J OPERATOR THEORY 1(1979), 55-108 © Copyright tight by INCREST, 1979 by
HOMOGENEOUS C*-EXTENSIONS OF C(Y)@K(W) PART I
M PIMSNER, S.POPA and D VOICULESCU
The remarkable work of L.G Brown, R.G Douglas and P A Fillmore ({10], [12}) on extensions of the ideal of compact operators by commutative C*- algebras has stimulated further research concerning more general extensions ([1], [3], [4], [9], [13], [16], [20], [26], [34—39], [41 47]) This is motivated in part by the desire to extend the Brown-Douglas-Fillmore theory so as to provide a tool for analysing the structure of C*-algebras
In particular, such a development might lead to a better understanding of the structure of type I C*-algebras
Also we should mention the general program for the study of extensions sketched by L G Brown in ref [9]
A class of extensions to be studied, as suggested in ref [26], are those of C(X) @ K(H) Among these, the homogeneous extensions, considered here, seem to be more tractable Let us explain what the homogeneity requirement means Roughly speaking, an extension of C(X) ® K(#) by a C*-algebra A (separable and with unit) gives rise, for each xe X, to an extension of K(H) by some quotient AjJ, of A The map which associates to x e X the ideal J, will be called the ideal symbol of the extension The extension is called homogeneous if J, = 0 for all xeéX Under a suitable equivalence relation and with some additional conditions on X and A (X finite-dimensional and A nuclear), the homogeneous extensions yield a group Ext(X, A), which will be the main object of our study For X reduced to a point, this is just the Brown-Douglas-Fillmore group, but the consideration of the more general Ext (X, A) will be seen (in Part IT) to be also of some interest for the study of the usual extensions by K(#)
Trang 256 M PIMSNER, S POPA and D VOICULESCU composition series with quasi-diagonal quotients (this includes the type I C*-
algebras)
In more detail, the content of the six sections of Part I is as follows § 1 contains general definitions and some preliminaries
In § 2, assuming that the ideal symbol of the extension satisfies some lower semicontinuity requirement and X is finite-dimensional, we prove the existence of trivial extensions and a generalization of the Weyl-von Neumann type theorem of [45]
Beginning with §3 we consider only homogeneous extensions We use the Choi-Effros theorem [16] to show that Ext (X, A) is a group when _X is finite-dimen- sional and A nuclear Also in § 3, we prove, under the same requirements, that in each equivalence class of Ext (X, A) there is an extension which can be realized using the norm-continuous L(H)-valued functions on YX
In § 4 the short exact sequence in the A-‘‘variable” (A-nuclear) for Ext (X, A) is proved This generalizes the exact sequence in [10] as well as the subsequent generalization in [9]
In § 5 we deal with homotopy-invariance for Ext (X, A) both in the X-“‘variable” and in the A-“‘variable’ Both homotopy-invariance properties are proved for nuclear quasi-diagonal C*-algebras via an adaption of the argument of Salinas [42] and then using § 4 extended to more general C*-algebras Let us also mention a brief discussion of quasi-diagonality in C*-algebras, an adaption of the notion due to P R Halmos [27]
In §6 we prove a short exact sequence for Ext (X, A) in the X-“variable”’ Finally we should mention that Part II of this paper is concerned with topo- logical properties of homogeneous extensions of C(X) ® K(A)
The authors gratefully acknowledge helpful advice from S Stratila and A Verona
§ 1
Let H be a complex, separable Hilbert space of infinite dimension Let L(H) denote the bounded operators on H, K(H) the ideal of compact operators and
nm: L(H) + L/K(H) = L(H)/K(H)
the canonical homomorphism of L(#) onto the Calkin algebra
For X a compact metrizable space, C,(¥, K(H)) denotes the C*-algebra of K(#)-valued continuous functions on X, where K(H) is endowed with the norm topology Similarly, C„(X, L(H)) is the C*-algebra of L(H)-valued continuous functions on X, where the continuity is with respect to the *-strong operator-topology on L(H) (of course, the C*-norm is the sup-norm) Clearly, C,(X, K(H)) is a closed two-sided ideal of C.,,(X, L(H)) By p we shall denote the canonical homomorphism
Trang 3HOMOGENEOUS C*-EXTENSIONS OF C(X)@ K(H) I 57 For A a separable C*-algebra with unit, an extension of C,(X, K(H)) by A, is a short exact sequence (*) 0 ¬ C,(X, K(H)) > B ¬> A—¬0 where B is a C*-algebra with unit, p and o are *-homomorphisms, o being unit- preserving For D aC*-algebra and M c D let us denote Ann (M; D) = {ye D; My = 0}
In order not to complicate our study of extensions it is natural to eliminate a certain trivial part of B, by considering only the extensions satisfying the requi- rements of the following
1.1, DEFINITION An X-extension by A is an exact sequence (*) satisfying the additional requirement:
Ann (p(C,(X, K(H))); B) = 0
The following folklore-type proposition, in fact about multipliers of C„(X, K(H)), gives a more concrete realization of X-extensions by A
1.2 PROPOSITION, Let (*) be an X-extension by A Then there is a unique *-homomorphism
po: B> C,(X, L(A)
such that @ ° p = i, where i denotes the inclusion
CX, KU) > Cy (4% L(H})
Moreover @ is injective and unit-preserving
Proof The closed two-sided ideal p(C,(X, K(H))) of B, being isomorphic with C,(X, K(H)), has a natural faithful non-degenerate »-representation on the Hilbert space:
EX; H) = {Œ)xex: hy eH, » ||5„llÊ < + oo}
Moreover this representation is in the commutant of the natural representation of 2°(X) on £7(X; H) By [21, Prop 2.10.4] the representation of p(C,(X, K(H))) has a unique extension to a representation of B (which is unit-preserving), still in the commutant of the representation of ?°(X) This yields unit-preserving *-homo-
morphisms g,: B + L(H) such that for be B and fe C,(X, K(H)) we have
Trang 458 M PIMSNER, S POPA and D VOICULESCU
where g €C,(X, K(H)) is given by g(x) = 9,(b)f(x) Moreover,
ø.(Œ)) =0)
Clearly, we may defđne ø() by (0()©) = @,(6) provided we prove that Xa xt o,(b) 6 L(A)
is strongly continuous (for »-strong continuity consider b* € B) For & € H, ||é|| = 1, let P denote the projection of H onto Cé and fe C,(X, K(H)) the constant function equal to P Then
(x) = 0@„()ƒ(x) = œ„(b)P
is an element g e C;(X, K(H)) and this is equivalent to the continuity of the map X35 xr 9,(b)é € H, ie the desired conclusion
The uniqueness of @ follows from Ann (i(C,(X, K(H)); C„,(X, L(H))) = 0 Indeed, if g’ is another +-homomorphism with ø'°sø = ¡, then forbe 8 we have ø(b) — ợ'(b) e Ann ((C,„(X, K(H)); C„(X, L(H))) = 0 Also, 1f ø() =0, then j(ø(ƒ)) =0 and since ừ(ƒ) e ø(C„(X, K(H))) we Infer bp(f) = 0 Thus Ker g < Ann (p(C,(X, K(H))); B) = 0,
which gives the desired result about injectivity Q.E.D
1.3, REMARK Jn view of the preceding proposition it is clear that, from now on, we may and shall assume, for an X-extension (*) by A, that
C,(X, K(H)) c B c C„(X, L(H))
1.4 DEFINITION Two X-extensions by A given by exact sequences 0> CX, K(A,)) > B, 3 A> 0
0 > C,(X, K(Hz)) > B, 3 A > 0
are said to be equivalent, if there is a unitary
Uec,, (X, LUN, Ap)
such that
Trang 5HOMOGENEOUS C*-EXTENSIONS OF C(X)@ K(H) 1 59 1.5 PROPOSITION There is a one-to-one correspondence between X-extensions by A 0->C,(X, K(H))>+ B> A470 and unital *-monomorphisms +: A > Cy (X, 1(H)/CŒ,(X K())
In this correspondence B = p™1(t(A)) and o is obtained from the obvious isomorphisms between B/C,(X, K(H)), t(A) and A
Proof How + yields an X-extension by A is quite clear from above, for the converse also, remark that o gives an isomorphism between
BỊC,(X, K(H)) c C„(X, L(H))/C,(X, K(H))
and A, the inverse of which will give the *-monomorphism t Since Bc C,,(X, L(A)), it is obvious that
Ann (C,(X, K(H)); B) = 0 QED
1.6 REMARK Proposition 1.5 gives.an alternative way of defining X-extensions by A This will be frequently used in what follows referring to an X-extension as defined by some *-monomorphism t For a unitary UeC,,,(X, L(Hy, He)) let «(U) denote the isomorphism
Cy (X, LUM) 2 fr» Uf U* € Cy (X, LAD)
and let «(U) denote the isomorphism betveen C„(X, L(H))/C,(X, K(H)) and Cy (X, LU) C,(X, KCH2)) induced by aU) Then the X-extensions defined by
tị: 4 > Cy (X, L(H)))/C,(X, KU), (= 1, 2)
are equivalent, iff t, = «(U)° t, for some unitary Ue Cy,(X, L(H;, Hp) We shall use the notation t, ~ Tạ for the equivalence of the X-extensions defined by tị qnả tạ
Let now for xe X, p, denote the *-homomorphism
p„: C„(X, L(H))JC,(X, K(H)) > L/K(H)
which associates to p(f) the element zx(f(x)) of L/K(H) We shall also denote by I(A) the set of closed two-sided ideals of A,# A
1.7 DEFINITION Let t: A > Cy, (CX, L(A))/C,(X%, K(H)) be a *-monomor- phism Then the map
X32 x +>» Ker (p, 2 t) € I(A)
is called the ideal symbol of the X-extension by A defined by t The X-extension defined by t is called exact if
() Ker (p, 2 t) = 0
Trang 660 M PIMSNER, S POPA and D VOICULESCU
In case Ker(p,°1)=0 for all xeX, the X-extension defined by 7 is called homogeneous
It is easily seen that the equivalence of X-extensions preserves the ideal symbol and hence also exactness and homogeneity Given a map X3axre I,el(A) satisfying the exactness condition we shall denote by Ext (X; A, ()xex) the set of equivalence classes of X-extensions by A with ideal symbol X3 Xx r> r> Te 1,(A)
Clearly, the X-extensions considered are exact In case f,, = 0 for all x € X, we shall denote by
Ext (X, A)
the set Ext (X; A, (1,),ey)
We do not know what conditions the ideal symbol must satisfy in order that Ext (X; A, (,)xex) #9, although in §2 a certain lower semicontinuity for the ideal symbol will be considered which is necessary for the existence of trivial exten- sions with the given ideal symbol and which will be shown also to be sufficient provided X is finite dimensional
If t defines an X-extension by A with ideal symbol (J,),¢, then [rle Ext(X; 4, (I),ex) denotes its equivalence class Consider also
1¡: A4 = C„(X, L(H,))/C,(X K(H,)), (i = 1, 2)
two *-monomorphisms with Ker(p,ot;)=1,, (= 1,2; xeX) This yields a natural *-monomorphism
t, ®1+;: 4 > Cy(X, L(A, ® !I,))/C„(X., K(H, @ H,))
with Ker (7x s (r @ rs)) = l„ for xe X Moreover, it 1s easily seen that [t, ® 7] depends only on [t,], [t.] Thus,
[t1] + [te] = [t1 © 12]
Trang 7HOMOGENEOUS Ct-EXTENSIONS OF C(X)@ K(H) ữ 61
Let X, Y be compact metrizable spaces and let g: ¥ + Y be a continuous map This yields a *-homomorphism
G:C, (Y, L(A) > C,(X, LCA) defined by G(f)=feg for feC,,(X, L(H)) Clearly,
G(C,(Y, K(H))) c C,(X, K(H))
and we have and induced *-homomorphism
G: C„(Y, L(H))JC,(Y, K(H)) ¬ C„(X, L(H))(C,(X, KH)
Let+r: Á—>C„,(Y, L(H))/C,„(Y, K(H)) define an Y-extension by 4 with ideal sym- bol (J,) ey; then Got=g*(t) is a *-homomorphism oŸ4 into C (X, L(H))/C,(X, K(A)) and Ker (p,.o (g*(t))) = Jx) So, in case (\ J.) = 0, there is a well-defined
xEX map, still denoted by g*, [t] +> [g*(t)] = g*[t],
g*: Ext (Y; A, Uy)yey) > Ext (X3 A, Uycxy)xex)s
which is a homomorphism In particular, for A fixed, Ext (¥, A) becomes a contra- variant functor from nonvoid compact metrizable spaces to commutative semigroups
§ 2
This section is devoted to the study of trivial X-extensions with given ideal symbol In case X is finite-dimensional and the ideal symbol lower semicontinuous in an appropriate sense, we shall prove the existence of trivial extensions and also a generalization of the Weyl-von Neumann type theorem of [45] (see also ref [4]), which will show that Ext (X; 41, (/,).ex) is a semigroup with unit in this case
We recall that the compact metrizable space X is of dimension <n if for every covering by open sets of X there is a finite open covering refining it, that has order < # (Ch V of (29)
The appearance of finite-dimensionality requirements in the study of X- extensions should be traced back to a continuous selection theorem of E Michael [33], which is also used in the related subject of continuous fields of Hilbert spaces
(see 10.1.2, 10.8.6, 10.8.7 and 10.10.9 in ref [21])
From now on, throughout the rest of this paper it will be assumed that the compact metrizable space X has finite dimension
Trang 862 M PIMSNER, S POPA and D VOICULESCU compact metrizable space of dimension < # can be imbedded in R2**+!{[Thm V 3 in ref 29], easily yields the following useful fact, we shall record as:
2.1 REMARK If X has dimension <n, then every open covering of X has @ refinement each open set of which intersects no more than 3**+1 — 1 other open sets of it 2.2 DEFINITION For X a compact metrizable space, a map Xax>1,ef1(4) is called lower semicontinuous (abbreviated L.s.c) if for every convergent sequence (x, 81 X, lim x, = Xo we have #?—>OO c (1 lu = đụ n=)
Denoting for ae A and JeI(A) by a/J the image of a in A/J, it is easily seen that the l.s.c condition 2.2 is equivalent to the following: whenever x, — Xo and ze 4, we have
lim inf |la/Z,,|| > ||2/T„a|
(use the fact that |ja/ 7) Z.,,|| = sup,||a/Z,,, || and consider subsequences)
2.3 DEFINITION An X-extension by A defined by the +*-monomorphism +, with exact ideal symbol (I,).cy is called trivial if there is a unital «-homomorphism
wi A > C,(X, L(A)
such that po w= and Ker (d, ow) =I, for all xe X, where d,:C,,(X, L(A) > — L(A) is the map 4,(f) =f)
Tt is easily seen that po p = 7 implies
Ker (d, ow) c Ker (p, ° t) = Ï,
so, for homogeneous X-extensions by A, the condition Ker (d,° #) = 0 follows from the first condition
Trang 9HOMOGENEOUS C*EXTENSIONS OF C(X)@ K(#) I 63 To prove that for X of finite dimension and l.s.c ideal symbol there exist trivial extensions we need some preparations
For the next Lemma A is unital and separable (as always), E(A) is the state space of A and for Je I(A), E(A/J) will be considered as a subset of E(A) 2.4 LEMMA Suppose X has finite dimension and let X3x+> I, € I(A) be Ls.c Then given a state g of A such that 0|I,,= 0, there is a map X3 xt > w, € F(A), continuous for the weak topology on E(A), such that w,|I, =0 for all xeX and o,, = @
Proof The idea is to use the selection theorem of E Michael [33] for the set-valued map
Xaxt> E(A/E,) = E(4) To this end we give E(A) the metric
4#) — & *1fla,) — (2)
where ||a,|| < 1 and (a,)% is total in A Clearly d induces the weak topology on E(A) and E(A) is a complete metric space since E(A) is compact for the weak topology Moreover the balls with respect to d are convex, so that their intersections with the E(A/ZI,) are convex and hence contractible
Thus, the only thing still to be checked is the lower semicontinuity (in the sense of Michael) for X3 x» E(A/I,) < E(A) The lower semicontinuity con- dition is
given e > 0, ye X and fe E(A/J,) there is a neighborhood V < X of y such that
E(A/I,) 0 (g € E(A); Af, g) < s} #“Ø
for all xe V
This is easily seen to be equivalent for metrizable Y with whenever x, > y and fe E(A/I,), there are
f,€ E(A/Z,,,) such that f, > f weakly
Now, for this reformulation it is easily seen that it will be sufficient to prove it only for f in some subset of E(A/I,) Thus we may assume f= k-(g, + + g%) where g,¢ E(A/J,), (j= 1, ,%), and pure But this makes a second reduction possible, namely we may assume f is pure Then, considering xz, any represen- tations of A with Kerz, = J,,, we have f|(()\Kerz, = 0, since f|Z,=0 and
E> Ole, = Ker mạ
Trang 1064 M PIMSNER, S POPA and D VOICULESCU
Thus the Lemma follows by applying the theorem of Michael Q.E.D Let C(X, E(A)) denote the set of continuous maps from X to E(A), F(A) being endowed with the weak topology We consider on C(X, E(A)) the topology given by the metric
6(F, G) = sup a( F(x), G(x),
where d is the metric on E(A) considered in the proof of Lemma 2.4, Further consider the closed subset Q < C(X, E(A)) defined by
9 ={Fe CŒ, E(4A); F@)|1„=0, (V) xe X}
2.5 LEMMA Suppose X has finite dimension and Xa x>I,cl(A4) is Ls.c Then there is a sequence {w,\$., of continuous maps w,;: X - E(A) such that
Ầ Ker wx) = I, for every xe X j=l
Proof In view of Lemma 2.4, {a,}[21 may be any dense sequence in @ Thus all we have to prove is that Q is separable when the metric 6 is given But since Q is a closed subset of C(X, E(A)) it is clearly sufficient to prove that C(X, E(A)) is separable This can be easily seen as follows The space X being
compact and metrizable, fix a metric on X and consider {V{/)}24, open coverings
(jE N), by open balls of radius 1/7 Let further {o{\2%, be a partition of unit subordinated to {V}24), and @ c E(A) a countable dense subset of E(A) Then it is easily seen that the maps Fe C(X, E(A)) of the form
n{j)
F(x) =3 pe) (x) O,
(with je N, @,¢€@), form a countable dense subset of 2 Q.E.D
2.6 THEOREM For X of finite dimension and Xax > I, € I(A) ls.c., (-) I, = 0, there exists a trivial X-extension by A, with ideal symbol Xex +>
xEX
+> I, € I(A)
Proof Consider {@,}72, a sequence of £(A)-valued functions satisfying the conditions in Lemma 2.5, and where each term appears an infinite number of
times Let then x¥) denote the representation of A on HY with cyclic vector
Trang 11HOMOGENEOUS C*t-EXTENSIONS OF C(X) ® K(H) I 65 be the set of those (4,),¢ such that
h„ = @® (Š ø,6z()} Ey) =1 \7=l
for some neéN, gi €C(X), a,¢ A, (1 <i <n, jeN), and where @;; # 0 only for a finite number of 7 Clearly, I’) is a vector subspace and since
Wal? = FY Gey) (,L9Naka,) j=1 1<p, qn
we Iinfer that Xa x ||A,|| eR is continuous for (4,),cx € Ip Define now rc]In, xEX as the set of those (A,),ex such that for every e>0 there is (h{),cx €I'y satisfying sup ||; — hy || < “ex
It is easy to check that ((A,),cx, [) is a continuous field of Hilbert spaces ((21], 10.1.2) which is also separable ((21],10.2.1) Moreover if ae A and (h,),cy ET then also (z,(a)h,),cx ET
By (10.8.7 in ref (21]) the continuous field of Hilbert spaces ((H,),cx, I) is trivial ((21], 10.1.4) Thus there are unitary maps U,, of H, onto H such that the set of maps Yaxr> U,h, ¢ H, where (4,),cx runs over I’, coincides with the set of all continuous H-valued functions on X Moreover, for ae A the function h(a): X > L(A), (u(@))(x) = U,2,(a) UX, has the property that p(a)fe C(X, H) for every fe C(X, H) Taking also p{a*) into account, this gives (2) e C„,(X, L(N)) Then t = pop is a trivial X-extension by A with ideal symbol (/,),¢x as can be easily seen since Ker z, = J, and z, is of infinite multiplicity for every x ¢ X Q.E.D Our next aim is to prove the Weyl-von Neumann type theorem This will also require several steps
2.7 PROPOSITION Suppose X has finite dimension, let
0 > C,(X, K(H)) > B= A> 0
be an exact X-extension by A with ideal symbol Xaxt>I,¢I(A) and let we C(X, E(A)) be such that w(x)| I, =0 for all xe X Then given se > 0 and VCH,
le Wc B finite dimensional subspaces, there is he C(X, H) such that
[AO =1, AX LV, W)xeX,
\(o(x))(«(b)) — CB(x)AC), A(X) < e[b|, (WY) xEX, Whew
Trang 1266 M PIMSNER, S POPA and D VOICULESCU Proof By N we shall denote an integer N > 32"+1 where n is > than the dimension of X Let
be the «-monomorphism which defines our X-extension Since tog = p| B and nod, = p,op, we have Pyo toa =p, °(p| B) = xo (d,| B) It follows that (x o d,)(B) = (p, © 1)(A) Also since ' Ker (p, ° t) = J,, there is a'(x) € E((p, © t(A)) = E(Œ ° d,)(B)) such that @ (X)s „s1 = w(x)
Considering now the state w’(x)o z on đ (PB) c L(A) and using (11.2.1 in [21)), it is easily seen that we can find a subspace R, < H, dim R, = N dimW+ 1 such that R,_L V and
KKd,()E, > — (ox) M(H » d,)(5))| < > (|i
for every be Wand €e R,, ||€|| = 1 This can be also written: K(a)E, € — (LINO) < +16 for be W, Ee R,, ||E|| = 1 Consider also an open neighborhood G, of x, such that é b(y)š— b(@)#lL< —=—- IIb@)é (x)é || Sad yy ll II(o()Xø()) — (œ()(ø())|| < T lơi
whenever y€G;, be W, éeR,, llễ|| = I
Smce X has topological dinension <ø, there is a refinement (GŒ/)Ÿ ¡› G, c GŒ¿„ of the open covering (G;)„ex such that each Œ, meets at most N other G;’s
Trang 13HOMOGENEOUS C*-EXTENSIONS OF C(X) @ K(H) 1 67
The ¢,’s will be chosen by induction For ¢; we may take any vector ¢,€R,,, |€:!|=1 Suppose ¢,, , ¢; have been chosen, then consider 1 <i,<ip < < i, <j those indices for which G;, 0 Gj, # O Clearly m < N by Remark 2.1 It follows that
` dim (đu (W)3š¡,) < NdimW <dim Ry
s=1
so we can find ¢,4,¢ Ry (|¢;41|| = 1 and such that
d,, (W) oi, Low (Uo <s<m)
Consider now {g,}%_1 a partition of unity subordinated to the covering (G,jf.1 Then we define
A(x) = S @”(x)&
Since ¿,_L é¿ whenever ø/”(+)@j”(x) # 0, it follows that ||A(x)|| = = ] for all xe X It is also obvious that A(x) 1 V for all x ¢ X and that the linear span of {đ(x)}„ex is finite-dimensional We have |Á6(x)đ(œ), h(x)> — (œ(+)(ø(ð))| < SY PGP (OC) G„nG;+ø — ODEs & 1 + ` mm + ¥ Gd (OH) — oo)! < > =1 e\|bi) „ oh? 1/2 S (x)@j NL PHD Cte peor
+ ¥ onc) Pl + ¥ ooo Lil
_ 3 Fos gl (Be x09) < EN (Sar)
3 b
< fois gilt w+ ¥ exc) =elldlh
This ends the proof Q.E.D
Trang 1468 M PIMSNER, S POPA and D VOICULESCU Let also Cp(A, M,) denote the set of completely positive unital maps from A to M, endowed with the point-norm topology
2.8 PRoposiTion Suppose X has finite dimension, let
0> CX, KH) + BS A+0
be an exact X-extension by A with ideal symbol X 3 x ++ I,,€ I(A) and let ¥: K > — Cp(X, M,) be a continuous map such that ¥(x)| I, = 0 for all xe X Then given e>0Oand V c H, Le W ce Bfinite-dimensional subspaces, there is U: X -+ L(C*, #) a norm-continuous map such that
U*(x)U(x) = Ten, U(x) (C") LV, (VW) xe X,
|| O*(x)b(x) U(X) — (F(x))(o())|| < eb, (W) xe X, (W) be W,
and the linear span of {U(x)\(C")}.ex is finite-dimensional Proof There is a natural isomorphism ([17, 4])
A: Cp(A, M,) + Cp(A @ M,, C) = E(A @ M,)
given by
APY a ® &,) => ¥ ¥y(a,) tj %7
where M, are the componenfs of W; i.e W{2) = ¥ W,(2)e¡
aaj
Consider the exact sequence
c@idy,
0 CX, K(A)) ® M, > B® M,——_—"> 4@ M, > 0
Identifying C,(X, K(H)) @ M, and C,(X, K(H”)), this sequence can be viewed as an exact X-extension by A @ M, with ideal symbol Ya xt L ®@ M,é HA @M,) Consider then wm = A(W)e E(A © M,) and apply Proposition 2.7 This gives a continuous function = (A,, ,4,): X + H” such that
Trang 15HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H) I 69 where {e,, , €,} is the canonical orthonormal basis of C” Then we have S*(x)S(x) = n > A(x), hx) ) sụ = n > (1 @ e:)A(x), h(x) } 63; so that IS'G)86)—1u,|< m 3 | CÚ 6 6/)8G), A@))— Su = =n » IX @ ø¡j)h(>), h@œ)}È — (œ(3))(1 @ €;))| < ;.« e2 < H-H^- = —- lồn? 16 Supposing « < | (which means no loss of generality), we have J(S*%()5(x))s=1⁄2 — laz„|[ < 2 — 1/2 2 — 1/2 < max ('={'+#£) ‘((-%) —1] <<, 16 16 4 so that | Sx) — SQ)(S*O) Sx)? || < |S) I] 1a, — S*CQSCQ)™ || < < 2U + ai <= 4 16 3 U(x) = Sx)\S*) SO), Finally, if
then U(x) is an isometry and clearly depends continuously on x €¢ XY We have
|| U*(x)b(x) U(x) — (¥%))(o()) || <
< ||U@œ) — S()||I|bl[ (1 +-|l.S()|) +- Í|S*Œœ)ð(3)S(x) — (f())(ø(@))||<
e é 1
< (244) S164 ¥ Coenen ney) — 1 wyonew)|=
Trang 1670 M PIMSNER, S POPA and D, VOICULESCU
Also, since U(x)(C") = S(x)(C”%, it is obvious that U(x)(C") 1 V and the linear span of {U(x)(C")},cx is finite-dimensional Q.E.D
Let X3 x+» I, < I(A) be an 1s.c map with () 7, = 0 and let
xEX
0 > CX, K(H,))> B, > A> 0
be a trivial X-extension by 4 with ideal symbol Xaxr>7,e/(4) Let also ly: A => Bị c C,,(X, L(A)) be the *-monomorphism implementing the triviality of this X-extension by A (i.e o,°f, = id, and Ker (d,on)=T1,, (V) xe X) Consider also
0+ CX, K(A)) > B> A 0
an arbitrary X-extension by A with ideal symbol X3 x I, € I(A) With these notations, we have
2.9 PRoposirion There is Se Cy,(X, L(H;, H)) such that SOS) = 1, (Vv) xe X,
Sw(ø()) — bSe C(X, K(H, H)), = (V) be B
Proof There is an increasing sequence 0 = 4ạ <S 4) < 4; < , ||4„l| < 1, of elements of C,(X, K(A,)) which are constant on X and of finite rank such that
lim ||4jk — k||=0, (V) keŒ,@, K(H))
joe
lim ||4jb — bA,|| = 0, (VW) be By jroo
Since C,(X, K(H,)) has an approximate unit which is an increasing sequence of constant finite rank elements, this follows from [4, remarks after the proof of Thm 1] Consider also {5,}92,, b; = bj, a total sequence in B Then replacing {A,}s., by some subsequence, we may suppose that
Iian(o(b,)), (Aj — Aya") || < 22 for 1<k<j7
Consider further P; ¢ C,(X, K(H))) constant projections such that A,P, = Pj;A; = A;
Using Proposition 2.8 several times one can easily construct norm-continuous maps X35 x +> U,(x) ¢ L(A, H) and finite rank projections R; ¢ L(H), Ri\< Re< , , such that
(i) UFU, = P;, (jE N)
(ii) U(x)(A) c (Ryaa — RD, (V) xe X
(iii) || — Ry db.) R,|| < 2-7, () xe X and 1 <k <j
Trang 17HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H) I 71
The sum
Š U,(x)(A; — Aja)" = S(x)
j=l
is easily seen to be strongly convergent and S*(x)S(x) = I,, Also since the A,’s are constant and because of (ii) it is easily checked that the sum defining S(x) is uniformly *-strongly convergent on X, thus defining an element S € C,,,(X, L(M,, H))
Using (ii), (ii) and b, = b* we have » IIU#b,U,jjj =2 3, I|Uð,U;||< 1.7 2>ửJ <2 3 JU*b,U,I +23, 2 u~b 1<+,7<h +>7>] co
<2 5% lU*b,U,|+ 2W ¡20D < + cĩ 1<i,7<k tol
Also using (iv) we have
YC; — Ar P6.U; — salolb (A — Aj-a)#2 < + 00 + đ and using the inequalities for Jj[(4; — 4; 4), m(ø(j)]|| we have ¥ llan(o(b)NA; — 4-2) — (4) — 41)! (o(b,)(A; ~ 4; || < + 00 Z Thus we have 3 l4; — 4; 1)12UP9,U/(4;— 4;-1)13 — wn(o(b,) (A; — Aja] + j=l + x || (A; — Aja)? U#b,0,(A; — Aj1)? || < + 00 inj
This proves that
S*hyŠ — m(0(b,)) e C,(X, K(H,) for all KEN Since {5,}., is total in B we infer
Trang 1872 M PIMSNER, S POPA and D VOICULESCU But this is equivalent with
bS— Sm(0ø0)) e C„(X, K(H;, H)) Q.E.D
2.10 THEOREM Suppose X has fite dimension qnd let X 3 x c> Ï,e€ l(A) be an exact I.s.c ideal symbol Then the trivial X-extension by A with ideal symbol X3x<4f,€I(A) are all equivalent and their class is a neutral element in the semi-
group Ext(X; A, Œ)xex)
Proof Let 1,1: A > Cy,(X,L(H))/C,% K(H)) be smonomorphsms de- fining X-extensions by A with ideal symbol ¥ 3x — I, €1(A) Then assuming that t, defines a trivial X-extension by A with the given ideal symbol, we shall prove that [c @ t,] = [t] This will show that [z,] is a neutral element for Ext(¥; A,J),ex) and since two neutral elements must coincide, also the other assertion of the theorem
will follow
Consider the exact sequences
0 ŒC,(X, K(H)-> B > A0
0 ¬ CX, K(H))> B, > A — 0
corresponding to the X-extensions by A, defined via t and 1, Denoting by Hj, the Hilbert space H®H@ , by py: A > Cy,(X, L(M,)) the «monomorphism
(H2(@))(x) = (n(2))) @ (¡(2)()) ® ,
and by B, the C*-algebra
By = yx(A) + C,(X, LUM) < Cy(X LU),
we obtain an exact sequence
0 ¬ Œ,(ŒX K(H))) ¬ By 2 A> 0
defining a trivial X-extension by 4 with iđeal symbol X a x -> l, e 1244) By Proposition 2.9 there is SeC,,,(X, L(H,, H)) such that
Sua(ø(b))— bSe C,(X, K(H, H)) for all be B Denote by VeC,,(X, L(A) the constant isometry
V(5)(h @ hạ @ ) = 0 @ đh¡ @ hạ @ and by PeC,,(X, L(H,, H)) the constant co-isometry
Trang 19HOMOGENEOUS C?*-EXTENSIONS OF C(X) ® K(H) 1 73 Clearly V commutes with (A) and hence with B, modulo C,(Y, K(H,)) Similarly, P intertwines p and y, Consider then U(x): H > H @ H defined by
U()0Œ) = (I — S(>)S*(2))h + S()V*()S*()h) @ P@)S*G@)h Then ¡s unitary, Ứe C,(X, L(H, H @ H)) and ã(Ũ)st = r @+ Q.E.D
§ 3
Beginning with this section we shall consider only homogeneous X-extensions by A Assuming that A is nuclear, we shall apply the Choi-Effros completely posi- tive lifting theorem ([16], see also [4, 46]) to prove that Ext(X, A) for finite-dimen- sional X is a group Using this fact we shall also prove that every homogeneous X-extension by A is equivalent to the one for which
C(X, K(H)) c B c CV, L(H))
3.1 LEMMA Le( W: 4Á ¬ C„(X, L(H)) be a completcly positive map Then there exists a separable Hilbert space H, > H and a unital *-monomorphism H: A C,,(X, LG)) such that
(Ÿ(2)Xx) = P(u(a)\(x) | H for every ae A, xe X (P denotes the orthogonal projection of Hy, onto H)
Proof For each xe X let ¥,: A > L(H) denote the completely positive map
¥(a) = (P(a))(x)
Trang 2074 M PIMSNER, S POPA and D VOICULESCU which is clearly a continuous function of xe X It is easy to check now that (HC H),ex, T) is a continuous field of Hilbert spaces (10.1 1 in [21}) Since X is finite-dimensional and this field is separable and each Hy, © H is separable infinite- dimensional, it follows by ([21}, 10.8.7) that we have a trivial field Hence there are unitary operators U,: H; © H — H,such that the set of functions Xa xt+-U,h,cH, where (h,),cx runs over C(X, H’, OQ A), is just the set of all continuous H,-valued functions C(X, H;) Defining A, = H@A; V,.:H @(H, OH) > HO HA;,
(=I, @®U,, and (0(2))(x) = V,B,(3)Vÿ, we shall see that
X 3x +> (u(a))(x) € LUM)
has the desired properties Indeed, since f(a) maps C(Y, 4) @ TI into itself, it fol- lows that (a) maps C(X, H)) into itself which entails the strong continuity of X ex >
> (/2))(x) Also the dilation property of y is quite obvious Q.E.D
3.2 THEOREM Suppose A is nuclear and X finite-dimensional Then Ext(X, A) is a group
Proof The proof is the same as that outlined in ([3]), only one must use Lemma 3.1 instead of the Stinespring dilation theorem
Indeed, let
t: A > Cy(X, L(A))/C,(X K(H))
define a homogenous X-extension by 4 By the Choi-Effros theorem there is a com- pletely positive map
W: 4> C¿(X, L(H)) such that po¥ = +t Using Lemma 3.1 for W we get
H: A —> Caf X; L(A,)), A, > H,
dilating Y Let ® denote the completely positive map ®:A—> C,,, (X, L(A, © A))
which is the compression of ¡ to H, G H Then [(pc 6) ®t], where rạ is any trivial homogeneous X-extension by A, will be an inverse for [t] Q.E.D
Since Ext(X, A) is a group, it is time to mention that keeping X fixed we get a contravariant functor from the category of separable nuclear C*-algebras with unit, the morphisms being the unit-preserving *-homomorphisms to the category of abelian groups This depends in fact on Thm 2.10 For ¢: A > B a unit-pre- serving +#-homomorphism, É,: Fxt(X, B) + Ext(X, 4) is defined by
Cult] = [Œ s Ơ © to]
where rạ is any trivial homogeneous X-extension by A
Trang 21“HOMOGENEOUS C*t-EXTENSIONS OF C(X) ® X(H) I 75 define a homogeneous X-extension by A Then there is tị: 4 > C„(X, L(H))ICUX, K(H)) such that [r] = [tn]| and r4) =< C,(X, L(H))JG(éX, K(H)) c Cx(X, L(H))/C,(X, K(H)) Proof Consider a completely positive lifting ®: A > C,,(X, L(A)) for t and consider also WA Cy(X, L(H’))
a completely positive lifting for some inverse of [t] so that there is a unital*-homo- morphism p: A > C,,(X, L(H @ A’)) such that
p(a) — (a) đ Ơ(a)e CX, K(H â H’))
for every ae A Let P and P’ be the projections of H @ H’ onto H and respec- tively H’ Consider ð: A — C„(X, L(H @ (H @ H') @ (H @ H’) @ )) defined by H(a) = #(a) @ pla) @ p(4) ®@ By Thm 2.10 we have [t] =[po 9] Consider also p: A > Cy(X, L(H ® H) @ (H @ H) @ )) defined by p= op @p@ , and let GeC„.(X,L(H @ H)@(H@ H)@ ,.,H@(H@H)@(H@ H)@ ) be the constant unitary operator such that
(G())(Œh; ® đ) © (hy @ hz) @ ô ) = hy đ (Ae â hy) đ (hg â hz) ©
The map
Trang 2276 M PIMSNER, S POPA and D VOICULESCU defined by (Œi(2))(+) = (4)\(+) — (P((3))œ) P° + P()Xx) P) @ ® (P(p(a)) (x)P’ + P'(p@)@)P) @ + + G*(x)[0 ® (P(eP@)x)P' + P'(p(@)O)P) ® ® @)X+)P'° + P(p(4)X+x)P @ ] 02) is such that [poy] = [t] Indeed, ((@)(x)— G*(>) (®(2)\œ)G() = = (Ø)\(+)— [Œ@(@)Xx) Pˆ+ P'(o())G)P) @ ] + + G*@)I0 @ (P(p(@))(x)P’ + P'(p(@))(x)P) @ « ] Gx) — — G*(x) [(P@)(x) ® (P(o(@))P + P'(p(@)))P’) @ ] 0(x) — — G*(x) [0 ® (P(p(@)(x)P’ + P'(p(@))(@)P) @ .] Gx) = = (P(p(a))(x)P + P'(3)\(x)P') © (P(p(@))P + P'(p@y@)P') ® — — B(x) + P'(p@)(X)P') © (P(p(@)()P + P'(p(a))(X)P’ @ « ) = = (PW(4)(x)PT— #(x)) @ 0@0 , so that clearly n(a)—G*#(a)G e C,(X, KH @ H) @ (H @ H) @ )) Consider a *-monomorphism
po: A> Cy(X, LH)
which is constant (po(a) is constant for each ae A) and such that
po A) N C,(X, KC) = 0
Clearly, [p> po] = 0 and p)(A) < C,(X, L(A)) By Thm 2.10, there is a unitary UeC,(X, L(A ® H', H)) such that
Trang 23HOMOGENEOUS C*-EXTENSIONS OF C(X) @ K(H) 1 77 To prove the theorem it will be sufficient to show that Un(a)U* ¢ C,(X, L(H @ H ® )) We have U(x\(n(a)) (X)U*(x) = U(x) ((@\a) U*G) — — ŨG) [(p(4))(x)(P) + P(p(a)\(X)P) @ ] U*(x) + + U(x)G*[0 © (P(p@)(x)P’ + P'(p(a)\(x)P) @ .] G U*(x) = = U(x)(p(a)(x)U*(x) © UX p@)X)U*(X) @ — —[UGXP(p(@))Œœ) P' + P(p()\X+) P)U*(x) @ ]+- + ỨG)G*[0 @ (P(p(4)\(x)P' -+ P((4))(x)P ® ]G Ú*Q@ Since Up(a)U* € C,(X, L(H)), it is clear that the first term is a norm-continuous function of x Also, p(a) — O(a) © (4) e C„(X, K(H @ H’)) implies that Pp(a)P' + P'p(a)P € C,(X, K(H ® H’)) Since U is »-strongly continuous it follows also U(Pp(a)P’ + P'p(a)P)U* € C,(X, K(H))
Trang 2478 M PIMSNER, S POPA and D VOICULESCU
hence
Ux)G*[0 @ (P(p(3))@)P' + P'(p(@)(x)P)® .] GU*(x) =
= (U(x)P'(p(a))(X)PU*(x) @ Ux)P'(p(a))@)PU*(x) @ .) 0 S* + + (U(X) P(p(@)Q)P'U*(x) ® U(X) P(p(a))(x)P'U*(x) @ ) s S
where Se L(H ® H @ ) is the shft Sứa @ h; @ ) =0@/;@®⁄z@ Since we have seen that U(x)P’(p(a)Xx)PU*(x) and U(x)P(p(a))(x)P’U*(xp
are norm-continuous functions of x eX, this ends the proof Q.E.D
Note that for t defining a homogeneous X-extension by A each p, ot defines an extension of K(H) by A and denoting by Ext(A) the Brown-Douglas-Fillmore: Ext for A, the preceding theorem implies the following corollary:
3.4 COROLLARY Suppose X is finite-dimensional, A nuclear and
t: A> C,(X, L(A))/ CX, K(H)) defines a homogeneous X-extension by A Then the map X 3x +> [p,,c t]€ Ext(A) is continuous
For what follows we shall also define Ext(X, x9; A) where (X, xo) is a pointed compact metrizable space, as the set of those [t] € Ext(X, A) for which [p,, ° tT} =0 Clearly this is a semigroup and, if X is finite-dimensional and A nuclear, it is a group
§ 4
This section is devoted to the proof of the following theorem
4.1 THEOREM Let A be a nuclear C*-algebra with unit, JA a proper (1 ¿J} closed two-sided ideal and (X, Xo) a pointed finite-dimensional metrizable compact space Consider J=J+C-1 a and i: J— A, q: A> AfJ the canonical *-homomor- phisms Then the sequence
Ext (X, x9; A/J) > Ext (X, x9; A) + Ext (X, Xo; J) is exact
The proof is quite long and will be carried out through a sequence of lemmas
First some remarks
Since A is nuclear, A/J and J are nuclear [48] so that the considered Ext’s are groups
Remark also that the non-pointed version of Thm 4.1, trivially implied by
Trang 25HOMOGENEOUS C*-EXTENSIONS OF C(X)® K(H) 1 79 Assuming the non-pointed version of Thm 4.1 holds, there is [o] Ext (X, 4/2) such that q,[o] = [zc] But then
_ [ø]— (B* o o*) [5] € Ext (X, xo; 4/J) and g„(ø] — (ÿ* s #*)[ø]) = [t] — 9,((B* © #*) [ø) = = [r]Ì— (B* s #*) q;Ïø] = = [r]— (Ø* sa") tr] = [x] which is the desired result Thus let 1:A= C„(X, L(H))JC\X, K(H))
be a *-monomorphism defining a homogeneous X-extension by 4, such that /„[t] = = Q All we must prove is the existence of [o] e Ext (X, A/J) such that q,,{o] = [rt], and this will be achieved in the remaining part of this section
Since A is nuclear there is a unital completely positive map Y: A> C,,,(X, L(A)) such that po ¥ = ¢ ((16]) Moreover, since i is injective, i,,[t] = [t°/] so that using Thm 2.10 and replacing [t] by some equivalent homogeneous X-extension we may assume there is a constant *-homomorphism implementing the triviality of [ro /], i.e there is aconstant «-monomorphism
0o: J — C„(X, L(H))
such that
po(a) — Y(a)e C,(X, K(A)) for all aeJc A
Consider also p: A > C,,,(X, L(H)) the constant, possibly non-unital, *-homo- morphism generated by (p,|J) with the same null-space as (9 |J) ({21], 2.10.3)
Let
O0<m <wm< , uj] < 1, be an approximate unit of J such that
Hy 11, = THƠ, (7eN)
Consider EÈ; e C„.(XY, L(H)) the constant element which is the spectral projection of p(u;) for the set {1} Since p(u;.,) p(u;) = p(u;) we infer E,,, p(u;) = p(u;) Also clearly p(uj)E; = E; and E; < p(uj;) <S Ej¿
Trang 2680 M PIMSNER, S POPA and D VOICULESCU we may replace {u;};e~ by some subsequence so that
S
@Œ) lp(a) — ø(œ)Ej| <2? for 1 <k<j (2) I\(Œ— E;+) øb2)È,j|<2? for L<k <j
Also it is clear that if E is the strong limit of the constant projections E;, then (I— E) is the orthogonal projection onto the null-space of p, in particular
(I—E)p(A) = 9
Let also
P; = E; — Ej-1; (Ey = 0),
and consider
Y, = p(b¿) — x (— E;+1)0 (by) Đ, — ` P,p/)— Eis 1):
Trang 27HOMOGENEOUS C*-EXTENSIONS OF C(X) @ K(H) 1 St 4.3 Lemma There are constant finite rank projections Q; < P; such that (f@j) — p(b))(E —3, Qj) e CŒ, K(H) J>1 for all keN Proof Since (fŒ,) — p(b))P; = (f(bj) — p(b,))0(w)P; = = [f(J(p)— W@)) + (f(0j)W()— ¥ xu) + + (f(uu) — p(buu))]P; € C,(X, KH), there are finite rank constant projections Q; < P, such that I(f() — p())(P;— Ø)|\ < 27 for 1< k <j It follows that the series YP) — py; — 9) is norm-convergent to (Pb) — pb IME —Y¥ Ø) and so (P(x) — pi) (E — x Q;)€C,(X, KH) Q.E.D J
Trang 2882 M PIMSNER, S POPA and D VOICULESCU
which are constant elements of C,,,(X, L(H)) Then O< B< O' < O” < E, and (f— QO") ¥,0’ € C,(X, K(A)) for all ke N Also clearly Q, Q’, QO” are projections
4.4 Lemma [Y,, Ble C, (X, K(A)) for all ke N Proof Consider
S bj Ri; — Trị (Ro,; = 0)
Then the S, ; form a family of pairwise orthogonal selfadjoint constant finite- rank projections Also B can be written as: B= s (SE *} j=1 Ví=1 J Note that Sar¥iSij = SsP¥iP)Si.jp so that S,1¥,S;,; = 0 whenever |t—jj| > 2 Also, if max (i +j,s+t)2k+2 and ji—s|>2 then S¡¡ Y,5Š,„ = 0 Indeed, since (S,-¥.S;,))* = SN
it will be sufficient to prove this only in case i—s 22 Now, if i+j<s+4, then t—j > 2 and the assertion follows from the preceding discussion Thus we are left with the case when i—s 22, i+j>s+t and |t—j| <1 But then Œ—2)+r+2>i-Lj—1>k and hence Si iViSse = Si ViRi-e Sst = = S; ARi-y, 141 + Ria + On đan = 0 Moreover, Y,S;,; = VR ii Si =
Trang 29HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H).I 83
Trang 3084 M PIMSNER, S POPA and D VOICULESCU
we have
J=K+5 =1 i=l—a
œ i J+8
C= ¥ (x %uT„— ¥ risTus) + Dap
where D, , is a finite sum of constant finite-rank elements and hence clearly Dag E € C,(XK(A)) Now remarking that T;,; = 0 unless i > 1—« and i > 1, it follows that so 1—2 Cup — Dap = ` ( ` G„—z)7/,] + i=max(t,1—«) co 7 j+8
Tài ( 3 Suu— 3 n7) | i=j-1
The first sum defines an element of C,(XK(H)) since (s;, j—Fị,j)T:,; are constant finite-rank, (i,j) # Œm, n) = TỶ,T„„ = T,,T⁄„=0 and J;;—r;;)7;;| > 0 whenever 1 <i <j and i+ j— +00 The same kind of argument shows also that
the second sum is in C,(X(K(H)) Q.E.D
We introduce now the following notations:
O=(—F)+Q Ø=(q—E)+0, Õ'=q—E)+0Œ,,
B=(q—E)+B
The properties of these elements are summarized in the following lemma
4.5 LEMMA We have:
(i) p(J)O" € C,(X, KU);
Gi) I—E< Ỗ << Ộ < Q” and O, 0, Q” are selfadjoint projections;
(iii) (I— 0") ¥(A)O' € C(X(K(A));
(iv) (P@— p(@)U— O)e CX, K(H)) for every ae A; (») [p(A), Bl < C,(X, K(A));
(vi) [¥(A), B] < C,(X; K(A))
Proof (i) By the first part of Lemma 4.2, we have p(J)Q” € C,(X, K(H)) Moreover, p(J)({— E) =0, which makes our assertion obvious
(ii) follows immediately from the fact that Q, Q’, Q’’ are selfadjoint projec- tions and
Trang 31HOMOGENEOUS C*?-EXTENSIONS OF C(X) ® K(H).I &5
(iv) is a transcription of Lemma 4.3, since
Œ—ÐØ=1——E)—0=E~—Q
(iii) By the second part of Lemma 4.2 we have (p(b,) — Y,)Q’ € C,(XK(A))
Also we know that (I— Q”)Y,0’ € C,(X, K(H)) so that (I—O")p(b,)Q’ € C,(X,K(H)) Since (I— E)p(A) = p(A\I—E)=0, we infer I— Ơ')j(b)Ỡ e CW, K(HM) and since {b,},en is total in A it follows that ((—O")p(A)O' c C,(X, K(A)) Since 0<I—O” <I—O, it follows by (iv) that U — O"\(p(a) — W()) s
€ C,(X, K(A)) for all ae A Hence we have
(I— 0") ¥(A)O" & C,(X, K(A)) (v) We have [o(b,), BI = [p(b,), B] = [p(b,) — Y;, B) + [Yp B] = => [o(y) — Ÿụ, @W?Ø'}] + [T; B] so [p(b,), Ble C,(X, K(H)) by the second part of Lemma 4.2 and by Lemma 4.4 (vi) We have
[¥(@), 5] = [(@) — p(a), B) + [p(4), B] = [o(@ — ¥@), I— B) + [p(a), đ],
where ae A Since I— B= (J— QO) (I— B) (— 0), assertion (vi) follows from (iv) and (v) Q E D
Using now the Choi-Effros theorem, there is a unital completely positive map @: 4/7 > A such that go @ = id,,,; Consider also the completely positive map
® =feo: AjJ > Cy (X LUM)
Since (q(a)) —aeJ for every ae A, using Lemma 4.5 @) we have
(*) B(q(a)O" — ¥(a)Q" € C,(X, K(H))
(recall also that ø(z) — W(a) e C,(X, K(H)) when ae)
Trang 3286 M PIMSNER, S POPA and D VOICULESCU
Consider By, Oy, O', ớt c€C¿,(X, L(H))) the constant elements deđned as follows: B, = WBW*, Õ,= WÕW*, ,= WÕNW', 0", = WO"'W* = WWw* Note that O,, 0", O"7 are projections, 0, < B, < 0, < 0", and that: (#8) BW = WB, W*B, = BW* 4.6 LEMMA We have (I— Øw(AJ)Ơt c C,(X, K(H,)), [u(A/J), 8] c C,(X, K(H))) Proof We have: (Œ— 0) u(4(2)) 1)* (I — 01) n(g(@)) 07) = = ÕII(4(2*2)) Ơi — ÕiH(4(a*)) Ị:'(4(2)) Of = = WƯ %(4(2*a)) Ị'W*— WO 0(4(2*)) Ð '®(4(2)) O'W*
Since đ(q(a)) 0" Ơ(a) O” € C,(X, K(H)) and Ở' < Ở”, we infer that
(Œ— Ø7) w4(2) O1)* (I — G4’) (42) Ởi) — WO'K(a*a) O'W* +
+ WO'V(a*) O"'¥(a) O'W* € C,(X, K(H,))
But W(a*a) — W(a*)W(a) € C,(X, K(H)), and so
WÕ W(a*a) Ị'W*— WÕ W(a*) O''P(a) O'W* =
Trang 33HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H) I 87
Since (J — Qy') u(q(a)) O; € C,(X, K(H,)), we get u(4(4)) Ơi — WW(a) W*Õ; =
= — ÕI') u(4)) Ơi + Ốt'#(4(2)) Ơi — W#(a) W*O; =
= (I— 61) u(4(2)) Õ( WÕ '®(4(2)) Ị'W*— WÕ'W(a) ð W*S—
— W(I— 0") Va) Ÿ W* e C,(X, K(H,))
by (+) and Lemma 4.5 (iii)
Since B,= Õ,B)Ội, 1t follows that
[u(q(a)), By] —[W¥(a) W*, Ble C,(X, K(A))),
and using (**)} the second assertion follows from Lemma 4.5.(vi) Q.E.D Let GŒc C„.(X, L(H, H,)) be the constant element G = B)”W(q— B)`” In view of (*#) we get G = BY? — 5)'2W = WBq— B)'2 = (I — B,)'? we”, so that B G* Q = € C„(X, LA ® H,)) G I—B, is a constant selfadjoint projection Note also that Ø ®0„, < 9 < QO" Oly, 4.7 LEMMA We have () (¥ @ nog) (A), Q) < CX, KA ® A,)); Gi) (Ơ @ueg@(r đ ko 4) (4)) (— @) e C,(X, K(H @ H,)) for all ắ A; (Œ) @Œ @nse4)()9 c G(X, K(H @ H)))
Proof Since I —Q < (I— Õ) @I Hy assertion (i) follows from Lemma 4.5.(iv) Also since Q < QO” @® 1, assertion (iii) follows from Lemma 4.5.(i) and the fact that
Trang 3488 M PIMSNER, S POPA and D VOICULESCU
In view of Lemma 4.6 and of Lemma 4.5.(vi), in order to prove assertion (i) it will be sufficient to show that
G¥(a) — (ug) (a) Ge C,(X, K(H, H,)) for all ae A
But
GY¥(a) — (ue gq) @) G = WB'"?— B)'? ¥(a) — (uo g) @ WB? I — B)'? = = WB? I — By, W(a)]+ W#(a)B'^(I— ð)'?— (u s 4) (ø) WẼ (-— BY”,
so that in view of Lemma 4.5.0i) and Q'ð'?= B!2 ịt wil be sufficient to prove that
W¥(a) ð — (u s 4) (3) WỠ' e C,(X, K(H, H,))
But this can be seen as follows:
W¥(a) O' — (n° 9) @ O' = WUI— Õ'') (@) ð — (I— Õi) (ws 4) (4) WỠ +
+ WØ"W(4) Ø — i0 s g) (@) WO! =
= Wq— Ø') Ÿ@) Ở — (I— Õi) (ws 4) (4) Ø1W +
+ WQ''(f(a) — #(4(a))) Ở' e C,(X, K(H, H,))
by Lemma 4.5.(1), Lemma 4.6, and (+) Q.E.D
The next lemma will be the final point in the proof of Thm 4.1 4.8 Lemma There is [o]e Ext (X, A/J) such that
4;Ìø] = Ïr]
Trang 35HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H) I 89 the unital completely positive maps defined by:
(xa(2)) %) = Qo(¥(@) @ (uo g) @) (%) | Qo)
(x2(@)) (x) = (— 20) (P(@) ® (ue g) (@)) (x)| W— 20) (H,) (9:(@)) (x) = Do — F— 20) (2(4)) (| (Đẹ — (— 9) (H,) (82(2)) (x) = (— Qo) (B(@)) (x)| Œ — Qo) He)
By Lemma 4.7.() it follows that poy, and po x, are +-homomorphisms and by Lemma 4.7 (ii) p 0 8, and hence also po @, are also *-homomorphisms
Moreover by Lemma 4.7 (iii), (p ° 1) J) = 0 Since
Dy — I— Qo) < Do ~1 + Qo’ ® In, = Qo’ @ 9,
it follows by Lemma 4.2 that
PJ) (Dp — FZ — 2o)) € CX, K(Do(A2)))
and hence
(p s9,) (7) = 0
Let y, 64, d2, Yi, Yo be the homogeneous X-extensions by A determined by P°P, P° Xt, P? Xe, P24, Po Os, i.e the homogeneous X-extensions by A obtained by adding to each of the above «-homomorphisms a trivial homogeneous X-exten- sion by A
We have then in Ext (X, A):
Trang 3690 M PIMSNER, S POPA and D VOICULESCU § 5
This section deals with the homotopy invariance properties of Ext (X, A) both in the X and in the A-“variabile’’ In fact these two homotopy-invariance properties are related and their proof reduces in the case of quasidiagonal C*- algebras to an adaption of the argument of N Salinas ((42]) for the usual Ext-groups The short exact sequence for Ext (X, A) in § 4 enables us to improve the result of Salinas: A may be any C*-algebra having a composition series with quasidiagonal quotients In particular, A may be any GCR-C*-algebra
First we need a few facts about quasidiagonality in C,,,(X, L(A)), but since this seems a rather awkward intermediate degree of generality, we prefer to digress a bit, considering a more general situation
Let L be a unital C*-algebra (not necessarily separable), K < L a closed two- sided ideal and p: L > L/K the canonical homomorphism (These notations will not cause any confusion since in our applications L = C,,,(X, L(A)) and K = = C,(X, K(A))) We will make the following assumption about K:
there is an increasing sequence P; <P, < of selfadjoint projections in K,
which is an approximate unit of K
The set P(K) of selfadjoint projections of K is not filtering in general, but has a weaker property For e > 0 and P, Q € P(K) we shall write
P<Q iff \|P—QP| <e
Then our special assumption on K implies that for any Q,, Q,¢ P(K) and ¢ > 0 we can find Q, € P(K) such that
Q, < Qs, Q; < Qs
For a bounded function ƒ: P(X) > R we define lim inf f(P)
PéP(K)
as the greatest lower bound of those re R, such that for every Pe P(K) and ¢ > 0 there is Q € P(K) such that /(Q) <r and P < Q Also we define
lim sup f(P) = — lim inf (—/f(P))
PeP(K) PEP(K)
For a finite set Y < L the modulus of quasitriangularity q(Z) is defined as q(Z) = lim inf (max ||(I — P) aP}))
PEP({K) a€s
and the modulus of quasidiagonality qd(Z) is defined as
Trang 37HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H) I 91 Remark that gd(z) = lim int inf (max ||[P, all) de» also since for ke K we have lim sup Jl(— P)kP| =0 PeP(&
we easily infer that
Iq((ai-1) — 4(/3ï~¡)| < max ||p(a; l<ign — a) ||, | qd((a,)?-1) — 43(;)ï-1)| < max |jp(4¡ — aj)| l<i<"n
5.1 Lemma Let {Q;};en © P(K) be such that Q; < Qj4,, then there are
1074
{O?};enc(P(K) such that
Ĩ; <S Ø7 (€N), and lim ||J@;— Øj| = 0 jco
Proof Consider first two projections P,Qe¢P(K), P< QO, ¢< 1/2 Then {1—e) P < POP < P so that we have the polar decomposition QP = wa where a= (POP)? and w= OP(7— P)+ POP), Then w*w = P and ww* e P(K), ww* <Q, Also ||w — P|| < 3e and hence ||ww* — P|| < 6e Denoting by E(P, Q) the projection ww*, we thus have
E(P, Q) < Q and |EỨ, Ø) — P|| < 6e
Using this we define recurrently {Q,; ;}1<;<; ¢ P(K) so that
0;,; = OF and 0;,; = E(Q;,j-15 Oi+1,)) (dl <i<j—1)
Clearly then Q; ; < Q,41,; and it is easily seen that
l@:;— Ø.,„;.ill < <6/11-1 10- i
It follows that
® lØi;— Qis+all < YH 6/10) = (5/2) 6/57
j=l j=l
Hence for j > + 00, Q;; converges to some projection Q; Clearly Q; < Qi41 and lim ||Q; — Q; || = 0 Q.E.D
5.2 LEMMA Consider a subset Q.< L, such that p(Q) is separable Then the following assertions are equivalent:
(i) For every finite subset Y < Q we have qd(x) = 0;
(ii) There is an approximate unit {Q;\;en ¢ P(K) such that Q; < Qj+1;
(eN), and lim |[O,, a] | = 0 for all aeQ + K
Trang 3892 M PIMSNER, S POPA and D VOICULESCU Proof That (ii) => (@ is immediate
For the converse it is clear, assuming (i), that for {a,},en © @ a sequence with {p(a,)},en dense in p(Q), there is a sequence {Qj};cen © P(K) which is an approximate unit of K, such that lim || [Qj,a,]||=0, for all keN, and
jroo
Q;, < Qi41 Then (i) follows using Lemma 5.1 and the fact that lim | [Q;, 5] || =0
10-* jroo
for every be K Q.E.D
A subset Q < L, with p(Q) separable will be called almost diagonal if it satisfies the equivalent conditions of Lemma 5.2
5.3 DEFINITION A homogeneous X-extension by A, defined by t:A—> > C,,(X, LGA))/C,X, K(A)) is called quasidiagonal, if p"\(<(A)) is almost diagonal with respect to the ideal C,(X, K(H))
It is easily seen that if t,,7, define equivalent homogeneous X-extensions by A, then 1, is quasidiagonal if and only if t, is Thus we can speak about quasidiagonal elements of Ext (X, A)
Also, as for the usual extensions by K(A), it is obvious that the quasidiagonal elements of Ext (X¥, A) form a semigroup
Recall from ({42]) that a unital separable C*-algebra A is called quasidiagonal if there is a *-monomorphism p: A > L(H), p(A) 1 K(H) = 0 such that p(A) is almost diagonal with respect to the ideal K(H) (i.e., in the usual sense)
In view of Thm 2.10, if A is quasidiagonal and X đnite-dimensional, then any trivial homogeneous X-extension by A is quasidiagonal Moreover it is also clear that the existence of a trivial homogeneous X-extension by A which is quasi- diagonal, insures the quasidiagonality of A
5.4 PROPOSITION Let A be a nuclear quasidiagonal C*-algebra and X a finite-dimensional metrizable compact space Let [t]e Ext (X x [0,1], A) be such
that i*([c]) = 0, where
ig: X x {0} + X x [0, 1]
Trang 39HOMOGENEOUS C*-EXTENSIONS OF C(X) ® K(H).I 93
Fix @,, ,@,€A and e>0 Since o(a;)eC,(X x [0, 1], L(H)), there is
a natural number 7 such that :
I(p(a)) & 4) —(9@)) tO <e (<i<m), whenever |f— ?'| < 2/n Using Thm, 3.2, Thm 3.3 and the Choi-Effros theorem, there is a completely positive map ni A> CX x [0, 1], L(H)) such that poy defines a homogeneous (X x [0, 1])-extension by A and [p se (@ Gn] =0 The completely positive map 6: A5C, (xx u2 (e8 mg) n-times pefined by k
(0@) % N=) &).® â (OC) (= +] đ (nla) (» *)}
determines an extension, and
[pO] = tr] +- 3) (Êu»/#„) 6] + [ø s n) = [ah
Trang 4094 M PIMSNER, S POPA and D VOICULESCU where 8;=ƒ, U<j<n); 8 =fixn O<jF <k—2); Sua = (k —ntyPf, + (nt —k + 1p? f,; & = —(t—k + 1)!2f, + (k— n1? fn; 8,=—ƒ, (kK+1<j<n)
It is easy to see that U, depends continuously on te[0, 1] Consider also the unitary vec, (1x 01,4 : n-times ne See) defined by V(x, t) = U, With these definitions it is now easy to see that
|V6(a) V*¥ — Ya) <Se for <<
Since p o y defines a homogeneous (X x [0, 1])-extension by A and [po W] = 0, it follows because A is quasidiagonal that Gd(W(ay), > Wa,)) = 0 and hence q4(6(m) Ø(a„)) < max | VO(a;) V* — W(a;)|| < Se But since [po 8] = [7] = [po], we infer
Gd(p(ay), - «5 P(An)) = Gd(O(Ay); - «5 HAm)) < Se
Hence since ¢ > 0 was arbitrary we must have
gd(p(a,), 43 ọ(2„)) = 0,