')k.K , where K = I UJ and where A= (Ak) if ke-. (we assume I nJ = #), of generators of