Báo cáo toán học: "Homogeneous C*-extensions of $C(X) otimes K(H)$. Part II " doc

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J OPERATOR THEORY 4 (1980) 211-249 (@â Copyright by INCREST, 1980 ont

HOMOGENEOUS C*-EXTENSIONS OF C(X) @ K(A)

PART II

M PIMSNER, S POPA and D VOICULESCU

Tn the first part of this paper (J Operator Theory, 1 (1979), 55-108) we began studying a generalization of the Brown-Douglas-Fillmore theory of extension, in which the ideal K(f) of compact operators is replaced by C(X)đ K(4)

For X a finite-dimensional compact metrizable space and A a separable nuclear C*-algebra with unit, the equivalence classes of certain extensions which we called homogeneous extensions, of C(X)@K(H) by A, gave rise to a group Ext(X, A)

Based on the results about Ext(X, A) obtained in the first part of this paper, we shall develop here further topological properties of Ext(XƠ, A) This includes the study of a certain K(X)-module structure on Ext(X, A), the long exact sequences for each of the two variables, the periodicity theorem and a result showing that taking suspensions in one of the variables has the same effect on Ext(X, A) as taking suspensions in the other variable

Much of the material in this paper is derived from standard techniques in algebraic topology and from the adaptions of these techniques due to L G Brown [9] for extending the Brown-Douglas-Fillmore theory from commutative to non commutative C*-algebras

Since we have chosen to make this paper rather selfcontained, some of it is almost expository

Before passing to a more detailed description of the content of the present paper let us briefly recall how far we had come in studying the properties of Ext(X, A) in Part I

After dealing with the Ext(X, A) is a group question, we obtained results on homotopy-invariance and short exact sequences Thus we proved that for a nuclear C*-algebra A and a two-sided closed ideal J of A there is an exact sequence

Ext(ŒY, 4/7) > Ext(X, A) > Ext(x, J)

Similarly, for Y a closed subset of X we obtained an exact sequence

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For the homotopy-invariance properties besides the usual assumptions on the C*-algebras and compacta we had to assume generalized quasidiagonality (abbreviated g.q.d.) of the C*-algebras

If A is g.q.d and p,: A > B(k = 1,2) are homotopic unital +-homomorphisms we proved that the corresponding group-homomorphisms:

Pag? EXt(X, B) > Ext(X, A) (k =1,2)

coincide

Similarly, also for A g.q.d., we proved that if f,: Ơ ~ Y (k = 1, 2) are homotopic continuous maps, then the group-homomorphisms

fo: Ext(Y, 4) > Ext(X, A) (k = 1,2) coincide

Part If has three sections Đ 7-Đ9, with numbers continuing those of the first part The list of references, for the readers convenience, is a concatenation of the list of references of Part I and of an additional list of references

In more detail the content of the three sections of the present paper is as follows

In Đ7 it is shown that there is a natural isomorphism Ext(X, C(S}) 2 K(X) and there is a homomorphism K(X) > Ext(X, A) related to weak equivalence

We also exibit a natural K(X)-module structure on Ext(Ơ, A) and we show that the action of fiber-preserving automorphisms of C,(X, K(#)) on Ext(X, A) corresponds to multiplication by line-bundles

In Đ 8 using the short exact sequences and the homotopy-invariance established

in Part I, long exact sequences for Ext(X, A) are derived In the X-variable this

is absolutely standard and the proofs are omitted For the A-variable the proofs are given, but this is only a more detailed exposition of L G Brown’s adaptation for the non-commutative case of the usual proofs

The reader who wants some intuitive background should read the derivation of the long exact sequence given here in parallel with the derivation of the long exact sequence for the usual Ext in the commutative case in [12] and think of how the constructions at the level of spaces translate into constructions at the level of the corresponding C*-algebras of continuous functions on those spaces

In Đ9 the periodicity theorems in the X-variable and in the A-variable are obtained We show that there is an interchange isomorphism which expresses the fact that taking suspensions in the X-variable or in the A-variable has the same effect on Ext(X, Xo, 4)

This makes the periodicity theorems in the two variables equivalent and our proof will be half in the X-variable and half in the A-variable

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HOMOGENEOUS C*-EXTENSIONS 213

We should mention that it had been noted by L.G Brown in [9] that the proof of the periodicity theorem in [12] could be adapted for the non-commutative case The assumptions under which we give the periodicity theorem are that X be a pointed finite-dimensional compact metrizable space and A be a nuclear g.q.d C*-algebra, having a rank-one homomorphism

A curious corollary of the periodicity theorem is given in 9.12

We would like to mention that further topological properties of Ext(X, 4) have been obtained by C Schochet [59]

On the other hand, G.G Kasperov has announced in the short note [50] results for a related much more general two-variables Ext-functor, both variables of which are non-commutative C*-algebras His results are obtained, in part, by connecting the Ext-functor to a generalization of his previous work on K-homology

[51]

Đ 7

In this section we discuss certain relations between Ext(X, A) and K(X)

First, we consider the K(X)-valued index for unitary elements of C„ (X, L(A) /C(X, K()), an adaption of Atiyah’s results in the Appendix of [5] This index gives a natural isomorphism Ext(X, C(S')) ~ K(X) and also a natural homomorphism K(X) > Ext(X, A) defining the weak equivalence relation on Ext(X, A) Second, a natural K(X)-module structure on Ext(X, A) is defined It is shown that the action of fiber-preserving automorphisms of C,(X, K(#)) on Ext(Ơ, A) can be expressed by means of the action of the multiplicative group of classes of line-bundles in K(X) The material in this section consists, to a large extent, of adaptations of knonw facts, included for the sake of some completeness

By Po(X, H), P(X, H) we shall denote the orthogonal projections in C,(X, K(H)) and respectively in C,,(X, L(A))

7.1 Lemma Let P,c P(X, H), (= 1,2), be such that dimP{x)H = co for all x e X, Ă = 1, 2 Then there is V € Cy(X, L(A) such that V*V = P,, VV* == Ps, Proof The projections P; determine continuous ‘fields of Hilbert spaces (POOF) ex,T 3) where I; H Œ@)) is the set of (P,(x) f(x)) where f

xe

runs over C(X, H#) Then by ((21], 10.8.7.) these two continuous fields of Hilbert spaces are isomorphic So there are unitary operators W, from P,(x)H to P.(x)H such that

xex

{(W,h,)xexI(h⁄)xex e ry} =F)

Define V(x) Â L(A) by V(x) A = W,P,(x) A Then it follows easily that V = (Ya x -

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For Pe P,(X, H), the subset YD ({x} x P@Q)A) of Xx H together with the natural projection onto Y defines | a Tocally trivial vector bundle over X

Let Vect(Ơ) be the semigroup of isomorphism classes of locally trivial vector bundles over X endowed with the direct sum operation For Pộ P,(X, H) the equivalence class of the corresponding vector bundle will be denoted by [P] Vect(X) and the stable equivalence class by [P], = K(X)

The next two lemmas are quite standard; their proofs will be omitted

7.2 LEMMA The map P(X, H)2 P > [P]e Vect(X) is onto Moreover for Đỡ, Dạ 6 P(X, A) the following conditions are equivalent:

(i) [Pi] == [Pol

(ii) there is a unitary U el - CX, K(A)) such that UP,U* == Po il) there is VEC,X, K(H)) such that V"V = P, and VV* =: Py

(iv) there is WeECs(X, L(A) such that W(x) P(x) H = Pix) Hand Ker W(x) 0 Pi(x) H =: 0 for al xe X

7.3 LEMMA Let P,, P2€ P(X, H) be such that ||P, — P,) <1 Then we have [P,] == [P.]

The next lemma enables us to define the K(X)-valued index

7.4 LemMa Let UVeC,,(X, LUA)YC(X, K(A)) be unitary Then there is a partial isometry WeC,,(X, L(H)) such that p(W) = U Moreover

[J— W*W], — [ẽ— WW*], â K(X)

is independent of the particular choice of W (i.e depends only on U)

Proof Consider P;Â P)(X,H), P, < Py < , an approximate unit of CX, K(A)) and let Ve C,,(XƠ, LCH)) be such that p(V) = U Then

(— P) V*V{ — Pj) = (I= P) + UT — P) (V*V — DU P)

Since V*V — Ie C,(X, K(H)) there is some j € N such that id — Pj) (V*V — I)(— PẠI| < I1 Set

W = V — P)) (P, + (ẽ— Pj) V*V(— P))~12

which is a partial isometry with p(W) =

For the second assertion, let W,, W, be partial isometries such that p(W,) = == p(W.) = U Let jy €Â N be such that for j > j, we have

lỚ— W#W)(I— P) <1 G=1,2)

Then for j > jo, P; I - P) W.W,(1S— P,) will be invertible and we may define partial isometries Ly= = W* W1 — P) (P; + (I— P,) WEW UI — P))~!? and pro-

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HOMOGENEOUS C*-EXTENSIONS 215 Then W,, = W,E,; will be partial isometries with the following properties: P(W;;) =U lim |/Ê;; — (I~ P)|| = 0 jJ—co fim || Wy; — Wo;il = 0 foo Then for j 2 jp great enough, we have WW, — WEW2,\| < 1 and || Wỡ,W?$ — W›;,W#|| < 1 so that Lemma 7.3 gives [f— WW ilk —_ [f— W;W}x = [f — WW;;]x — [J — Wi Wiilk- Thus it will be sufficient to prove that: U — WiW gle — WW = (+) = [I — W Wilk —H— MW,M?} Definind Ry = W*W,— M??W,, S,= W,W*— W,W?, we have R¿ ij

Si, € P(X, H) since E,, < WW, Then W;R,; is a partial isometry ij with {W,Rj)*(WM,R,) = Rụ and (M,R,)(W,R,)* = Sj; so that by Lemma 7.2 we have [Rj] = [Su] Mow (đ) folows from:

[Ứ— WHEW) = [i — w* Wi] + [Rj]

[[— WW =~ Wwe + [Su] Q.E.D

Now we can define the index oƒa unitary element U s C„(X, L(H))/C„(X, K(H))

by

index U= [7 — W*M], — [l— WM*]u where W is any partial isometry with p(W) = U

The next lemma gives the main properties of the index

7.5 LEMMA The index-map from the unitary group of Cy,(X, LUA))/C,(X, KC) to K(X) is onto Also, index U = 0 if and only if there is a unitary V € Cy {X, L(H)) such that pV) = U For U,, Uy unifaries ớn C„.(X, L(H))/C,U(X, K(H)) we have:

(i) index(U, @ U2) = index U, + index U, (ii) index U,U, = index U, + index U,

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Proof Given a € K(X), byLemma 7.2 there are P,, P, <Â P,(X, H) such that a [Pilk ~ [Pelk- Then because of Lemma 7.1 there is Ve C,,(X, L(H)) such that

VeV =: [ P,, VV* = I — Py Clearly p(V) is unitary and index p(V) = ô If V is a unitary of C,(X,L(M)) then index p(V)=0 Conversely let UeCz,(%, L(H))/C,(X, K(H)) be unitary with indexU = Oand let We C,(X, L(A)) be a partial isometry with p(W) = U In view of Lemma 7.2 there is Q € P(X, H) such that J — W*W]+ [Q] = U— WW*]+ [Q] Using Lemma 7.1 there is

S€C,,(X, L(H)) such that S*S = I and SS* = W*W Then V, = W( — SQS*)

is a partial isometry, p(V,;)=U and [J — VỀH;]=[I1— W*M] + [O]=

:= [J— WM*] + [@] = — V.V?'I

Hence using Lemma 7.2 there is L € C,(X, K(H)) such that J— V?V, = L*L, I— V,V == LL* Defining V=-V,+ L we have p(V)=U and V is unitary Concerning assertions (i)-(iii) we remark that (i) is quite trivial and we shall first prove (iii) and then (ii)

To prove (iii) we shall first prove that ||U,—U,||<1/2 implies that indexU, = = indexU, Indeed by the proof of the first part of Lemma 7.4 there are partial isometries W (i =: 1, 2) with p(W,) = U;, and |W, — W,||<1/2 Then |l(— WẩW))—

—(I — WFW,))\| <1 and | — WW) — U — W,W3)|| < 1 so that index U, =

= index U, follows from Lemma 7.3

Now f ly — ¿|| < | we have {U,U* — J} < Land hence there isa hermi-

tian element A Â C,,,(X, L(H)/C,(X, K(H)) such that exp(id) = U,UF so that U,

and U, may be joined by the continuous curve exp(itA)U, (t €[0, 1]) But in view of the previously proved fact the index is locally constant and hence index U, = == Index U4

To prove that indexU,U, := indexU, + indexU, in view of (i) and (iii) it will be sufficient to prove that U,U,@J and U,đU, can be joined by a norm- continuous curve of unitaries This is done with the usual trick:

U(t) = h 0 (oY — sint ( 0 cost sin/

0 i} sint se} O F}\— sint cost

t €[0, 2/2]

Identify S1 with the unit-circle {z Â C| |z| == 1}; the function y € C(S), given by x(z) = z, is unitary and generates C(S*)

The following proposition is an immediate consequence of Lemma 7.5 7.6 PROPOSITION The map Ext(X, C(S}) 3 [t] index t(y) € KCXD) is well defined and is an isomorphism of Ext (X, C(S')) onto K(X)

We pass now to the discussion of the weak equivalence of homogeneous X-extensions

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is aunitary Ue C,,(X, L(H))/C,(X, K(A)) such that Ut,(a) = t.(a) U for all ae A To emphasize the distinction between weak equivalence and equivalence the latter will be also called strong equivalence The semigroup of weak equivalence classes of homogeneous X-extensions by A will be denoted by Ext,(X, A) and there is a natural homomorphism Ext(X, A) > Ext,,(X, A) We shall write [t],, for the weak equivalence class of t

Assume [t,] € Ext(XƠ, 4) (=1, 2) are weakly equivalent and let Ve, (X, L(A))/C,(% K(H)) be a unitary implementing the weak equivalence Using Lemma 7.5 it is easily seen that the strong equivalence class [t,] depends only on [z,] and index U Since the class of trivial X-extensions by A is a natural element in Ext(X,A) it follows that [t,] and [t,] are weakly equivalent if and only if there is [o] weakly equivalent to the trivial extensions such that [t,] + [o] = [t,] Assume now {o] € Ext(X, A) is trivial and let Ue C,,,(X, L(H))/C,(X, K(#)) be unitary and define o,(a) == Ua(a) U* (acc A); then ,since [o,] depends only on indexU, there is a map e: K(X) > Ext(X, A) such that ô (indexU) = [o,] In view of the properties of the index, Â is a homomorphism and the diagram

K(X) > Ext(X, A) > Ext,(Ơ, 4) >0

iS an exact sequence, in the sense that [t,],, = [te], if and only if [t,] = [t,] +- e(a) for some a € K(X) Of course if Ext(X, A) is a group then Ext,(X, A) is also a group and exactness of the above sequence has the usual meaning

if A, B are C*-algebras with unit and f: A > B is a unit-preserving #-homomorphism then it is easily seen that the diagram ey A) F Nett, B) K(X) is commutative

In particular if A has a one-dimensional representation then ô=O and hence Ext(X, A) = Ext,,(X, A)

The next question we shall discuss is a natural K(X)-module structure on Ext(X, A) Of course we shall assume A is nuclear, so that Ext(X, A) is a group In the remaining part of this section, for T; € Cy,(X, L(H)) (i = 1, 2) we shall use the notation

T, @ T, = (Xax > T(x) @ T,X) € LH @ H)) € Cy (X, L(H @ H))

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ForTeC,,,(X, L(A)) define

u(T) = V(T đ@ P) V* EC, CX, L(A) Then y is a unital ômonomorphism of C,,,(X, Ê(A)) into itself and

u(T)€ C,(X, K(A)) = Te C,(X, K(A))

It is easily seen that for some other P’ < P,(X, H), with [P’] = [P]and some other V’ corresponding to P’, the corresponding homomorphism uy’ differs from u by an inner automorphism of C,,,(X, L(A)), i.e there is a unitary U e C„ (X, L(H)) such that p(T) = Up(T) U* for ail T'€ C,,,(X, L(A)) Note also that for Q € P,(X, H) we have

[u(Q)] = [P @ Q)

Let y be the unital ô-monomorphism of C,,,(X, L(A))/C,(X% L(A)) induced by uw Thenfor t: A > C,,,(X, L(A))/C,(X, K(A4)) defining a homogeneous X-extension by A it is easily seen that ji°t also defines a homogeneous X-extension by 4 and [jiot] depends only on [t] and [P] Also for Ue C,,(X, L(H))/C,(X, K(A)) a unitary we have

index (UV) = [P], indexU Thus for P € P,(X, H) with P(x) # 0, (V) x eX we may define

[P]}-[t] = [fo7]

Jt is quite standard to verify that for P;ÂP)(X, H), P(x) #0 (V)xeX = 1,2) and [t,] € Ext(X, A) (i = 1,2) we have

[Pa]-[ta] + [Pe)-[t1] = (Pa + (Pe) Ta] PHI: u] + (te) = (Pi) Eta) + [Pi)- [te]

[Pi] -([P2}-[t2]) = LPị @ PP] - [ra] [P]-s(3) = Ê(Pik -3)

From these properties we immediately infer that ([P], [t]) > [P]-[c] can be uniquely extended to a bilinear map K(X) x Ext(X, A) > Ext(X, A) which defines a K(X)-module structure on Ext(X, A) Moreover e: K(X) > Ext(X, A) is a homomorphism of K(X)-modules

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shall use, are well known and can be found for instance in Section 2 of [53] Thus, every automorphism of C,(X, K(4)) is the composition of an automorphism induced by an automorphism of the base space X and a fiber-preserving automorphism; i.e an automorphism acting trivially on X (the spectrum of C,(X, K(H)) The group of fiber-preserving automorphism will be denoted by Autcx)(C,(X, K(A))) since is is easily seen to consist of those automorphism which preserve the C(X)-module structure of C,(X, K())

We will be interested only in the action of Autc,y)(C,(X, K(H)) on Ext(X, A) and we shall point out below that this action can be expressed in terms of the K(X)-module structure of Ext(X, A)

Since the inner automorphisms, i.c the automorphisms Inn(C,(X, K(H))) induced by unitaries of C,,,(X, L(H)), act trivially on Ext(X, A) it will be actually the factor group

Out(X) = Autcx(C(X, K(M)))/Inn(G,(X, K(H)))

which will act on Ext(X, A)

Now locally, every fiber-preserving automorphism ô is given by a unitary That is, there is an open cover {w,} ;Â, of X and there are unitaries U; € C,,,(X, L(A))

such that œ(7) = U,TU* for Te C,(X, K(H)) with supp T < @; Moreover for

xXEW, 1 @;, UF(x) U(x) = 4,(x)f where A,(x) eC, |;(x)| = 1 We get thus a l-cocycle (Ai;)o:nw, & O The automorphism ô is inner if and only if the cohomology class of this cocycle in H4(X,T) is zero The product a°f has as cocycle the product of the corresponding cocycles Thus we have an injective homomorphism of Out(X) — H1(X, T)

This is also surjective because of the contractibility of the unitary group U(H) endowed with the *-strong topology

Now there is also a bijection from H1(X, T) to equivalence classes of line bundles over X In fact, if (4,;) is the cocycle obtained from a fiber-preserving automorphism ô, then a corresponding line-bundle is the line-bundle given by ô(P,) where P, is some constant rank-one projection

Since there is a cocycle corresponding to both a and to the automorphism constructed from o(Py), we infer that these automorphisms differ only by an inner automorphism Thus we have a commutative diagram

Out(X) ———>——ằ HX, 2) > K(X)

End(Ext(X, A))

In summary, the elements [7,], [rạ] Â Ext(X, A) are conjugated by some fiber- preserving automorphism if and only if [t,] = ft] where B e K(X) is the class of some line-bundle

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Đ 8

Using the short exact sequences and homotopy-invariance results of sections 4, 5, 6 we shall obtain in this section one-sided long exact sequences for Ext(X, Xo; A) For the X-variable this is standard algebraic topology and the corresponding result will be mentioned without proof at the end of this section The same techniques were used for commutative 4 by Brown-Douglas-Fillmore, and L.G Brown [9] has supplied the necessary definitions for suspensions, mapping cylinders etc., to make the same machinery work also in the non-commutative case Since the presentation in [9] is somewhat sketchy, we give below for the reader’s convenience a more detailed presentation of the proof of the long exact sequence in the A-variable Let A, B be two unital C*-algebras and p: A ~ Ba unital *-homomorphism Then we shall consider the unital C*-algebras:

Z(p) = {€ đ x < C0, !]), 8) @ 4| š() = p@œ)}

and , `

C(p) = tế đ xe C(0, 1], 8) @ 4 lộ() = ứ@), š(0) € C- ig}

These C*-algebras are the analogous of the mapping cylinder and of the mapping cone from algebraic topology

Consider further

CA = {€ € C((0, 1], 4) [€) € C14}

and, ,

= {Ê e C{0, 1], 4) l¿(0) e Cl„, (1) e C1„}

which correspond to the cone and suspension

Since we shall need the short exact sequence in Theorem 4.1 all C*-algebras’ in this section will be assumed nuclear: Clearly Z(p), C(p), SA, CA will also be nuclear

Further, in order to use the homotopy-invariance results in Đ5 we shall consider C*-algebras A having composition-series (J,)o<)<2 With quasi-diagonal quo- tients JnailDps a property we shall call generalized quasidiagonality (abbreviated g.q.d.) Also, throughout this: section all C*-algebras will be assumed to be g.q.d

It is easy to see that.direct sums and subalgebras of g.q.d C*-algebras are still g.q.d Also for A g.q.d we have that C((0,1}, A) is g.q.d (consider the composition series (C((0,1], L,)o<p <a): AS a consequence we infer that Z(p), C(p), SA, CA will also be g.q.d

One caution is necessary: wher A is g.q.d., it does not follow that a quotient AJ is also g.q.d (in fact every separable C*-algebra is a quotient of a quasidiagonal C*-algebra), so A/T will be assumed in what follows to be 8 g.d It is also easy to see that if A/J and J are g.q.d then A is also g.q di

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HOMOGENEOUS C*t-EXTENSIONS 221

8.1 LEMMA Let (X, x) be a pointed, finite-dimensional compact metrizable space p: Á B a unital *-homomorphism Then

Ext(X, x; B) “ Ext(X, x9; A) “> Ext (X, x9; C(p))

is an exact sequence, where q: C(p) — A is the natural projection Proof We shall consider the C*-algebra Cy(p) = {€ @ x € C(p)| €(0) = 0} and the following +-homomorphisms: i: C,(p) + Z(p) the inclusion, p: Z(p) > A_ the natural projection, r: Z(p)—> B defined by r@ @x) = š(0) for š @+xeZ0), k: A-> Z(p) given by k(x) = ƒ„ đ x, where ƒ„ € C ([0,1], 8) is the constant function equal p(x) Then the diagram a 2(p) me <4 | a

is commutative (i.e rok = p, poi = q)

Applying Theorem 4.1 to the exact sequence

Co(p) > Z(p) > B+ 0

we obtain an exact sequence

Ext(X, X93 B) > Ext(X, xạ; Z(p)) “> Ext(X, x93 C(p))

(Note that C,(p) is C(p).)

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Thus, consider G,: Z(p) + Zp), s€[0,)) the ôhomomorphisms defined by GLE @ x) = Â, Ox where €,(t) = (1 —(i — 5) (1 — 4)

Then Gy = idzijp, G, = ksp and obviously G, depends continuously on s in

the point-norm topology Q.E.D

8.2 Lemma Let (X, Xo) be a pointed finite-dimensional compact metrizabie space, assume the C*-algebra A is unital and J < A is a closed two-sided ideal such that A[J is contractible (of course A, J, AjJ are nuclear, g.q.d.) Then the inclusion i: J > A induces an isomorphism

ist Ext(X, xạ; 4) > Ext(X, xo; J)

Proof Let p: A> A/J be the canonical surjection By homotopy-invariance we have

Ext(X, x9; A/J) = {0} and hence by Theorem 4.1 we infer that i is injective

By Lemma 8.1 there is an exact sequence

Ext(X, x9; A) + Ext(X, x93 JJ> Ext(X, x9; C(i)

So to prove that i is surjective it will be sufficient to prove that

Ext(X, x9; C(i)) = {0}

Consider @:C() — S(A/J) defined by o(€@x) = f, where C(t) = p(E(t)) for tộ[0,1] It is easily seen that @ is a surjection Thus, there is an exact sequence

Ext(X, x9; S(A/J)) > Ext(X, x93 C(i)) > Ext(X, x9; Ker)

Tt follows that it will be sufficient to prove that S(A4/J) and Ker @ are contractible C*-algebras

It is easily seen that S(A/J) is contractible (suspensions of contractible C*-algebras are contractible) Indeed, the contractibility of A/J means that there exists a continuous family (đ,)sero, 1) of *-homomorphisms of A/J into A/J such that đ) = := id;„ and đ, is one-dimensional Then defining Ơ,: S(A/J) > S(A/J) by (Ơ,Â)(t) = = Ởđ,(ÊŒ)) for 0 < s < 1/2 and (Ơ,6)() = O,(E(2(1 — sf) for 1/2 <5 <1 we see that (Ơ,),er, 1) implements the contractibility of S(A/J)

To show that Ker 9 is contractible remark first that Kerg consists of all elements of the form €@x, where € C(0,1], 4), x eJ,x = €(1), €(t) eJ for all te [0,1] and &(0) = 0 Thus Kerg is isomorphic to

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Defining G,: B > B by Ge + â = Je + &’ where &(t) = &(st) for t € [0, 1] it is easily seen that (G,),e,o, 1; implements ‘the contractibility of B OED

Consider now A, J and g: A > A/J as in the preceding lemma and let us define c„: CÁ > A by c,(&) = E(1) and C4 LJ C4/7 = {Ê@(c C4@€C4/J | qex() = ca(Q} Consider further the diagram r CA LJ C4/Jj ——S4AjJ A (*) is Sự ƒ SA <————- SA where sf) = EOC) for ESA Œ()Œ)=š—?) for €€SA, te [0,1], and r(ộ)=Al,@ộ& where EeSA/J and ộ(l) = Al 4j3-

With these preparations we can now state the next lemma

8.3 LEMMA The diagram (*) is commutative up to homotopy (i.e sef is homotopic to r°Sq)

Proof Let H,:SA => CA \J CA/J, s [0,1], be defined by H,(Ê) = 6.06,

where E(t) = (1 —st), ŠŒ)= 4((Q —3)?)), re[0,I] Then Hy = re Sq,

A, = s°ƒ,

8.4 THEOREM Let (X, X9) be a pointed finite-dimensional compact metrizable space and let J be a closed two-sided ideal of a unital C*-algebra A(J, A, A/J are nuclear, g.q.d.) Then there is a natural exact sequence

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the +-homomorphisms given by the projections We have:

Kery = {(0@ €€A @ C(O, 1], A/J) | €0) € Cl yz, E(1) = 0} Kerụ = {x @ 0e44 @ C0, l], 4/7) | x c2}

Hence đefủning Sạ8 = {ế e C(0, I], Đ) | €(0) eC-1,, E(1) = 0} we have obvious isomorphisms Kerụ ~ J, Kery ~ S,A/J Thus there are exact sequences

0+S,4/J > ALJ CAI A 0 0>7->A4LJC4/j> C4J7 —ơ 0

Moreover y° k is just the inclusion i of J into A Also, since CA/J is contractible, Lemma 8.2 shows that k, is an isomorphism between Ext(X, 2x; AULJCA/J) and Ext(X, x9; J) With these preparations we can now define the connecting homomorphism by 0 = h,,okx!' This gives a commutative diagram

Ext(X, x9; A) ——> Ext(Ơ, x9; ALU CA/J) ——> Ext(Y, xạ; Š A/J)

Ks

Ext(X, x3; J)

Since k,, is an isomorphism, remarking that S,A/J is SA/J and that exactness of the top row implies exactness of the bottom row, we have thus proved exactness of the long sequence at Ext(X, xy; J)

We pass now to the exactness at Ext(X, x); SA/J) We shall use here the notations of Lemma 8.3 Consider also the *-homomorphism

I CA U CAN +AU CAI, (ERD = C(O OC

Then /°r =A Using Lemma 8.3 we have that the triangles in the diagram xt(X, xạ; CÁ( JCA/2)

(sof de

(#e) Ext(X, x; AUC AJ) re Ext(X, xạ; S4)

he Ext, x9; SA/J) (Sa)

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HOMOGENEOUS C*-EXTENSIONS 225 Now r,, is an isomorphism because there is an exact sequence

0 > 8g;4/7 > CAL CA/I + CA 30

where CA is contractible

Moreover denoting SA = /(Sạ4) we have the exact sequence

0384 2%, CAU CAI > AU CAI > 0

Thus the top row in diagram (#*) is exact andr, being an isomorphism the bottom row will also be exact Since 0 = A,,° kx! where k, is an isomorphism it follows that the exactness of the bottom row in (**) is in fact equivalent to the exactness at Ext(X, x9; SA/J) of the long exact sequence

We haven’t mentioned until now exactness at Ext(X, xy; A), which is the content of Theorem 4.1

In order to obtain exactness also for the rest of the sequence from what has been already proved, there is still a point to be established, namely that the inclusion KerS"Â c S"J, induces an isomorphism between Ext(X, x9; S7) and Ext(X, x9; Ker S”2) But thớs follows from Lemma 8.2 applied to the exact sequence

0 ơ Ker S4 — S7 —> S“C —› 0

_ Thỉs ends the proof of the exactness of the long sequence

The naturality of the long exact sequence refers to x-homomorphisms p:A >A’ and ideals J <Â A, J’ < A’ such that ứ(7) < J’ Then p induces also a +-homomorphism A/J > A’/J’ and the naturality property is the commutativity of the diagram

Ext(X, x9; A’/J’) + Ext(X, x9; 4’) > Ext(X, x9; J”) > Ext(X, x9; SA’/J") >

| ị | |

Ext(X, x9; A/J) > Ext(X, Xạ; 4) — Ext(X, xạ;J) > Ext(X, x9; SA/J) >

The easy verification is left to the reader Q.E.D

A consequence of Theorem 8.4 which we shail use is given in the next lemma 8.5 LEMMA Let (X, xo) be a pointed finite-dimensional compact metrizable space and leth: A — B be a surjective *-homomorphism (A, B nuclear, g.q.d.)._ Assume moreover there is a *-homomorphism j: B > A such that hej = idg Then we have the split exact sequence:

A, ~

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Proof Since hej=idg it follows that j,° hy, = idgxyx,x,; 28) and (SJ„)° (SH)„ == idextix, x9; sa) Thus h,, and (SA), are injective and the Lemma follows

from Theorem 8.4 Q.E.D

Since we want to discuss reduced suspensions, we shall consider ‘‘pointed”’ C*-algebras and their “smash-product’’,

But first we need some remarks concerning the fact that the tensor product of nuclear g.q.d C*-algebras is still nuclear g.q.d

It is known that the tensor product of two nuclear C*-algebras is still a nuclear C*-algebra Moreover if an exact sequence

0 + / > Á— Aỉj +0

is tensored by a nuclear C*-algebra, the new sequence will again be exact (use [58}) Now, the spatial tensor product of two quasidiagonal C*-algebras is easily seen to be again a quasidiagonal C*-algebra If the quasidiagonal C*-algebras are also nuclear, then there is a unique tensor product, which must then be quasidiagonal Consider A, Bnuclear g.q.d C*-algebras with composition series (1,)1eÂ,(Jp)ge 5 where #, Ơ are well-ordered sets Then A@B is nuclear and is also g.q.d as can be seen using the composition series J, @B + In41@Js)a,ae.sx9 Where Ơ x F# has been given the lexicographical order

The “‘pointed’’ C*-algebras we shall consider will be unital C*-algebras A together with a:specified one-dimensional unital ô-homomorphism 7:4 ~ C Then it is natural to define the “‘smash-product”’ of (A,, x) and (Ap, x2) as (Ker x@Ker z,) together with the one-dimensional +-homomorphism x such that Ker y=Ker 7,đ@

@đKer yo

Consider now unital nuclear g.q.d C*-algebras A, A,, A, and unital z-homomorphisms y,: A, > C (k = 1,2) Consider further the following four +-homomorphisms corresponding to natural inclusions

J›: 1@44;> 4@4Ă@4; Jo: A@A, > ABA, @Ay i: A@đ(Kery, @ Kery,) > 4@4i@ 4;

j: A> A@(Kery, @Kery2)

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8.6 LemMA, The map iy gives a natural isomorphism of Kerj,x \ Ketjox onto Ker jy

Proof Applying Lemma 8.5 to hyo j, == id4ea, we see that, denoting by k, “—————~ * the inclusion of 4@4,đ Ker y, into A@A,@A,, the map k,, gives an isomorphism ~~ ——————— of Kerj,,, onto Ext (X, x9; Ker ft.) = Ext (X, x); A@A,@Ker 7) ——_~ _>xrxưn ,

Denote by /Ă the inclusion of A@Ker y, into Ađ@A,@Ker y, and by Aj: A@A,@Ker x, > A@Ker x, the left inverse of jj which is the restriction of hy

Let also k, be the inclusion of A@Ker y,@Ker y, into A@A,@Ker yz Applying

Lemma 8.5 to (4, jj) and remarking that Kerh; = 4@Ker 7,@Kery, it follows that k,, gives an isomorphism of Kerjj, onto Ext(X, x9; A@Ker 7, @Ker 7) Thus we infer that (k,° k,), gives an isomorphism of Ker(k, oj})y Ker jay, onto

Ext (Ơ,x9; A@Kery, @Kery,) :

On the other hand, applying Lemma 8.5 to (4, /) and denoting by k the inclusion of Kerh = 4@Kerz@Kerz, into 4@(Kery,@Ker x.) we have that ky gives an isomorphism of Kerj, onto Ext(X, x9} A@Kery,@Ker yx)

Since io k == k, ok, and k, |Ker j,, is an isomorphism, it follows that in order to conclude the proof it will be sufficient to show that

Ker(q s 71) ủ Ker joy = Ker ji, N Ker joy Consider the split exact sequence

0 > A@Ker yy, — A@A, = 4 — 0

where Â, rare the canonical inclusions and s=1,@y, Then (k o j;),.=(Ci° ty Thus, if w € Ker(k, o ji), we have j,,% = 5,8 where B = (jer)„x Now, j,oris the natural inclusion of A into A @.A; @ Ag Hence, if ais also in Kerj, then (j,°r)4ô% = 0,

so that we get the desired conclusion Q.E.D

Let us also indicate a generalization of Lemma 8.6, the proof of which can be based on using Lemma 8.6 and Lemma 8.5 several times and which will be omitted

8.7 LEMMA Let A be a g.q.d nuclear C*-algebra and let (A,, 1%) (k = == 1, ,m) be “‘pointed” g.q.d nuclear C*-algebras Consider j,, the inclusion of 4@4i@ @AÁ, ;@4,,Ă@ @4, ữo A@4y@ @Aạ, ÿj the inclusion of A into A@(Ker ⁄(@ @ Ker x,) and i the inclusion of A@(Kery,đ @Ker y,) into A@A,@ @A, Then iy gives a natural isomorphism of Ker jy 1 1

n Ker j, onto Ker jy

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For (A, x) a “pointed” nuclear, g.q.d C*-algebra we shall consider the “reduced suspension” S(A, yx) of (A, x) which is:

S(A, x) = {feSA | xf) is constant} or equivalently

S(4, 0 = {fe C01], 4) | xƒữ)=0, Vze[0,1],/(0)=/() =0}

which makes S(A, x) a ‘‘pointed’’ C*-algebra The ‘‘reduced suspension”’ of (A, x) defined in this way coincides with L G Brown’s suspension [9] of kerxy with a unit adjoined Denoting by J the ideal

{fe C((0,1], A) | xflt) = 0, V re [0,1], f) = fl) = 0},

we have an exact sequence

0>7— S4 > C(0,1})) + 0

and since C({0,1]) is contractible and J =S(A, x) it follows that the inclusion S(4,x) c < SA gives a natural isomorphism of Ext(X, xạ; SA) and Ext(X, x9; S(A, y)) More generally it is easily seen that this holds also for iterated suspensions, i.e there is a natural isomorphism between Ext(X, x); S"A) and Ext(X, x9; S"(A, 7)) Thus, for “pointed”? C*-algebras, as faras only Ext is involved, we can always replace the usual suspensions by reduced suspensions

Now, the ‘“‘reduced suspension’’ can be viewed as a ‘“‘smash product’’, Indeed, consider (C(S1), e) where Â: C(S*) > C is any character of C(S1) Then there is a natural isomorphism between S"A and Kery@Keređ @ Kere Consider n-times hg: C(S")đ .C(S)—> A@C(S!}) @ @ C(S, n-times a-times the natural inclusion and consider also h,: A@C(S)@_ @C(S) ơ A@C(S9@ @C(S) (1 <j<m) —— _—>—— (a—1)-times n-times

the injection obtained by omitting the j-th C(S")-factor Then we have the following consequence of Lemma 8.7 (see also the remarks after this lemma)

8.8 COROLLARY There is a natural isomorphism of Ext(X, x9; S"(A, x)) onto Kero, 1 Kerfyg N A Kerhy,

This concludes our discussion in this section concerning the A-“‘variable” and we shall now briefly summarize the coresponding facts for the X-“‘variable”’ (without proofs)

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Then for a nuclear, g.q.d C*-algebra A we have:

1°, g*: Ext (SX, 0; A) > Ext (SX, 09; A)

ộs an isomorphism (0, oo are basepoints) 2° There is a natural exact sequence

Ext (Y, xạ; 4) â Ext (X4 x9; A) Ext (X, Y; 4)

ô Ext (SY, xạ; 4) Ext (SX, xạ; 4) â Ext (SX, SY; 4) —

“~

3° The group Ext("X, xạ; A) is naturally isomorphic to the subgroup of Ext(S" x X, (6, X9); A4) (whereơe S", xạcX are basepoims) which have trivial

restrictions to both S" x {x9} and {a} XX

Đ 9

This section deals with periodicity for Ext (X, A) First we establish some properties of a certain clutching construction Besides its use in the proof of the periodicity theorem, this yields the fact that, roughly speaking, taking suspensions in the X-variable or in the A-variable has the same effect on Ext (X, x9; A) This makes the periodicity theorems in the X-variable and in the A-variable equivalent

The proof of periodicity that we give has two parts Half is an adaption of a half of the proof for K-theory in [5]; half is almost a repetition of half of the proof

for the usual Ext given in [12] ,

In addition to the usual assumptions: X finite-dimensional, A nuclear, g.q.d we will obtain our results under the additional assumption that A has a one-dimensional representation

For the clutching construction, consider X a finite-dimensional compact me-

trizable space, S* the one-dimensional sphere identified with {z ÂC| |z| = 1} and let

r: Xx[0,1] > X, r(x, h) =x

s: Xx[0,1] ~ Xx S}, S(x, A) = (x, exp(2ziA))

t:X>XxS4, (x) = @, 1)

Ă„:X > Xx [0,1], i,(x) == (x, A), ủ € [0,1]

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Consider U € Cy, (XX [0,1], L(A) any unitary such that U(x, 0) == U(x) and

U(x, 1) = I (such unitaries exists since the unitary group of L(H) is contractible in the *-strong topology) Since U(x) p(4)U-1{x) — p(a) e C(ẫ, K(H)) we can find R(a) € C,,, (íx S1, L(H)) such that

R(a) es — U(p(a)or) U4 CX x [0,1], K(A))

Clearly p(R(a@)) depends only on ứ and U and the map a > p(R(a)) is a -homomorphism which defines a Xx S1-extension by A

Let us now make some remarks concerning the preceding construction: 1) Let U’e C,,,(X x [0,1], L(H)) be another unitary such that Ữ ‘(x, 0) = U(x) and U’ (x, 1) == and let R’(a@) be defined in the same way as R(a) using U’ instead of U Then there is a unitary VEC, (x@S, L(H)) such that Vos =: U' Ũ-1, We have Vẹ(a) — R(a)V< C„(Xx S1 K(H)), so that the XXx $1! -extensions by A defined by a — p(R’(a)) and a — p(R(a)) are equivalent Thus the class of the Xx S}-extension by A defined by a — p(R(a)) depends only on p and U Denote it by a(p, U)

2) Assuming that p’: A > C,,,(X, L(A)) is another unital ô-homomorphism such that pop = pop’, it is straight-forward to check that a(p, U) == a(p’, U) 3) Consider p and U as above and let U’€C,,, (X, L(H)) be a unitary such that U’ - UeC,(X, K(A)) To prove that a(p, U') = a(p, U) we shall need the fact that p has a one-dimensional representation Consider W = U-! U' el +

Œ(X, K(H)) _ _

We shall construct a unitary WeC,, (Ơx[0,1], L(A)) such that W(x, 0) =-

=< Wx), Wx, 1) =: Land [W, p(a)or]e C(Xx 0,1], K(A)) for all ae A This

will then easily give the desired conclusion by taking U=0 W, so all we have to do is to construct W Since Wel+ CX, K(H)) there is a projection Pe € C,,,(X, L(A)) with dimP(x) = co and dim(/ — P) (x) = œ for all x eX, such that [|W — (PWP + J — P))I < 1/2 Define W(x, h) for he[0,1/3] the unitary obtained from the polar decomposition of (1 — 34)W(x) + 3A(PQX)W(OX)P() +

+ (I — P(x)))

Consider now y: A > C a one-dimensional unital *-homomorphism It follows from Theorem 2.10 that there is a projection Q € C,,,(X, L(HM)) with dimQ@)= =: dim( Q) (x) =: eo for all xe X, such that ứ(2) — x@Q€C,(X, K (A)) for alla Â A Using Lemma 7.1 we easily constructa unitary V € C„.(X, L(H)) such that VPV* = Q and because of the contractibility of the unitary group of L(H) with respect to the *-strong topology there isa unitary Ve Cy, (Xx [1/3,2/3], LCA)

such that V(x, 1/3) = J, V(x, 2/3) = V(x) Define W on Xx [1/3, 2/3] by W(x, A) = =V(x, h)W(x, 1/3)V*(x, A) Remark that W(x, 2/3) I~ O(x) == ƒ— Q09

By Lemma 7.1 there is some Se C,,,(x, LCA)) such that SS* = Q, S*S = J Then S*(x) W(x, 2/3) S(x) is unitary Let Me Cys (XX [2/3,1], LCA)) be any unitary such

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by W(x, h) = S()M(x, A) S*(x) + UI — Q(9) It is now easy to check that W has the desired properties Thus we have proved that

a(p, U)= a(p, U’)

4) For (p’, U’) another pair with the same properties as (p, U) it is immediate that a(p@p’, U@U') = a(p, U) + a(p’, U’) Also if VeEC,,(X, L(A)) is a unitary such that Vp(a) = p'(a)V, (V)aeAand VU = U’D itis quite straightforward that a(p, U) = ô(p’, U’) This Jast remark together with 2) and 3) shows that more gene-

rally if there isa unitary Ve C,,(X, L(A)) such that Vp (a) — p'(a)V € C,(X, K(A))

then a(p, U) = a(p’, U’) (of course we assume that A has a one-dimensional representation)

The pair (p, U) defines a x-homomorphism of A@C(S?) into C,,(X, L(H))/ /C,(X, K(A)) Tf this +-homomorphism defines a trivial X-extension by A@C(S!) it is immediate (in view of the additivity of ô and of the equivalence of trivial extensions) that a(p, U) = 0

5) Consider 7: A > A@C(S!), my: C(S1 + 4@C(S1) the natural homomorphisms and let [ứ] Â Ext(X, A@C(S*)) be such that zĂ„[ứ] = 0, z;„ẽứ] = 0 Let (p, U) bea pair as above defining o Then 4) implies that a(p, UV) depends only on [co] Thus we may write ô([o]) for such up, U) and by 4) it follows that ô is a homomorphism from

{lứ] < Ext(X, 4@C(S1)) |m„[ứ] = 0, z;„[ứ] = 0} into Ext(X@ S1, A) Also clearly 1*a([o]) = 0

Moreover if (p, U) is any pair for which we did define a(p, U), then (p, U) gives rise to a unital s-homomorphism of A@C(S?) into Cx, (X, L(H))/Œ,(X, K(A)) For (p’, U’) a trivial X-extension by A@C(S1) we may consider a(p@ p’, UGU’) By 4) it follows that a(p, U) = a{p@p', U@U') Thus we may add to (p, U) some trivial X-extension by AđC(S"*) and this leaves ô (9, U) unchanged

6) We shall prove that

#:{[ứ] c Ext(X, 4@C(S9) | zi¿[ứ] = 0, ~;„[ứ] = 0} — —{r]e Ext{Xx S1, 4) |i*[z] = 0}

is injective

Proof Assume a([o]) = 0 and let (p, U) be a pair defining [oc] In view of the construction of a(p, U), this means that there is a unitary Ve Cys (Xx [0,1], LUT) such that Vix, 0) = V(x, 1) for all x e X and

VU(p(@) or) Ù-1~! — p(a) ôre C(Xx[0,1], KU)

for all ae A Since U(x, 1) = Z, denoting V(x) = Vix, 0) = V(x, 1) we have

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Hence defining U, = (Ver)-1 VO we have \(0(4)* r)r' — p(4)ằ r e C,(X [0,1], K(H)) for ae A and

U(X, 0) == U(x), U(X, = 1

Since [U;, p(a)° r]€ C,(Xx[0,1], K(H)), it follows that U, and the p(a)° r (ae A)

define a unital s-homomorphism of A @ C(S}) into C,,(X~x[0,1], LCA))/C (xx x[0,1], K(H)) and by direct sum with some trivial X [0,1] -extension by: Ađ C(S*) we get an element [6] e Ext(X x[0,1], A @ C(S*)) Since U,(x,0) = U(x) and U,(x, 1) = Jit follows that ij[6] = [o] and i*[6] = 0 Thus by the homotopy invariance property we infer [o] = 0

7) We shall prove that

a: {[o] € Ext(X, A đ C(S*) | m.[o] = 0, =„[ứ] = 0} > — {Ir]e Ext(Xx $1, 4) | 7#[z] = 0}

is onto In view of 6) this will show that ô is an isomorphism

Proof Assume [zt] € Ext (Ơ x $1, A) is such that Â*[t] = 0 and for each ae 4 let R(a) € Cy,(X X S!, L(A)) be such that z(2) = p(R(a)) Since t*[t] == 0 there is a unital homomorphism ứ: 4 + C,,(X, L(H)) defining a trivial X-extension by 4' such that p(a) — R(a)e te C,(X, K(A)) for all ae A Also, from s° ig =: t we infer i‡(s*[r]) = 0 and hence by homotopy s*[t] = 0 ,

Thus there is a unitary V Â Cy,(Ơ x[0,1], L(H)) such that Pip(a) ° r) Ÿ~! — (R(4)°s) e Œ(X [0,1], K(H)) Because of (p(a)°r)°i, — (R(a@)°s)°i,EC,(X, K(H)) ths implies [V, p(đ]C c C(X, K(H)) where V=f2j In a similar way we have also [Ve ip, p(A)]CC,(X, K(A)) Hence defining U = V(V> r)~tand U = Ue ig We have: U(p(a)e r)T-* — (R(a)e s) € C(XX{0; 1], K(H)), [U, p(A]eC,(X, K(E), Ữ:zj=TI x

These relations show that [t] = a(p, U), which proves our assertion

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Summing up the preceding discussion, we have proved: 9.1 THEOREM The above defined homomorphism a: {[ứ]c Ext(X, 4 @ C(SĐ) | m4[0] = 0, 224[6] = 0} > — {[r] e Ext(Xx S1, 4) | /*[r] = 0} is a natural isomorphism In view of Corollary 8.8 there is a natural isomorphism {[ứ] e Ext (X, 4 đ@ (59) | m„ẽứ] = 0, z;„[ứ] = 0} ~› Ext(X, S(4, X))

Assuming we have a pointed space (X, xằ) and denoting by yo the basepoint of SX, using Đ 8 and the functoriality of ô we easily get:

- 9.2, COROLLARY The isomorphism ô induces a natural interchange isomorphism (still denoted by a):

a: Ext (X, X93 S(A, X)) > Ext (SX, yo; A)

For the proof of the periodicity theorem we shall need the following technical result:

9.3 “LEMMA, Let nộe Ext i (XX S1, A đ > C(S%)) be such that 1 Mn = 0, ma„n = 0, tụ = 0 qT hen there is-a unital -homomorphism p: A > Cy (XX S', L(H)) defining a trivial XX S'-extension by A and a unitary U € Cy, (XX S', L(H)) with [U, p (A <

<C, (Xx S1, K(H)) such that the following properties hold: — (i) p is constant with respect to Xx S1,

(ii) U(x, 2) with x eX, z € S is norm continuous in 2, uniformly with respect to xeX

(1) ứ and U define a Xx S1-extension by A đ c(SY) of class ne

Proof Fort a tensor product of unital C*-algebras Ay @ A, â Ag, we shall denote by 7; and z,; the natural injections of A, and 4, đ A; into 4Ă @ 4; @ 4; Since /*# = 0 it follows from Theorem 9,1] that ‘there is some class

Oe Ext (x, Ađ C(S*) @ CCS)

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that U, is norm-continuous and [Us (a)? r] = 0 for all ae A In view of the construction of ứ(ỉ) the norm-continuity of Ủ; implies (ii) and the commutation [Os H(a)°r]==0 together with the fact that ằ is constant with respect to x

implies (i) Q.E.D

Consider now: p:A—> C,,,(Xx S', L(#)) a unital +-homomorphism constant with respect to Xx S! defining a trivial Xx S'-extension by A and let Uec,,(XxS', L(H)) bea unitary such that [p(4), U]cCC,(Xx<đ1!, K(H)) Then p and U determine a unital ô-homomorphism of A @ C(S*) into C,,,(X x S', L(H))/ (C,(X x S!, KCA)) By taking the direct sum of this homomorphism with some trivial XX S!-extension by A @ C(S1) we get an Xx S-extension by A đ C(S*), the -equivalence class of which will be denoted by T(p, U) Clearly 2,,7(p, U) = 0, Tz.0(p, U) = 0 In fact every element 4 € Ext(Xx S', 4 @ C(S?)) with 7, = 0, No34 = 0 is of the form T(p, U), where p can even be supposed fixed in view of ‘Theorem 2.10

In case GEC,,(XxS', L(H)) is an invertible element and [p(A), Glc <C,(Xx S1, K(H)) we may consider T(p, w(G)), where w(G) = G(G*G)"Ơ* is the unitary part in the polar decomposition of G Two such invertible elements G, € Cy,(Xx S', L(A)) Gj = 1,2) with [p(A), G] oC, (xX x S!, K(H)) will be called homotopic if there is an invertible element GeC,,(Xx S!x[0,1], LGH)) with G(x, z, 0) = G,(x, z), G(x, z, 1) = G(x, z) and tổ, 0(4)1<Œ,(Xx S1x{0, 1), K(A)) where (p(a)) (x, z, 4) == p(a) (x, z) It is immediate that if G,, G, are homotopic then Tứ, œ(G)) = Tớ, œ(G,)) Also, Ăf [p(4), G] = 0 then Tứ, œ(G)) = 0 9.4 Lemma We have T(p, @(G,G,)) = T(p, @(G2G,)) = T(p, @(G,) o(G,)) = = Tớ, œ(G;) @(G)) = T(p @ p, œ(Œ, @ G,)) = = Tớ, œ(GŒ,)) + Tớo, œ(G,))

Proof It is sufficient to prove that T(p, @(G,G.)) = T(p đ p, a(G,@G,)) We

have T(p, o(G,G2)) = T(p đ p, o(G,G, @ 1) Thus it will be sufficient to prove that G,@ G, and G,G, â J are homotopic This is established by the usual trick:

G,(x, z) 0 cos sin m I 0 cos — sin

G(x, 2, hy = n "

7 Tt 7h nh

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9.5 Lemma Let 4 € Ext(Xx SS}, A @ C(S4) be such that mH = 0, Toy = = 0, /* = 0 Further, let p: A > Cy(X x S', L(H)) be a unital ằ-homomorphism, constant with respect to X x S), defining a trivial X x S'-extension by A Then there is some integern and Dy, D,, , D,€Cy(X, L(H)) satisfying [D,, p(a)e the C,(X, K(A)) forj =0,1, .,n and for alla A, such that

G(x, Z) = y 2’ D(x) j=0 is invertible in C„(Xx S1, L(H)) and

T(p, o(G)) = 4

Proof The fact that p in the statement of Lemma 9.3 can be given in advance is a’ consequence of Theorem 2.10 Thus using Lemma 9.3 there is a unitary UeC,„(Xx đS!, L(H)) satisfying the conditions specified in the statement of Lemma 9.3 so that 4 =T(p, U) Let o 20 be a scalar Cđ-function on S! with ( @(Š) dÃ(6) = 1 (đ4-Lebesgue measure) and define G’ Â C,,,(X x S1, L(H)) be the

st

convolution:

Gey =| UG, é ứŒš”Đ dÃ(9 AY

Then, because of property (ii) in the statement of Lemma 9.3, if the support of @ is in some small enough neighborhood of 1 € S1 we shall have ||U — G’|| < 1/2 Also it is easily seen that [G’, p(A)] < C,(Ơx S!, K(H)) and Ư and Œ' are homotopic so that y = T(p,@(G’)) Since G’ is a convolution by a Cđ-function, its Fourier series: G'(x,z) = 3 z/D;(x) JeZ with D; €C,,,(X, L(H)) is uniformly absolutely convergent, i.e & Dll < 0 JcZ Moreover it is immediate from the formulae giving the Fourier-coefficients Dj that [Dj, p(a)° 1] C,(X, K(H)) for ae A Defining G'(x, 2) = SY zDj(x) j==—m

for m great enough, in view of the absolute convengộnce of the Fourier series of G’ we shall have [|G — U]j < 1/2! so that T(p, 0G") = = ny Defining

Gx, 2) = FID)

J=0

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we shall prove that

Tứ, œ(G)) = '

Let us consider Z{(x, z) = zl Then [ứ(4), Z”] =0 so that Tứ, Z") = 0 Using Lemma 9.4 we have

Tụ, œ(G)) = T(p, œ(Z"G”)) = T(p, Z") + T(p, o(G")) = = T(p, o(@")) = 1

Q.E.D For the next two lemmas p: A > C,,,(% x S!, L(A)) will be a given unital *-homomorphism, constant with respect to Xx S! and defining a trivial Xx S1-extension by A

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Let I denote the matrix in the first term of the above identity; we have I(x, z) = P(x) + 20(x) where Ip, P, € Cy(X, LC @ @ H)) and [T;, (op @

đứ)(@))° Ho CX, K(A đ @ H)) for j= 0,1 and aeA The first and third matrix in the second term of the identity are of the form /-+ nilpotent, and hence homotopic to 7 Thus we have:

T(o @ Op, OF) =

=T(p@ @p,aG, @Iđ OD) = T(p, o(Gy)) = 7

By the non-commutative Weyl-von Neumann type theorem there is a constant unitary VeC, (Xx S}, L(A, H đ đ H)) such that

V-(p đ Bp) (a) V — pa) Ee CXx S', K(H))

for a Â A, Then we may take G= V-!PV, D,= V-F,V, (j=0,1) Q.E.D

9.7 LEMMA Let n € Ext(Ơx S}, A @ C(S)) be such that 149 = 0, Togn = 0, t*y =0 Then there are orthogonal projections Py, Py ộ Cy(X, L(A)) satisfying

[P;, p(a)° tle C,(X, KA) P; = PƠ = P?, Pp + Pp =I

for 7 =0,1 and aộ A, so that for G(x, z) = Po(x) + zP,(x) we have T(p, G) = 4 Proof In view of Lemma 9.6 we have 9 = T(p,(G’)) where G'(x, z) = == Do(x) + zDi(x) Defining Dj(x, z) = Dj(x), we infer from *7 = 0 that

T(p, (Dg + D})) = 0

Hence using Lemma 9.4 we have

T(p, o((Dy + Di)-1 GY) = 0

Thus taking D,; = (Dj + Di)~'D} (G=90,1) and G(x, z) = D(x) + 2D,(x) we have T(p, o(G")) = and D)+ D,=T The invertibility of Dy + 2D, for all zộC with |z| = 1 is equivalent to that fact that the spectrum of Dg does not meet {€ eC} Ref = 1/2} Let then Q, denote the spectral projection of Dy for {€ eC [Ref > 1/2} and Q, = I — Qp Since Q, is an idempotent we have

(QoQ5)? — QoQs = QoT — Qo)*T — Qo) OF = 0

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— PạOy(I — Pa) is invertible and SP,S-1=Q,) We defủne Œ”(x,z)=: ểs(x) +

+ 20,(x), Pạ= I— Po, S(x,z)= S(x) and G(x,zZ) = Pa(x) + zP.(x) Taking

T(, z, ủ) = (L — h) DgGũ + hOs(â + z((L — A)-D,(x) + AO)

it is easily seen that IT gives a homotopy connecting G’’ and G’” so that =

== I(p, o(G")) = T(p, o(G'"))

Using Lemma 9.4 we have

t Tứ, œ(SŒ”“ S”*)) = T(p, a(G’"))

so that ?? = 7(ứ, œ(G)) : : _Q.E.D

We define now the maps

L,: Ext(Ơx S1, 4) > Ext((Xx S1, 4 @ C(S))

in the following way: for [z] e Ext(Xx S1, A), t and Z* determine a homomorphism of A đ C(S) into C,,,(Ơ x S1, L(H))/C,(X x S*, K(A)) which after adding a trivial

Xx S'-extension by A @ C(S) determines a Xx S'-extension by A @ C(S})

The class of this Ơ x S!-extension by A đ C(S') is easily seen to depend only on [7] and will be denoted by L,[t] Then L, isa homomorphism, 7,,.2,[t] = [t], ta4Ê,[t]- 0 and Â*Z,[t] does not depend on k cú

Denoting by :Xx<S; > X, the projecHion z(x,z)=—=x for [t] € Ext(X, A) we define - ‹ B[r] = Lu#d — Luw*[d It is immediate that „B[r] = 0, m;„ff[r] = 0, :*ỉ[r] = 0 So we have a homomorphism

B: Ext(X, A) > {n e Ext(Xx $1, 4 @ C(SĐ) im = 0, toy = 0, t*y = O} Now Lemma 9.7 is equivalent to the fact that ỉ is surjective Indeed, with the notations of Lemma 9.7, taking P¿ = Pạ â @0, PĂ=: P:@0@1, ỚŒQ,z) =

== Pq(x) + 2(Py(x)) we have T(p đ p @ p, G’) = T(p, G) + T(p, 1) + Tip, Z)=

== I(p, G) = n and considering [to], [t,] the X-extensions by A obtained by “‘restricting” t*(p đp @p) to Py and P; it is immediate that [t)] + [t4] =0 and

= Tip Op @p, G) = Bir)

The periodicity theorem is equivalent to the assertion that f is an isomorphism Since we have already proved that is.surjective we must now prove that it is also

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To this end, we shall use a map

y: {y € Ext(X, A @ C(S1) @ C(S"))| tay = 0} > Ext(X, A)

related to the periodicity map for Ext, given in [12]

We shall prove that (y° ô~*e B)[t} = [zt] for all [ct] â Ext(X, A) which in particular shows that f is injective In fact after finding a suitable description of (a-1e B)[t] the proof of (ye a1 f) [t] = [t] will be an ad literam repetition of the surjectivity of the periodicity map for Ext

Let g: S'xS'— S*đ be obtained by collapsing (S1x{1}) uU({1} x Sto a point and let P’ be a projection in C(S") @ C(SĐ @ Mạ ~ C(S!x S') @ M, corresponding to the pull-back from S? to S'x S$! via q of the Hopf line-bundle (an explicit realization of P’ will be given later), Let further P= 1, @ P'’€Ađ @ C(S1) @ CUS!) @ Mz and let [n] € Ext(X, A â C(S) đ C(S')) be such that To3% [q] = 0 Then [y] gives rise to an element [y @ idy,] € Ext(X, A @ C(S!) @ @ C(S*) đ Mz) such that (7 @ idy,) restricted to 1, @ C(S*) đ C(S!) @ M, is trivial Then (y @ idm) (P) is a projection in Cy,/C, which lifts to a projection in C,, Since moreover (7 đ idw,) (P) commutes with (4 @ idaz,) (2,(A)) it follows that we can define a homogeneous X-extension by A, by restricted (y @ idy,)° 7, to (n @ idm,) (P)

The class of this X-extension by A in Ext(X, A) will be denoted by y([y]) Of course, since the construction of y([y]) implies the choice of a lifting of P, y([n}) will be defined only up to weak-equivalence, but in view our assumption that A has a one-dimensional representation, weak and strong equivalence for homogeneous X-extensions by A coincide So the map y is well-defined and is, of course, a homomorphism

We pass now to the description of (a—!e ỉ) [r]

Thus consider t: A > C,,(X, L(A))/C,(X, K(A)) defining an X-extension by A and let further

tT: A > Cy (X, L(Mp))/C,(X, KCHp))

and By = Be € C,,,(X, L(A)) be such that t) and p(B,) define a trivial homogeneous X-extension by A @ C([— 1, 1]) Consider also RQ), Ro(a) such that p(R(a)) = = t(a), P(Ri(a)) = 1,(a) for ae A Let further R(a), Be Cy (X x [0, 1], L(H) be defined by (R(a)) (x, h) = (R(a))\(x), B(x, h) = = (2h ~ 1)7 and let similarly R,(a), ủeC, x:(Z X[0, 1], LCH,)) be defined by (Ro(a)) (x, A) = (R,(a)) (x) and B(x, h) =

=

= Bo(x) By 7, t) we shall denote the Ơx([0,1} extension by A defined by the R(a) and R,(a) Because of homotopy we infer that the Xx{0, 1) | extensions of Ađ c— 1,1) defủned by (Z @ To, p(B â Bo)) and (7 đ 7%, pl đ Bo)) are equivalent Hence there is a unitary Ve Cy(Xx[0, 1], LF â Ho)) such that

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and

VU đ By) — B â B) Vec,Xx[0, 1], K(H @ Hy)

Defining U == (Ve ig) (Ve )~! we have:

[U, R(a) đ Roa] e C,X%, K(H @ H,)) UT â By) — (— D đ B) U < CÚ, K(H @ HẠ)

Consider now W == — exp(xi(I đ By)) € C„(X, L(H @ H,)) The triple (t @ tọ, P(W), p(U)) defines then a unital ôhomomorphism of A đ C(S*) @ C(S‘) into C„,(X, L(H @ Hạ)/CU,(X, K(H @ H,)) which defines (after adding a trivial extension) an element [o] € Ext(Ơ, 4 @ C(S1) @ C(S9)

Consider also k: A â C(S4) â C(S1) + A the unital *ô-homomorphism such that k(a â f đ g) = af(1) gC) With these notations we shall prove the foliowing lemma

9.8 Lemma We have:

(a-te B) [t] = [o] — k, [x]

Proof Of course this is equivalent to proving that a([o] — k,,[t]) = Blt], which in turn can be seen as follows

Let R_(a) €C,,,(X, L(A)) (a € A) be elements defining (—[t]) Then z;;„([ứ] —k,,[t]) is defined modulo a trivial X-extension by A đ C(S") by the elements R(a) đ @đ R,(a) @ R_(a), (a€ A), and by unitary J @ (— exp(ziB,)) â I Putting together the first and third summands of these elements it is seen that 72,.({o] — k„[z]) =0 Also modulo the same trivial extension, the unitary to be used in the construction of ô is U@đIJ and its extension to Xx[0,1] can be taken (V(V> ie r)-) OL But then (VY: her)-) OD, (R@ @ Ra) đ R-(@)° rle €C,(Xx(0, 1], KH @ Hy đ H)) and (WV? ier) @D(W @Deor) (WV ye r)-? @ D7 — ~ (— exp(ni(B â B))) @ Le Œ,(Xx[0, I], K(H @ Hạ @ H))

In view of the fact that B(x, h) = (2h ~ 1)I and B(x, h) = B,(x) it is now immediate

that ô({o] — k,[c}) = ple} Q.E.D

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defini-HOMOGENEOUS C*-EXTENSIONS 241 tion of the domain of ô and from the fact that 2,,f[t] = 0 and 2,,8[t] = 0 In particular since t5,(k,[t]) = 0 it follows that 23,[o] = 0

Thus in order to prove that (ys œ~1s ổ) [r] = [c], remarking that y(k„|z]) = [c], 1t follows that it wIl be sufficient to prove that y[ứ] = 2{[r

Let us now recall the realization given in [12] for the projection P’ e C(S1) â đ C(S*) @ M,, corresponding to the pull-back of the Hopf line-bundle and which is used in the definition of y Consider 7: C(S") @ C(SĐ @ Mĩ; > C([— 1, 1) đ đ C(S!) đ@ M, the *-homomorphism j = /j’ @ id đ@ id where j’ is induced by the map [— 1, 1] 3 t > (— exp(zit)) e S1 Then /(P') can be described as the matrix- valued function on [— 1, 1] x S" given by

| t2 —(#z-+-tÊr)Jù— "

—(ttz-241-)Jl—# I—?

where 7# = (1/2) (| +& ?)

In view of the description we have given of [ứ] (modulo a trivial extension which we shall no longer mention) we sce that putting B= 1@ B, and Bt = == (1/2) (|đ + ỉ) we have that (ứ đ 1đm,) (P) corresponds to

B? —B+(I — B*)'?U—B- (I — BY)'2\ 1 Pp HỆ —B+(— B!2U-1— B-(I — B*)!? I— B J \ Hence y[ứ] will be given by restricting the elements (“ @đ 1) (4) 0 0 (t đ %) ứ) to this projection

Let now Q, denote the constant projection in C,,,(X, L(H @đ H,)) which pro- jects H @ Hy onto 0 đ Hy We shall prove that y[o] = 2[t] by showing that the restrictions of 4sa- (950 0 0 (t đ 1a) (2) to p(E) (o đ idy,) (P) and (I — p(E,)) (o đ ida, (P) where Ey = lM ) 0

are both equivalent to [t] Of course we must first check that p(Eo) and ( @ idx,) (P) commute This amounts to proving that p(Q,) commutes with p(B?) (which is obvious)

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Now, [p(U), p(F*)] = 0 and F*Q, = Q,F* = F* which immediately give the desired commutation We_have (I — p(Eo)) (6 @ idas,) (P) = _ ( —p(0) 0 0 0 which shows that the restrictions to this projection of the elements ( (t đ %) (a) 0 0 (Œ @đ tạ) (4)

defines an extension equivalent with [ct]

For the other restriction, to be shown equivalent, consider

| UBtY+B- 0

D= AE

—(— B>)12 0

Using the definitions of U, B+, B-, B and the relations connecting them it is not difficult to see that

p(D)*p(D) = ( 0) p(Đ) p(é)* = p(E) (+ @ id,) (P)

[n> seằ ứ)]=°

and that

These relations give the desired equivalence of the restriction to p(Eq) (ứơ đ @ idys,) (P) with [t] This ends the proof of y[o] = 2[t] Summing up our discussion and taking into account Theorem 9.1 we have proved the following proposition which is equivalent to the periodicity theorem

9.9 PRoposiriOon We have natural isomorphisms:

Ext(X, 4) -> {n e Ext(Xx S1, 4 @ C(SY) Iman =0, myn = 0, ty = 0} +

T5 {uc Ext(X, 4 @ C(S)) @ C(S9) Imsyu = 0, mạyu =0, mạạyu = 0} >

—+ Ext(X, A)

and ye ante B is the identity automorphism of Ext(X, A)

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HOMOGENEOUS C*-EXTENSIONS 243% Fort the A-variable, since A has a one-dimensional representation, Corollary 8.8 gives an isomorphism of Ext(X, S24) and

{u € Ext(X, A @ C(S") @ C(S")) [me = Ú, may —= ễ, Tay = 0) Then y, via this isomorphism, induces an isomorphism

Ext(X, S24) > Ext(Ơ, A)

which we shall denote by Per,, This map coincides with the straightforward genera-: lization of the periodicity map in [12] The apparent difference in the constructions consisting in the “‘multiplying” by pull-back of the Hopf line-bundle instead of the Hopf line-bundle minus a trivia] line-bundle is inessential, since replacing in the definition of y the Hopf line-bundle by a trivial line-bundle yields a map which is Zero on

{uw € Ext(X, A @ C(S") @ C(S9) ÍfisyH = Ú, ray =O, Tes ft = O} In particular when X is reduced to one point we get the same map as in [12] We want also to remark that in the constructions we did, the projections to which we did restrict extensions did lift to projections, a fact which is not true in general

For the periodicity in the X-variable it is better to consider a pointed space: (X, x9) In view of the naturality of B, we see that B gives an isomorphism:

Ext(X, xp; A) > {9 e Ext(Xx S1, A đ C(S) |mgt = 0, toy = 0, d*y = 0} where d is the inclusion of (Xx {1}) u ({x0} x S4) into Xx %1,

Using Đ8 and Theorem 9.1 we see that aằ 8 gives an isomorphism

Ext(X, x9; 4) > Ext(S2X, *; A)

(where * denotes the basepoint)

This isomorphism will be denoted by Per* and we shall show that it is the obvious generalization of the periodicity map in K-theory Let 6,: Ơx S1x Si! + X be the projection and by 0.: Xx S!x St S? the projection onto S'x St composed with the map q: S'x S! > S? we have already used We want to prove that

Per*[z] = (F(L] — (1))- OF lx],

where Ext(S2X,,ô; A) has been identified,in the usual way with a subgroup of Ext(X x S1x S!; A) As usual, Z is the Hopf line-bundle

This can be seen as follows:

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t= f*ty and UeC,(XxS'x[0, 1], LLY @đH@A)) such that U(x, z, 1) = a Tạ Hạn and al 0 0 | Ủœ,z,0)=|0 â 0 0 0 Z1, Then the elements p() (Œ @1 @đ@1) (2) p(ệ)-! are the images of elements in Cy (Xx S)x S}, L(A đ A @đ A)YC (Xx S'x S1, K(H @ H @ H))

which define, modulo a trivial extension, an extension equivalent with

o(Lyu*[e] + Lou*[y] + L_yw*[te]) =

== (Lyu*[t] — Lou*[r]) = (we B) [r]

To prove our assertion remark that U can be chosen of the form (a;(x, 2, t))1<i,;<3@J, where the unitary 3 x3 matrix V(x, z, f)= (4;/(+, Z Đà <Ă,/<3 has scalar coefficients Since the projections 1 0 0 V(x, z, ofo 0 | V~1(%, z, t) 0 0 0 0 0 0 V(x,.Z, t) f 1 | V-(x, z, t) 0 0 0 0 0 0 V(x, 2, of 0 | V-*(x, z, t) 00 1

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IHOMOGENEOUS C*-EXTENSIONS 245

Thus we can formulate the periodicity theorem as follows:

9.10 THEOREM Let (X, x9) be a pointed finite-dimensional compact metrizable space and let A be a unital nuclear, g.qg.d C*-algebra having a one-dimensional representation Then the maps considered above

Per*: Ext(X, x93 A) > Ext(S2X, +; 4) Per„: Ext(X, xạ; S24) > Ext(X, xạ; 4)

are natural isomorphisms and

Per* > a~*> Pery = idextcx, x; 4)-

To conclude this section we will give two corollaries of the periodicity theorem The first corollary is that we can now compute Ext(X, x); C @ C) a fact which clarifies the problem of lifting projections from C,,,(X, L(A))/C,(X, K(A)) to projections in C,,(X, L(H)) Indeed, we have isomorphisms Per* œ1 Ext(X, xạ; C @ C) 5 Ext(S2X, +; C @ C) aut = Ext(SLY, *#; C(S})) ~ K(SX) ~ K-1(X) 9.11 COROLLARY There is a natural isomorphism Ext(X, x9; C đ C) ~> K~1{X)

The second corollary is related to the injectivity of the map Ê; it will be stated for the sake of simplicity only for the case when X is a point

Let t: A — L(H)/K(A) be an injective unital x-homomorphism Consider the group:

UC(t) = {U € L(H) |U unitary, [p(U), t(A)] = {0}

Clearly UC(t) up to isomorphism depends only on [t] On UC(t) we shall! consider two topologies: the norm-topology and a second topology we shall call the commutators-topology The commutators-topology is defined as the weakest topology for which the following maps are continuous:

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where Â runs over H, XƠ runs over p~(t(A)) and H and K(H) are given the norm-

topologies

Consider also the following loop denoted by ọ: S'3 2 —> zIc UC(t)

2.12 COROLLARY Let A be a unital nuclear, g.q.d C*-algebra having @ one-dimensional representation and let t: A + L(H)/K(H) be a unital *-monomorphism Then the following assertions are equivalent:

(i) [7] = 0

(ii) @ is homotopic to zero for the norm-topology on UC(t)

(iit) @ is homotopic to zero for the commutators-topology on UC(t) Proof Clearly (ii) => (iii)

That (i) = (ii) can be seen as follows Since UC(t) depends only on the equivalence class of t, we may assume H = H, đ H, (where H,, H, are infinite-dimensional) and t = pe (p â Jx,) where p is a unital *-homomorphism Then Â(z) == = In, đ (2ly,) and by Kuiper’s theorem the loop z — Jy, @ (2/u,) is zero-homotopic in Jx, đ U(H,) which is contained in UC(z)

Next, we prove that (ili) > (i) Let wy: S'x[0,1]— UC(x) be continuous for the commutators-topology and assume /(z, 0) = zẽ and Ứ(z, l) =ù Then is a unitary element of C,,(S!x[0, 1], ZCH)) such that [(g*r) (4), p)] = 0 where ứ: S1x[0, 1] — {+} and [t] is viewed as an element of Ext({*}, A) Then g*t and wp determine a (S!x/[0, 1})-extension by A @ C(SĐ which restricted to S'x {0} and S'x {1} gives extensions equivalent to L,u*[c] and Lou*[t], where u: Sl {+} By homotopy it follows that L,u*[t] = L,u*[t] which means A[t] = 0 By the injectivity of B it follows that [zt] = 0 Q.E.D

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2 APosror, C., Quasitriangularity in Hilbert space, Indiana Univ Math J., 22 (1973), 817—825

3 ARVESON, W B., A note on essentially normal operators, Proc Roy Jrish Acad Sect A., 74

(1974), 143-146

4 Arveson, W B., Notes on extensions of C*-algebras, Duke Math J., 144 (19771), 329—355

Š ATIYAH, M F., K-theory, Benjamin, 1967

6 BERG, Í D., An extension of the Wely-von Neumann theorem to normal operators, Trans Amer Math Soc., 160 (1971), 365 —371

7 Brown, L G., Operator algebras and algebraic K-theory, Bull Amer Math Soc., 81 (1975),

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