THE CORONA FACTORIZATION PROPERTY AND APPROXIMATE UNITARY EQUIVALENCE

Transcription

1 Houston Journal of Mathematics c 2006 University of Houston Volume 32, No. 2, 2006 THE CORONA FACTORIZATION PROPERTY AND APPROXIMATE UNITARY EQUIVALENCE DAN KUCEROVSKY AND P.W. NG Communicated by Gilles Pisier Abstract. We study Rørdam s group, KL(A, B), and a corona factorization condition. Our key technical result is a lemma showing that approximate unitary equivalence preserves the purely large property of Elliott and Kucerovsky [10]. Using this, we characterize KL(A, B) as a group of purely large extensions under approximate unitary equivalence, generalizing a theorem of Kasparov s. Then we prove the following: Let B be a stable and separable C -algebra. Then the following are equivalent (for absorption of weakly nuclear extensions): i) The corona algebra of B has a certain quasi-invertibility property, which we here call the corona factorization property. ii) Rørdam s group KL 1 nuc(a, B) is isomorphic to the group of full essential extensions of A by B. iii) Every strongly full and positive element of the corona algebra of B is properly infinite. iv) Every norm-full extension of B is absorbing, with respect to approximate unitary equivalence. v) Every norm-full extension of B is absorbing, with respect to ordinary unitary equivalence. vi) Every norm-full extension of B is absorbing, with respect to weak equivalence. vii) Every norm-full trivial extension of B is absorbing, with respect to unitary equivalence. viii) A K-theoretical uniqueness result for maps into M(B)/B. We show that if X is the infinite Cartesian product of spheres, then C(X) K does not have the corona factorization property. We apply our technical lemma to study quasidiagonality and weak quasidiagonality of extensions Mathematics Subject Classification. Primary 46L85; Secondary 47C15, 46L05. Key words and phrases. K-theory, C -algebras, extensions of C -algebras, noncommutative topology. Funded by AARMS and NSERC. 531

2 Introduction: KL and absorbing extensions Early in the history of KK-theory, a generalized bivariant K-theory group for C -algebras, Kasparov showed that the group KK(A, B) could be regarded as a group of absorbing extensions under unitary equivalence, where an absorbing extension is defined to be one that is equivalent to its own sum with an arbitrary trivial extension. The exact statement is: Theorem 1.1 ([12, lemma 1.1, p.560]). If A and B are separable C -algebras, with at least one of them being nuclear, then the group KK 1 (A, B) is given by unitary equivalence classes of absorbing extensions. The unitary equivalence in the theorem is taken to be with respect to unitaries coming from the multipliers of the stabilization of the algebra B. It follows from this foundational result that two absorbing extensions are unitarily equivalent if and only if they define the same element of KK 1. But then the following two questions should be asked: i) How may we determine if a given extension is in fact absorbing? ii) How may we decide if two given extensions define the same element of the associated KK 1 -group? The first question was recently answered by work of Elliott and Kucerovsky [10], who gave a simple algebraic criterion for an extension to be absorbing (that is, to be equivalent to its own sum with a trivial extension). Their theorem is reproduced as theorem 3.1 on page 538 below. At first glance, the second question is answered for a very large class of algebras by the Universal Coefficient Theorem (UCT) or the more recent multicoefficent UCT of Dadarlat and Eilers: Theorem 1.2 ([31, 7]). For algebras in the bootstrap class, the following exact sequences hold: 0 Ext 1 Z(K (A), K (B)) KK(A, B) Hom(K (A), K (B)) 0 and 0 Pext 1 Z(K (A), K (B)) KK(A, B) Hom Λ (K (A), K (B)) 0, where K stands for the union of the mod-p K-theory groups. The above exact sequences are quite important and useful, and in general one would prefer to use the second of the two, since Pext (the set of pure extensions of one group by another) is a smaller set than the set of all group extensions, Ext 1 Z,

3 CORONA FACTORIZATION PROPERTY 533 that appears in the first sequence. The first appearance of Pext in the context of KK-theory is probably Salinas s proof[32, 5.3] that the closure of the equivalence class of the identity in a KK-group with respect to a certain natural topology is Pext 1 Z(K(A), K(B)). However, for certain applications, in particular to classification type problems, we are given an element of Hom(K(A), K(B)) or Hom(K(A), K(B)) rather than a KK-cycle, and would like to find a corresponding element of Hom(A, B). Naturally, one must restrict this problem in some way to have a hope of solving it. Rørdam found that it was useful to simplify the multicoefficient UCT by taking the quotient by the subgroup Pext 1 Z(K(A), K(B)), or equivalently, the quotient of KK 1 (A, B) by the closure of the equivalence class of the identity, and using that group instead of KK 1 (A, B)! Before discussing this group, written KL 1 (A, B), we give a generalization of the main theorem of [10]. Specifically, we use our earlier characterization of absorption to prove an if and only if characterization theorem for absorption with respect to approximate unitary equivalence. We should mention that there is a regrettable ambiguity as to the exact definition of approximate unitary equivalence, since under some definitions, approximate unitary equivalence is actually stronger (finer as an equivalence relation) than unitary equivalence. In this paper we use the definition found in [26, 5], reproduced as i below : Definition. i) Two extensions φ, ψ : A M(B)/B are said to be approximately unitarily equivalent, if there exists a sequence {u n } n=1 of unitaries in the corona algebra of B such that u n φ( )u n converges to ψ( ) pointwise. We pause for a few moments to also define some of the other terminology that we shall use: ii) A -homorphism φ : A M(B)/B is said to be norm-full, if for every nonzero positive element a A, φ(a) is a norm-full element of M(B)/B. (Equivalently, the ideal generated by φ(a) in the corona is everything.) iii) An extension, of B by A, is said to be norm-full if the corresponding Busby invariant τ : A M(B)/B is a norm-full -homorphism. iv) A nonzero, positive element c M(B)/B is said to be strongly full if C (c) does not nontrivially intersect any proper ideal of M(B)/B. We have engaged in a bit of historical revisionism here: Rørdam s definition of the KLgroup actually predates the proof of the multicoefficient UCT, and hence his construction was quite a bit more involved than our description would suggest.

4 534 DAN KUCEROVSKY AND P.W. NG v) A positive element d M(B) is said to be strongly full if its image in the corona algebra is strongly full. vi) An extension is said to be weakly nuclear if it has a completely positive weakly nuclear splitting map s : A M(B). vii) A map s : A M(B) is said to be weakly nuclear if it becomes nuclear after being cut-down by arbitrary elements of B: viz. a b s(a)b. viii) An extension is said to be nuclearly absorbing if it is equivalent to its own sum with an arbitrary weakly nuclear trivial extension. In particular, if an extension is weakly nuclear, then it is semisplit, and hence is a so-called invertible extension. All extensions in this paper are assumed weakly nuclear, which of course is equivalent to invertibility if one of A or B is nuclear. Furthermore, every extension that is absorbing in the usual sense is necessarily norm-full. As defined here, norm-fullness is a strong condition. For example, a nonzero, positive element in the image of a norm-full extension is strongly full. We define KK nuc (A, B) following Skandalis [33], replacing Kasparov s assumption of nuclearity of A or B by weak nuclearity of the extensions(see also [10]). We should note that if we want a group, the extensions are conventionally assumed to be unital or nonunital together, since if we try to mix unital and nonunital extensions, the sum of a unital and nonunital extension is always nonunital and we are then not able to find inverses within the group. Returning to our argument, the key step is certainly the following interesting lemma: Lemma 1.3. Let φ, ψ : A M(B)/B be two approximately unitarily equivalent weakly nuclear extensions of separable algebras. If one of the extensions is absorbing, so is the other. The short proof is deferred to page 538. From the lemma, we now obtain what is probably our most fundamental result: Corollary 1.4. Let A and B be separable C -algebras, with one nuclear, and let B be stable. Then, the following statements are equivalent: i) τ is absorbing with respect to approximate unitary equivalence, ii) τ is absorbing with respect to weak equivalence of extensions, iii) τ is absorbing with respect to unitary equivalence (by elements of the multipliers), iv) τ is purely large in the sense of Elliott and Kucerovsky [10]. Remark 1.5. Weak equivalence of extensions is by definition equality of Busby maps under conjugation by unitaries from the corona.

5 CORONA FACTORIZATION PROPERTY 535 Proof. Clearly, iii implies ii, and ii implies i. The equivalence of iii and iv is already known (cf. theorem 3.1), so we need only to show that i implies iv. But, given an extension τ that is approximately unitarily equivalent to its BDF sum with an arbitrary trivial extension, we can choose a trivial extension κ that is purely large in the sense of theorem 3.1 for example, the original absorbing trivial extension due to Kasparov [12]. Recall that [10, lemma 13] the BDF sum of two extensions is purely large if one of the extensions is purely large: hence τ is approximately unitarily equivalent to a purely large extension. But lemma 1.3 implies that τ is therefore purely large. Proposition 1.6. Let A and B be separable C -algebras. The closure of the identity in the group KK 1 nuc(a, B), represented as a group of weakly nuclear absorbing extensions, is the set of extensions that are approximately unitarily equivalent to Kasparov s absorbing extension. Proof. Let us suppose that some given extension τ is approximately unitarily equivalent to Kasparov s absorbing trivial extension κ. Rørdam showed [26, prop. 5.4] that approximately unitarily equivalent extensions define the same element of KL, and hence τ is trivial in KL. Thus, the absorbing extension τ is in the image of Pext 1 Z(K(A), K(B)) under the multicoefficient UCT sequence. But Salinas [32] (followed by refinements due to Schochet [35]) showed that the component of the identity in KK 1 (A, B), with respect to what is now called the Salinas topology, is in fact exactly this image. Thus, there is a sequence τ n of not necessarily absorbing trivial extensions converging to τ. Since the Salinas topology is compatible with addition of extensions, we can add some trivial absorbing extension κ to the τ n, obtaining a sequence of absorbing trivial extensions converging to τ + κ. Hence τ + κ is in the closure of the identity, but of course, by lemma 1.3, the extension τ is absorbing so that τ + κ is unitarily equivalent to τ. For the easier converse direction, if τ is in the closure of the identity [1] KK 1 (A, B), then there is a sequence of trivial extensions τ n going to τ in the Salinas topology. The definition of the Salinas topology is that it is given by the Hausdorff metric d(τ, τ ) := 2 i (τ τ )(a i ) / a i where a i is some arbitrary dense countable set in A. (Different dense countable sets will give equivalent topologies.) But, since this topology is exactly the pointwise convergence topology on Busby maps, and since τ n = u n κu n, we see that τ is approximately unitarily equivalent to κ. Since the group operation in KK 1 (A, B) is compatible with the closure operation, we have a very interesting corollary:

6 536 DAN KUCEROVSKY AND P.W. NG Corollary 1.7. If A or B is nuclear, then the group KL 1 (A, B) is the group of absorbing extensions under approximate unitary equivalence. This corollary could have been obtained more directly from theorem 1.4, but it is of interest for our study of quasidiagonality in the last section to have the result on closures of the identity. We can define KL nuc in much the same way as Skandalis s KK nuc, replacing nuclearity of A or B by weak nuclearity of the extensions [33, 10]. For the sake of clarity in exposition, we shall not overemphasize the distinction between KK nuc and KK, (or KL nuc and KL); however, we mention that the expected result is trivially true: Corollary 1.8. The group KKnuc(A, 1 B) is the group of absorbing extensions under unitary equivalence, and the group KL 1 nuc(a, B) is the group of absorbing extensions under approximate unitary equivalence. We note that since KL is defined entirely in terms of KK-groups and K- groups, it follows from the well-known equivalence of M(B)/B and B (up to a dimension shift) that KL(A, M(B)/B) = KL 1 (A, B). One should in fact take the viewpoint that KL (A, M(B)/B) is defined to be KL 1 (A, B), since it is preferable to avoid the highly nonseparable corona algebras for the purposes of KK-theory. The usefulness of our results clearly depends on having simple ways to identify or construct absorbing extensions. We shall find a simple necessary and sufficient condition for all full extensions of a given algebra to be absorbing. With this in mind, we state the following corollary of our corollary 1.4: Corollary 1.9. Let A and B be separable C -algebras, and let B be stable. Then, the following statements about weakly nuclear extensions are equivalent: i) Every full essential extension of A by B is purely large. ii) KL 1 (A, B) is the group of full essential extensions under approximate unitary equivalence. iii) The set of full essential extensions form a group, denoted KK 1 w(a, B), under unitary equivalence by corona unitaries. iv) KK 1 (A, B) is the group of full essential extensions under unitary equivalence by multiplier unitaries. In general, one does not expect the groups KL 1 (A, B) and KK 1 w(a, B) to have the same functorial properties (split exactness, stability, etc) as KK-theory, since Higson has shown that KK-theory is, under mild technical conditions, characterized by its functorial properties. There are examples (see [8]) to show that

7 CORONA FACTORIZATION PROPERTY 537 KK 1 w(a, B) is not equal to KK 1 (A, B). It is therefore interesting to ask what functorial properties KK 1 w(a, B) may be expected to have. In particular, does it satisfy some counterpart of the UCT? 2. The corona factorization property In this paper we make the assumption that all extensions are extensions of separable C -algebras by separable C -algebras. Definition 2.1. We say that a C -algebra B has the corona factorization property, or that it is an absorbing algebra, if it satisfies one (and hence all) of the following equivalent conditions: Let B be a stable and separable C -algebra. Then the following conditions will later be shown to be equivalent: i) Every norm-full extension of B is nuclearly absorbing. ii) Every norm-full trivial extension of B is nuclearly absorbing. iii) For every separable C -algebra A, Rørdam s group KL 1 nuc(a, B) := KL nuc (A, M(B)/B) is the same as the set of approximate unitary equivalence classes of essential, full, nuclear extensions of B by A. iv) For every separable C -algebra A, the group KK 1 nuc(a, B) is the same as the set of unitary equivalence classes of essential, full, nuclear extensions of B by A. v) For every projection P which is norm-full in M(B) there is an element x M(B) such that xp x = 1 M(B). vi) For every projection Q which is norm-full in M(B)/B there is an element y M(B)/B such that yqy = 1 M(B)/B. vii) For every positive multiplier element c not in B such that C (c/b) does not nontrivially intersect any proper, nontrivial ideal of M(B)/B, there is an element x M(B) such that xcx = 1 M(B). The conditions v, vi and vii were motivated by stronger conditions that were proposed in an earlier preprint [17]. Conditions v and vi are equivalent to P 1 M(B) and Q 1 M(B)/B respectively. We like the above formulation in terms f of quasi-invertibility since it makes clear that the corona factorization property is a generalization of the extremal richness property introduced by Brown and Pedersen [4], now the subject of a growing body of work. The term absorbing algebra had originally referred to algebras satisfying the noncommutative topology condition i above, whereas an algebraic condition

8 538 DAN KUCEROVSKY AND P.W. NG similar to v, vi, or vii was termed the corona factorization property. We now see that in most situations, the two terms can be used interchangeably. Finally, we note that the corona factorization property is related to interesting problems, discussed elsewhere, concerning stability and the structure of separable C -algebras. 3. Proofs and lemmas Elliott and Kucerovsky showed the following equivalence of three conditions: Theorem 3.1 ([10]). Let A and B be separable C -algebras, with B stable and A unital. Let 0 B C A 0 be an essential, unital extension with weakly nuclear splitting map ˆτ : A C. Then the following are equivalent: i) The extension is absorbing, in the nuclear sense. ii) The extension algebra has the purely large property, i.e. the property that, for every c C + that is not contained in B, the hereditary subalgebra that it generates, cbc, contains a stable subalgebra that is full in B. iii) The extension algebra has the following approximation property: For every c C + that is not contained in B, and for every positive b in B, there is an element r B making the norm of b rcr arbitrarily small. Moreover, the element r can be chosen to have norm bounded by one, if b and c/b both have norm equal to one. We can now give the short proof of the key lemma from section 1. Proof of lemma 1.3. Let C 1 be the extension algebra of the absorbing extension, τ 1, and let C 2 be the extension algebra of the other extension, τ 2. Since both extensions are essential, both C 1 and C 2 sit canonically inside M(B) and in particular contain B. Let π denote the usual quotient map. It suffices to establish that C 2 has the approximation property in 3.1 clause iii if C 1 does. Hence, suppose that we are given c 2, a positive element of C 2 that is not in B, b B + and a number ɛ > 0. Let us assume that π(c 2 ) and b both have norm one. The extensions τ 1 and τ 2 are approximately unitarily equivalent, so let (u n ) be a sequence of corona unitaries such that u n τ 2 ( )u n goes pointwise to τ 1. Let (ũ n ) be a sequence of multiplier contractions lifting these unitaries. Choose n and an element m of C 1 such that π(ũ n c 2 ũ n m) has norm less than ɛ/4 and m has norm one in the corona. Thus, there is an element b B such that The fact that the lift ũ can be an exact contraction is from [1, prop. 2.10]. Approximate contractions would do, but then the proof would involve one more step.

9 CORONA FACTORIZATION PROPERTY 539 ũ n c 2 ũ n (m + b) has norm less than ɛ/4. Note that m + b is in the algebra C 1. Since ũ n c 2 ũ n is positive, we can perturb m + b to a positive element c 1 C 1 + that is within ɛ/2 of ũ n c 2 ũ n. Thus, ũ has norm one, c 1 /B has norm one, and ũc 2 ũ is within ɛ/2 of c 1. By the approximation property for C 1, we can find r in B so that r 1, and rc 1 r is within ɛ/2 of b. Hence, rũc 2 ũ r is within ɛ of b. Moreover, the norm of rũ is less than or equal to one. This completes the proof of the results of the first section of our paper. In the next section we need the following result: Proposition 3.2. Let B be a stable σ-unital C -algebra. Let l be a nonzero positive element of M(B). The hereditary subalgebra, lbl, generated by l of the multipliers M(B) is isomorphic to a hereditary subalgebra generated by a multiplier projection P. Moreover, if l is a norm-full element of M(B) then P is also a norm-full element of M(B). Proof. The first part of the proposition is from Brown s theorem [3], since lbl K = B, there is a multiplier projection such that P BP is isomorphic to lbl. The key step in showing that the projection is norm-full if l is, is to notice that L(lB, H B )L(H B, lb) L(B, H B )l 2 L(H B, B), where lb is regarded as a countably generated Hilbert module over B and H B is the standard Hilbert module over B. One could stay in the framework of Hilbert modules and prove the first part using the Kasparov stabilization theorem. The next result is perhaps unexpected: Proposition 3.3. Let B be a stable C -algebra. If d is a norm-full element of M(B)/B, and if c is a positive lifting of d to M(B), then c is norm-full in M(B). Proof. Let x 1, x 2,..., x n be elements of M(B) such that 1 M(B)/B = n i=1 π(x i)dπ(x i ). Then let b be an element of B such that 1 M(B) = n i=1 x ic(x i ) +b. Now since B is stable, let S be a partial isometry in M(B) such that S bs has norm less than ɛ. Then 1 M(B) is within ɛ of n i=1 S x i c(x i ) S. If we chose ɛ to be small enough, then we can find R M(B) such that 1 M(B) = n i=1 R S x i c(x i ) SR. We need one last lemma, which can be derived from our previous work in [10]. The main point of this lemma is that it is possible to absorb trivial extensions even if the Busby map of the trivial extension is not injective. Naturally, the absorbing extension must always have an injective Busby map.

10 540 DAN KUCEROVSKY AND P.W. NG Lemma 3.4. Let A and B be stable, separable C -algebras. If τ : A M(B)/B is a nonunital nuclearly absorbing extension, then τ κ is unitarily equivalent to τ, for every trivial extension κ. In particular, the trivial extension κ is not required to be essential, and can be either unital or nonunital. Proof. In the unital case, this lemma is contained in the remark at the bottom of page 399 of [10], and then the nonunital case can be obtained by a unitization trick. Or one can reason as follows. First, consider the set of triples of isometries generating unital copies of O 3. The natural diagonal action of the multiplier unitaries on this set is transitive, so that BDF addition is associative. Now let φ be some trivial nonunital nuclear essential extension, such as Kasparov s absorbing extension. Then φ+κ is essential and nonunital, and is absorbed by τ. But on the other hand, τ +φ is equivalent to τ. Hence τ +κ = (τ +φ)+κ = τ +(φ+κ) = τ. Recall that all the extensions in this paper are extensions of separable C - algebras by separable C -algebras. Theorem 3.5. Let B be a stable, separable C -algebra. Then the following conditions are equivalent: i) Every norm-full extension of B is nuclearly absorbing. ii) Every norm-full trivial extension of B is nuclearly absorbing. iii) Suppose that A is a separable, nuclear C -algebra. Suppose that φ, ψ : A M(B)/B are two full -monomorphisms. Then [φ] = [ψ] in KL nuc (A, M(B)/B) if and only if φ and ψ are approximately unitarily equivalent. iv) Suppose that A is a separable, nuclear C -algebra. Then KKnuc(A, 1 B) is the same as the set of unitary equivalence classes of essential, full extensions of B by A. v) For every projection P which is norm-full in M(B), there is an element x M(B) such that xp x = 1 M(B). vi) For every projection Q which is norm-full in M(B)/B, there is an element y M(B)/B such that yqy = 1 M(B)/B. vii) For every strongly full and positive multiplier element c, there is an element x M(B) such that xcx = 1 M(B). Proof of theorems 2.1 and 3.5. The equivalence of i, iii, and iv is corollary 1.9. Also, i clearly implies ii. In what follows, let π : M(B) M(B)/B be the natural quotient map. We first show that ii implies v. Suppose that P is a norm-full projection in M(B). Consider the extension, of B by C, given by the Busby map ρ : C

11 CORONA FACTORIZATION PROPERTY 541 M(B)/B : α απ(p ). Clearly, this extension ρ is norm-full, nuclear, essential, and trivial. Therefore, by hypothesis, ρ is absorbing. Let θ : C M(B)/B be the trivial extension, of B by the complex numbers C, given by θ(α) = α1 M(B)/B. Then the BDF sum ρ + θ is unitarily equivalent to ρ by a unitary coming from M(B). Since B is stable, let S 1 and S 2 be isometries which generate a unital copy of O 2 in M(B). Hence, by the definition of a BDF-sum, we have that ρ(1) is unitarily equivalent to π(s 1 )ρ(1)π(s 1 ) + π(s 2 )θ(1)π(s 2 ). By changing S 1 and S 2 if necessary, we may assume that the unitary is the identity. Hence, π(p ) = π(s 1 P S 1 ) + π(s 2 S2). Since S 2 is an isometry in M(B), π(s 2 S2) is a projection which is Murray-von Neumann equivalent to 1 M(B)/B. Hence, since S 1 and S 2 have orthogonal ranges, there is a z M(B) such that π(z)π(p )π(z) = 1 M(B)/B. Hence, let b B be such that zp z + b = 1 M(B). Since B is stable, let S be an isometry in M(B) such that S bs has norm strictly less than ɛ. Hence, S zp z S is within ɛ of 1 M(B). Since we can make ɛ arbitrarily small, there is an x M(B) such that xp x = 1 M(B). We next show that v implies i. By proposition 3.2, for every full positive multiplier element c, there is a projection P in M(B) such that cbc = P BP. Since c is a full element of M(B), we can choose P to be norm-full in M(B). Using the hypothesis, P is equivalent to 1 (see lemma 4.3 in [27]). Hence cbc is stable, and by Theorem 3.1.ii, all full extensions must be nuclearly absorbing. To see that vii implies vi, we need only note that if Q is a projection which is norm-full in M(B)/B, and if c is any positive element of M(B) which is a lift of Q, then c is strongly full. Next, we show that vi implies v. Suppose that vi holds, and that P is a projection which is norm-full in M(B). By hypothesis, there exists an element y in M(B)/B such that yπ(p )y = 1 M(B)/B. Cutting down by an isometry (as in proposition 3.3), we can find an x M(B) such that xp x = 1 M(B). Finally, we show that i implies vii, which will complete the proof. Since c is strongly full, every nonzero, positive element of C (π(c)) is norm-full in M(B)/B. Hence, the natural inclusion map, ι : C (π(c)) M(B)/B, gives a norm-full, essential extension of B by C (π(c)). Since C (π(c)) is a nuclear C -algebra, it follows, from the hypothesis, that this extension ι must be absorbing. Suppose that s is a nonzero point in the spectrum of π(c). The map j : C (π(c)) M(B)/B, given by f f(s)1 M(B)/B, is the Busby map of an extension which is trivial, but not necessarily essential. Using lemma 3.4, we see that this extension is absorbed by the extension ι. Since B is stable, let

12 542 DAN KUCEROVSKY AND P.W. NG S 1 and S 2 be isometries which generate a unital copy of O 2 in M(B). Hence, ι(π(c)) is unitarily equivalent to π(s 1 ) ι(π(c))π(s 1 ) + π(s 2 )j(π(c))π(s 2 ). Note that π(s 2 )j(π(c))π(s 2 ) = π(c)(s)π(s 2 S2), which is a nonzero scalar multiple of π(s 2 S2). As in the proof of ii implies v, we can find an x M(B) such that xcx = 1 M(B). 4. The infinite property In this section, we prove a short but interesting result: that an algebra is absorbing if and only if all strongly full positive elements of the corona are properly infinite. We use to denote Murray von Neumann equivalence of positive elements (which are now not necessarily projections). The definition is that a and b are equivalent in this sense if and only if there is some element x such that a = x x and b = xx. The novelty is that the element x is not assumed to be a partial isometry. Definition. A positive element c is said to be strongly properly infinite if c q+r, where q and r are orthogonal positive elements such that q c and r c. For positive elements, this notion is apparently stronger than the notion of properly infinite which is found in [14]. For projections, however, properly infinite and strongly properly infinite are the same. Theorem 4.1. equivalent: Let B be a stable, separable C -algebra. Then the following are i) The algebra B is an absorbing algebra. ii) Every strongly full positive element of the corona algebra of B is strongly properly infinite. iii) Every full projection in the corona algebra of B is properly infinite. Proof. We show that iii implies i: Suppose that P is a norm-full projection in M(B)/B. Being properly infinite by hypothesis, it follows that the multipliers of the nonunital algebra P BP contain a copy of O, which by proposition 3.2 implies the same is true for the multipliers of cbc for any full positive element c M(B). But then any full extension of B is purely large, hence absorbing (by theorem 3.1 ). That ii implies iii follows since every full projection is automatically strongly full. We now prove that i implies ii. Let c be a strongly full positive element of the corona algebra M(B)/B. Since c is a positive element, let λ be the largest point

13 CORONA FACTORIZATION PROPERTY 543 in the spectrum of c. Let ι : C (c) M(B)/B be the natural inclusion map, and let τ : C (c) M(B)/B be the trivial extension given by τ : f f(λ)1 M(B)/B. Since ι is a norm-full extension, and since C (c) is a nuclear C -algebra, it follows, by our hypothesis, that ι is an absorbing extension. Hence, ι + τ is unitarily equivalent to ι. Now let S 1, S 2 be isometries generating a unital copy of the Cuntz algebra O 2 in M(B). Hence, S 1 cs1 + S 2 cs2 S 1 cs1 + c S 2 S2. But the last expression is unitarily equivalent to c, since ι+τ is unitarily equivalent to ι. We see that we can now add to our list the following equivalent conditions: Theorem 4.2. We can add the following equivalent statements to the lists in theorems 2.1 and 3.5: vii) Every strongly full positive element of the corona algebra of B is strongly properly infinite. viii) Every strongly full positive element, of the corona algebra of B is properly infinite. ix) Every full projection, in the corona algebra of B is properly infinite. One important consequence of being able to reduce the corona factorization property to a statement about projections is that it is now clear that corona factorization is a K-theoretical condition. To be precise, let B be a stable, separable C -algebra, and let V be the monoid of Murray-von Neumann equivalence classes of projections in the corona algebra M(B)/B. Then the following are equivalent: (a) B has the corona factorization property. (b) If r generates V as a monoid, then r t for every t V. (c) If r generates V as a monoid, then r is properly infinite (i.e. r r + r). 5. Examples In this section, we give examples of C -algebras with the corona factorization property. We also give an example of a separable, stable C -algebra without the corona factorization property. The material in this section started out as a separate preprint, but it seems appropriate to instead include it here. It was shown in [19] that if B is a stable, separable, type I C -algebra with finite decomposition rank, then B has the corona factorization property.

14 544 DAN KUCEROVSKY AND P.W. NG See also the classic paper [24] for the case where B has the form B = C(X) K, where X is a finite-dimensional, second countable, compact metric space. Next, a result about certain hereditary subalgebras of B which are generated by norm-full positive elements of M(B). Lemma 5.1. Let A be a a separable stable C -algebra containing a nontrivial full projection. Suppose that c is a full positive element in M(A). Then cac is a full hereditary subalgebra of A, with no nonzero unital quotients and no nonzero bounded traces. Proof. Let c be as in the hypothesis. Let C := cac be the hereditary subalgebra of A generated by c. We first show that C is a full, hereditary subalgebra of A. Now since c is normfull in M(A), we have that M(A)cM(A) = M(A). Hence, M(A)cM(A)cM(A) = M(A), and, AcM(A)cA = A. Hence, AcAcA = A, and C := cac is a full, hereditary subalgebra of A. Now we show that C has no nonzero bounded trace. Suppose, to the contrary that λ is a tracial state of C. Since C is a full hereditary subalgebra of A, by Brown s theorem, C K = A. We can thus take C to be embedded in A as the e 11 corner. The trace λ extends naturally to a lower-semicontinuous trace on A. This extension, in turn, determines a trace on the positive cone of M(A). We still denote the extension(s) by λ. Let J λ be the ideal of M(A) which is the norm closure of the set {a M(A) : λ(a a) < }. Since 1 M(A) is not in the domain of any such trace, J λ is a proper ideal of M(A). Now let {e n } n=1 be an approximate unit for A, consisting of an increasing sequence of projections, converging to 1 M(A) in the strict topology, which is possible since A = pap K, where p is the given full projection, and pap is unital. Then c = lim n c 1/2 e n c 1/2, where the limit converges in the strict topology. But since λ is a bounded trace on C, and since for every n, c 1/2 e n c 1/2 is an element in C with norm less than c, there is a positive real number α such that for each n, λ(c 1/2 e n c 1/2 ) α. Hence, by the definition of the extended trace, λ(c) = lim sup n λ(e n ce n ) = lim sup n λ(c 1/2 e n c 1/2 ) is a finite number. Hence, c 1/2 J λ ; and hence, c J λ. This contracts the norm-fullness of c in M(A). Finally, we show that C has no nonzero unital quotients. Suppose, to the contrary, that J is a primitive ideal of C such that C/J is unital (and nonzero). Since C is a full, hereditary subalgebra of A, and by [23, Prop ], let π be an irreducible -representation of A such that ker(π) C = J. Let D = π(a). Extend π to the (unique) strictly continuous surjective -homomorphism π : M(A)

15 CORONA FACTORIZATION PROPERTY 545 M(D). Since D is nonunital, M(D)/D is nonzero. Since π(c) is unital, and since C is a hereditary subalgebra of D, let p be a projection in π(c) D such that π(c) = pdp. Let {f n } n=1 be a countable approximate unit for D, consisting of an increasing sequence of projections. Then π (c) 1/2 f n π (c) 1/2 is in π(c) for all n. Hence, for all n, π (c) 1/2 f n π (c) 1/2 c p. But lim n π (c) 1/2 f n π (c) 1/2 = π (c) in the strict topology of M(D). Hence, π (c) c p. Hence, π (c) is an element of D. But since M(D) properly contains D, this contradicts the fact that π (c) is a norm-full element of M(D). Theorem 5.2. Suppose that A is a separable, exact, stable, real rank zero algebra containing a full element of finite spectrum. Suppose that either one of the following conditions hold: i) A has cancellation of projections and weakly unperforated K 0 -group, or ii) A is purely infinite. Then every norm full extension of A is nuclearly absorbing. In other words, A has the corona factorization property. Proof. Notice that A contains a full projection: it contains a full element which is a finite sum of projections, and we can use stability to find a new element which is an orthogonal sum of this finite set of projections. This new element will be a full projection. Suppose that τ : B M(A)/A is a full extension of A by a C -algebra B. Let b be a nonzero positive element of B. Let c be a positive element in M(A) which is in the preimage of τ(b) (under the natural quotient map M(A) M(A)/A). Let C be the hereditary subalgebra of A given by C = cac. By theorem 3.1, to show that τ is nuclearly absorbing, it suffices to show that C has a stable subalgebra which is full in A. Note that by lemma 5.1, C is a full hereditary subalgebra of A such that C has no nonzero unital quotients and no nonzero bounded traces. Suppose that A has cancellation of projections and weakly unperforated ordered K 0 group. Then, by [27, Proposition 3.4], C is stable. Suppose that A is purely infinite. Then by [27, Proposition 5.4], C contains a stable subalgebra which is full in A. Whichever the case, C contains a stable subalgebra which is full in A; and τ is nuclearly absorbing. We end this section by showing that a certain separable, type I C -algebra (with continuous trace) does not have the corona factorization property.

16 546 DAN KUCEROVSKY AND P.W. NG Proposition 5.3. Let Z be an infinite Cartesian product of spheres, n=1 S2. Then B := C(Z) K does not have the corona factorization property. Proof. Let p M 2 (C(Z)) be the Bott projection over S 2. Then p is a onedimensional projection such that the Euler class of the corresponding vector bundle (the Bott bundle) is nonzero. For each integer n, let p n be a one-dimensional projection in B = C(Z) K given by p n (x 1, x 2, x 3,...) := p(x n ), for every (x 1, x 2, x 3,...) Z = n=1 S2. Replacing each p n be an appropriate Murrayvon Neumann equivalent projection (in B) if necessary, we may assume that i. the projections p n are pairwise orthogonal, and ii. P = n=1 p n converges in the strict topology to a projection in M(B) = M(C(Z) K). It is shown in [28], Prop. 4.5, that the projection P is full in the multipliers but is not equivalent to 1 there. Thus, P is not properly infinite and hence our algebra cannot have the corona factorization property. 6. Quasidiagonality It has become clear that the study of Salinas components is intimately related to quasidiagonality, so it is natural to ask if our results about the component of the identity (from section 1) imply anything about quasidiagonal extensions. First, a definition: Definition. An extension τ : A M(B K)/B K is said to be quasidiagonal if there exists a quasidiagonal lifting; that is, a lifting ˆτ : A M(B K) and a quasicentral approximate unit of projections (p n ) B K such that for each a A, the commutator [ˆτ(a), p n ] goes to zero as n. Approximate units are just sequences (or more generally, nets) converging to 1 strictly. We note that quasidiagonal extensions need not exist, for a given A and B. If they do, we can say that the algebra A is quasidiagonal relative to B. From proposition 1.6 we have the following corollary: Corollary 6.1. Let A and B be separable C -algebras, with A bootstrap class and B stable. Then, the set of weakly nuclear absorbing quasidiagonal extensions of A by B, if not empty, is exactly the set of extensions approximately unitarily equivalent to Kasparov s (or Thomsen s) absorbing extension. We can remove the assumption of relative quasidiagonality by using a definition due to Manuilov and Thomsen, where the projections are allowed to come from the multipliers:

17 CORONA FACTORIZATION PROPERTY 547 Proposition 6.2 ([22]). An extension τ : A M(B K)/B K is said to be weakly quasidiagonal if there exists a weakly quasidiagonal lifting; that is, a lifting ˆτ : A M(B K) and a quasicentral approximate unit of projections (p n ) M(B K) such that for each a A, the commutator [ˆτ(a), p n ] goes to zero as n. The set of weakly quasidiagonal extensions of SA by B K is isomorphic to Pext 1 Z(K (A), K (B)). The set of weakly quasidiagonal extensions is never empty, so we can drop the caveat re empty sets that we had needed previously. Proposition 6.3. Let A and B be separable C -algebras, with A bootstrap class and B stable. Then, the set of weakly nuclear, absorbing, weakly quasidiagonal extensions of SA by B, is exactly the set of extensions approximately unitarily equivalent to 1 KK 1 nuc (SA,B). We point out the following interesting corollary concerning quasidiagonal extensions: Corollary 6.4. Let A and B be separable C -algebras, with A bootstrap class and B stable. Then, i) if B is an absorbing algebra, the set of weakly nuclear full quasidiagonal extensions of A by B, if not empty, is exactly the set of extensions approximately unitarily equivalent to 1 KK 1 nuc (A,B), and ii) if SB is an absorbing algebra, the set of weakly nuclear, full, and weakly quasidiagonal extensions of SA by B, is exactly the set of extensions approximately unitarily to 1 KK 1 nuc (SA,B). We will show elsewhere (using a noncompact PPV-type theorem) that if B is absorbing, then the suspension SB is also absorbing, thereby simplifying the hypotheses above. For bootstrap class algebras, the set of extensions in proposition 6.3, and in corollary 6.1, can be characterized as being exactly those that induce the zero map on K. The Künneth sequence for tensor products can be however used to show that extensions inducing zero on K-theory in fact induce zero on K, so we can conclude that: Proposition 6.5. Let A and B be separable C -algebras, with A bootstrap class and B stable. i) The set of weakly nuclear, absorbing, weakly quasidiagonal extensions of SA by B is the set of absorbing extensions that induce zero on K,

18 548 DAN KUCEROVSKY AND P.W. NG ii) The set of weakly nuclear, absorbing, quasidiagonal extensions of A by B, if not empty, is the set of absorbing extensions that induce zero on K, Question 6.6. By what operations is the corona factorization property preserved? Is the corona factorization property preserved by extensions? Is it preserved by crossed products by the integers Z? Is it preserved by simple inductive limits of simple C -algebras? We note that the corona factorization property is not preserved by general inductive limits. For example, take Z to be the countably infinite Cartesian product of spheres. We have shown that C(Z) K does not have the corona factorization property. However, C(Z) K is an inductive limit of C -algebras of the form C(X) K, where X is a finite-dimensional compact CW -complex. C -algebras of the latter form always have the corona factorization property by, for example, the classic Pimsner-Popa-Voiculescu theorem [24]. References [1] C. Akermann and G. K. Pedrsen, Ideal perturbations of elements in C -algebra. Math. Scand., 41 (1977), pp [2] B. Blackadar, K-theory for operator algebras. Mathematical Sciences Research Institute Publications, (1998). [3] L. G. Brown, Stable Isomorphism of hereditary subalgebras of C -algebras. Pacific J. Math, 71 (1977), pp [4] L. G. Brown and G. K. Pedersen, On the geometry of the unit ball of a C -algebra. J. Reine Angew. Math., 469 (1995), pp [5] M. D. Choi and E. G. Effros, The completely positive lifting problem for C -algebras. Ann. of Math, 104 (1976), pp [6] M. Dadarlat and S. Eilers, On the classification of nuclear C -algebras. Proc. London Math. Soc. (3), 85 (2002), pp [7] M. Dadarlat and T. A. Loring, A Universal Multicoefficient Theorem for the Kasparov groups. Duke Math. J., 84 (1996), pp [8] K. R. Davidson, C -algebras by example, Fields Institute monographs, AMS, (1996). [9] G. Elliott, Derivations of matroid C -algebras. II. Ann. of Math. (2), 100 (1974), pp [10] G. Elliott and D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption theorem. Pacific J. of Math., 3 (2001), pp [11] J. Hjelmborg and M. Rordam, On stability of C -algebras. J. Funct. Anal., 155 (1998), pp [12] G. G. Kasparov, The operator K-functor and extensions of C -algebras. Math. USSR Izvestia, 16 (1981), pp [13] E. Kirchberg, The classification of purely infinite C -algebras using Kasparov s theory. Preprint, (1995).

19 CORONA FACTORIZATION PROPERTY 549 [14] E. Kirchberg and M. Rordam, Nonsimple purely infinite C -algebras. Preprint, (1999). [15] D. Kucerovsky, Double commutants of hereditary subalgebras of the corona. J. Funct. Anal., 219 (2005), pp [16] D. Kucerovsky, Extensions contained in ideals. Trans. AMS, 356 (2003), pp [17] D. Kucerovsky, The nonsimple absorption theorem. Preprint, (2000). [18] D. Kucerovsky, Large Fredholm Triples, in press, J. Funct. Anal.. [19] D. Kucerovsky and P. W. Ng, Decomposition rank and absorbing extensions of type I C -algebras. Preprint. [20] H. X. Lin, Ideals of multiplier algebras of simple AF-algebras. Proc. Math. Soc., 104 (1988) no. 1, pp [21] H. X. Lin, Simple C -algebras with continuous scales and simple corona algebras. Proc. Amer. Math. Soc., 112 (1991) no. 3, pp [22] Manuilov and Thomsen, Quasidiagonal Extensions and Sequentially Trivial Asymptotic Homomorphism. Adv. Math., 154 (2000), pp [23] G. K. Pedersen, C -algebras and their automorphism groups. Academic Press, London, (1979). [24] M. Pimsner, S. Popa and D. Voiculescu, Homogeneous C -extensions of C(X) K, Part I. J. of Operator Theory, 1 (1979), pp [25] J. Raeburn and S. J. Thompson, Countably generated Hilbert modules, the Kasparov Stabilization theorem, and frames with Hilbert modules. Proc. AMS, 131 (2003), pp [26] M. Rordam, Classification of certain infinite simple C -algebras. J. Funct. Anal., 131 (1995), pp [27] M. Rordam, Stable C -algebras, Operator algebras and applications. Advanced studies in pure mathematics, (2003), pp [28] M. Rordam, A simple C -algebra with a finite and an infinite projection. Acta Math., 191 (2003) no. 1, pp [29] M. Rordam, Stability of C -algebras is not a stable property. Documenta Math., 2 (1997), pp [30] M. Rordam, Extensions of stable C -algebras. Documenta Math., 6 (2001), pp [31] J. Rosenberg and C. Schochet, The Kunneth theorem and the universal coefficient theorem for Kasparov s generalized K-functor. Duke Math. J. 55 (1987), pp [32] N. Salinas, Relative quasidiagonality and KK-theory. Houston Math. J., 18 (1992), pp [33] G. Skandalis, Une notion de nucleariteé en K-théorie,. K-Theory I (1998), pp [34] G. Skandalis, Some remarks on Kasparov theory. J. Funct. Anal., 56 (1984), pp [35] C. L. Schochet, The fine structure of the Kasparov groups, I. J. Funct. Anal., 186 (2001), pp [36] C. L. Schochet, The Fine Structure of the Kasparov groups III: Relative Quasidiagonality. J. Op. Theory, to appear. [37] C. L. Schochet, A Pext primer: Pure extensions and lim 1 for infinite abelian groups. NYJM Monographs, 1 (2003). [38] K. Thomsen, On absorbing extensions. Proc. AMS, 129 (2001), pp [39] N. E. Wegge-Olsen, K-Theory and C -Algebras. Oxford University Press, Oxford, (1993).

20 550 DAN KUCEROVSKY AND P.W. NG [40] S. Zhang, Certain C -algebras with real rank zero and their corona and multiplier algebras, I. Pac. J. Math., 155 (1992), pp Received June 24, 2004 Department of Mathematics and Statistics, UNBF-F, Fredericton, NB, Canada E3B 5A3 address: dan@math.unb.ca address: pwn@math.unb.ca