The purpose of this short note is to prove that a stable separable C*-algebra with real rank zero has the so-called corona factorization property, that is, all the full multiplier projections are properly in finite. Enroute to our result, we consider conditions under which a real rank zero C*-algebra admits an injection of the compact operators (a question already considered in [21])

Kucerovsky, Dan

Perera Domènech, Francesc

Universitat Autònoma de Barcelona. Centre de Recerca Matemàtica

Centre de Recerca Matemàtica

Diposit Digital de Documents de la UAB

REAL RANK ZERO ALGEBRAS HAVE THE CORONAFACTORIZATION PROPERTYDAN KUCEROVSKY AND FRANCESC PERERAAbstract. The purpose of this short note is to prove that a stableseparable C∗-algebra with real rank zero has the so-called coronafactorization property, that is, all the full multiplier projections areproperly infinite. Enroute to our result, we consider conditions underwhich a real rank zero C∗-algebra admits an injection of the compactoperators (a question already considered in [21]).Introduction and PreliminariesThe corona factorization property has connections with extension theoryand the internal structure of C∗-algebras. Its origins can be traced backto [6], and it has been subsequently studied in other papers, such as [9, 8,10, 12, 13, 15, 16, 18].Known examples of C∗-algebras satisfying this property include stabi-lizations of purely infinite simple C∗-algebras; stabilizations of exact C∗-algebras with minimal ranks and weakly unperforated K0 group; and stabletype I C∗-algebras with finite decomposition rank (see, e.g. [17]).There are various equivalent formulations of the corona factorizationproperty (see, e.g. [8, 13, 10]). We take as our definition the one below,which has a more K-theoretical flavour.Recall that a projection p in a C∗-algebra A is infinite if there is a (non-zero) projection q such that q < p, yet q ∼ p in the Murray-von Neumannsense. If p is not infinite, then we say that p is finite. The (non-zero)projection p is said to be properly infinite provided that p⊕ p . p (that is,Date: September 22, 2006.D. K. was supported by an NSERC Grant; F. P. was partially supported by the DGIMEC-FEDER through Project MTM2005-00934, and by the Comissionat per Universi-tats i Recerca de la Generalitat de Catalunya.12 DAN KUCEROVSKY AND FRANCESC PERERAthe projection matrix( p 00 p)is equivalent to a subprojection of p). As usual,we say that an element c in a C∗-algebra A is full if the closed two-sidedideal AcA is all of A.Definition 0.1. Let A be a stable C∗-algebra. We say that A has the coronafactorization property if all full projections in M(A) are properly infinite.Note that, because of the stability assumption, the above definition is infact equivalent to the requirement that all full projections in M(A) are infact Murray-von Neumann equivalent to 1M(A).Recall that a C∗-algebra A has real rank zero provided the self-adjoint,invertible elements form a dense subset of the self-adjoint elements. Equiva-lently, A has real rank zero precisely when every self-adjoint element can beapproximated arbitrarily well by a self-adjoint element with finite spectrum([3]). The most natural question about the corona factorization property isperhaps whether a non-unital C∗-algebra with real rank zero has this prop-erty. This basic question has been mentioned in a number of papers andconference talks, see for example Question 4.17 in [17] which asks whethersimple, separable, stable and nuclear real rank zero algebras have the coronafactorization property.A partial result in the positive direction was given in [14, Theorem 1.4],but there the authors require that the given real rank zero algebra admitsan embedding of an AF-algebra with finite dimensional trace space.We answer this question in general and affirmatively, without needingsimplicity nor nuclearity assumptions. Our proof reduces to the case wherethe algebra has compact primitive ideal space, for which we give a charac-terization in Section 1. We proceed in Section 2 to establish an intermediateresult that allows us to draw a generarization of a result in [19]. Our mainresult is proved in Section 3, where we also prove that if A is a separableC∗-algebra with real rank zero and compact primitive ideal space such thatsome matrix algebra is stable, then A contains a copy of the compact op-erators that trivially intersects every proper ideal of A (see [21], where thisquestion was also considered). The corona factorization property has deepconsequences in KK-theory, see for example Corollary 3.4.REAL RANK ZERO AND THE CORONA FACTORIZATION PROPERTY 31. A characterization of algebras with compact structurespaceProposition 1.1. Let A be a separable C∗-algebra. Consider the followingconditions:(i) The primitive ideal space PrimA is compact;(ii) The full elements of A+ in A form an open set.Then (i) implies (ii), and the converse also holds if A has moreover real rankzero.Proof. We first prove the implication (i)⇒ (ii). The Dauns-Hoffmann theo-rem [20, 4.4.4] shows that for each positive x ∈ A+, the map xˇ : PrimA→ Rgiven by t 7→ ‖x+ker t‖ is lower semicontinuous. If the full elements of A+are not an open set, then there exists a full element x ∈ A+ and a sequence(yn) ⊂ A+ that converges to x in norm, with none of the yn full. By prop-erties of quotient norms, the yˇn converge to xˇ uniformly. Each yˇn has azero. By a triangle inequality argument, the infimum inf xˇ is therefore zero.By one of the Weierstrass theorems, a lower semicontinuous function on acompact set attains its miminum, so that xˇ has a zero. But this contradictsthe assumption that x is full in A+.We now prove the converse, so hence assume A has real rank zero. Ifthe full positive elements form an open set, then by assumption we canapproximate some full positive element by an element with finite spectrum,that is moreover full. By the functional calculus, we see that there exists afull projection, say P ∈ A. By Brown’s theorem [2], we now have that theunital algebra PAP is Morita equivalent to A, and thus these algebras havethe same primitive ideal spaces [22]. But it is well-known (see, e.g. [4]) thatunital separable C∗-algebras have compact primitive ideal space. ¤2. An intermediate resultIn this section we prove a result that establishes the corona factorizationproperty under somewhat strong hypothesis. As a corollary, we obtain ageneralization of a result by E. Pardo ([19]), which says that under strong4 DAN KUCEROVSKY AND FRANCESC PERERAconditions on the algebra, multiplier projections that are not in the algebraare either properly infinite or finite.Theorem 2.1. Let B be a stable separable C∗-algebra. If, for every fullmultiplier projection P , the hereditary algebra PBP contains a full stablesubalgebra, then all full multiplier projections are properly infinite.Before proving the above, we recall two of our earlier results. These arebasically KK-theoretical results, stated here mostly without the languageof KK-theory. The first of the quoted results [9, Lemma 10] is1 :Lemma 2.2. Let A be stable and σ-unital. A hereditary subalgebra cAcgenerated by a positive element c of the multipliers of A is isomorphic tosome hereditary subalgebra PAP generated by a multiplier projection P . Ifc is full in the multiplier algebra, then P is also full in the multiplier algebra.The next result is from [8, Theorem 3.2] (see also [6, Theorem 6]).Lemma 2.3. Let B be a stable and separable C∗-algebra. The followingare equivalent:(i) Every full multiplier projection is properly infinite;(ii) Every weakly nuclear full extension of B by a separable C∗-algebra Ais absorbing;(iii) Every hereditary subalgebra cBc generated by a full positive element cof the multiplier algebra of B contains a stable full subalgebra.We now obtain the proof of Theorem 2.1 as a corollary of these lemmas:Proof. Let c be a full multiplier element inM(B). By Lemma 2.2, there isa full multiplier projection P such that cBc ∼= PBP . Our hypothesis thenimplies that cBc contains a full stable subalgebra.By Lemma 2.3, all full multiplier projections are therefore properly infi-nite. ¤1The statement on fullness follows directly from a calculation with unitary equivalencein a Hilbert module, see the proof of [8, Theorem 4.4].REAL RANK ZERO AND THE CORONA FACTORIZATION PROPERTY 5We have the following further corollary that more closely resemblesPardo’s result.Corollary 2.4. Let B be a stable simple C∗-algebra. If no full multiplierprojections are finite, then in fact all full multiplier projections are properlyinfinite.Proof. To deduce this corollary from the above theorem, it is sufficient toshow that if a full multiplier projection P is infinite, then PBP containsa stable subalgebra. To do this, let v be a partial isometry with v∗v =P and vv∗ ≤ P , with of course vv∗ 6= P . Now, C∗(v) is an abstractToeplitz algebra, and thus contains a copy K of the compact operators. Wenotice that K is contained in the hereditary corner generated by P in themultipliers, and therefore the same is true of KBK. Thus we have a stablesubalgebra within PBP . ¤Of course, it is not quite fair to call the above a generalization of Pardo’sresult, since his result is phrased in terms of individual projections, whereasours has a stronger hypothesis involving all full multiplier projections. Onthe other hand, this stronger hypothesis allows us to use much weaker con-ditions on the algebra.3. The corona factorization property for real rank zeroalgebrasLemma 3.1. Let A be a separable C∗-algebra with real rank zero such thathas a full projection. If Mn(A) is stable for some integer n, then there is acopy of K in A that does not nontrivially intersect any proper ideal of A.Proof. Let us first consider the special case where A is of type I. TheKirchberg-Winter decomposition rank of A is, in this case [23], equal to thetopological dimension of the primitive ideal space of A, but on the otherhand by the Bratteli–Elliott theorem [1] the real rank zero property impliesthat the primitive ideal space has a basis of compact open sets. In otherwords, the topological dimension of the spectrum is zero. In [5], it is shown6 DAN KUCEROVSKY AND FRANCESC PERERAthat the decomposition rank of an algebra is zero if and only if the algebrais an AF algebra. Thus, A is an AF algebra and thus is stable if and onlyif it has no bounded traces. Since a bounded trace on A would extend to abounded trace on the (stable) algebra Mn(A), we see that the algebra A isstable. We notice that p⊗K ⊆ A⊗K, where p is a full projection in A, isa full copy of K inside A⊗K ∼= A.Having disposed of this quite nontrivial special case, we now consider thecase where A is not of type I. We shall first show that there exists a copyofC :=∞⊕n=1Mn(C)inside A, and that this copy does not nontrivially intersect any ideal of A.Let PN be a full projection in A. The hereditary subalgebra PNAPN hasno finite-dimensional representations. Thus, by the weak divisibility resultfrom [21, Proposition 5.7], there is a unital ∗-homomorphism ιN : F →PNAPN where F is a finite-dimensional C∗-algebra whose factors can beassumed to be matrix algebras of dimension greater than 2N. Let us denotethese matrix algebras by F1, F2, · · · , Fk. As pointed out in [21] (see theremark after Proposition 5.7), we can suppose that one of the ιN (Fi) is fullin PNAPN . We may as well take i = 1.Now, let us define a map N : MN (C) → A by embedding MN (C) inF1 as the top left corner. Let us also define PN+1 := PN − n(1MN ), andnotice that this is a full projection in A because it majorizes, for example,ιN (e2N,2N ).We now have, by induction, defined a sequence of embeddings of Mn(C)into A as orthogonal full subalgebras. (The base case of the inductionbegins with simply choosing any full projection of A as P1.) We thus haveREAL RANK ZERO AND THE CORONA FACTORIZATION PROPERTY 7the intertwining(3.1)M1(C) −−−−→ Ay yM1(C)⊕M2(C) −−−−→ Ay yM1(C)⊕M2(C)⊕M3(C) −−−−→ A....................which gives a copy of C :=∞⊕n=1Mn(C) inside A as claimed.Now, writing the compact operators K as the closure of the algebraicdirect limit D := alglim~n(Mn(C), x 7→ x⊕ 0), we notice that there is a copyof D inside C. Since of course A is closed, we thus have a copy of K insideA as claimed. ¤The next theorem shows that under minor technical hypotheses, realrank zero C∗-algebras have the corona factorization property. We note thatstability is clearly a necessary condition if all full multiplier projections areto be properly infinite (the unit of the multipliers of a nonunital separableC∗-algebra is properly infinite if and only if the given algebra is stable2).Theorem 3.2. Let A be a stable separable real rank zero C∗-algebra withcompact structure space. Then every full multiplier projection is properlyinfinite.Proof. Let P be a full multiplier projection. Then, for some positive in-teger N , we have that 1 in M(MN (A)) is subequivalent to⊕N1 P (thatis, 1 is equivalent to a subprojection of⊕N1 P ). Since by stability 1 isproperly infinite,⊕N1 P is therefore also properly infinite. But the unit ofthe multipliers of MN (PAP ) is therefore properly infinite, which impliesthat MN (PAP ) is stable. Furthermore, we claim that PAP contains a fullprojection. Indeed, by the results in [2] and [22], the structure space of2To see this, notice that if the unit of the multipliers is properly infinite, we have aunital embedding of O2 into the multipliers, hence a unital embedding of O∞ into themultipliers, whereupon the main result of [7] gives stability8 DAN KUCEROVSKY AND FRANCESC PERERAPAP is compact. By Proposition 1.1, the full positive elements of PAP aretherefore an open set, and it is then easy to use the real rank zero propertyof PAP to find a full projection p ∈ PAP . By Lemma 3.1, there is nowa full copy of the compact operators K in PAP . Theorem 2.1 finishes theproof. ¤Our Lemma 2.3 gives a KK-theoretical corollary:Corollary 3.3. Let B be a stable separable real rank zero C∗-algebra withcompact structure space. Every full weakly nuclear extension of B by aseparable C∗-algebra A is absorbing.Further corollaries can be obtained, for example:Corollary 3.4. If B is a unital real rank zero C∗-algebra and A is aseparable C∗-algebra, then all elements of KK1(A,B) can be taken to befull weakly nuclear extensions τ : A→M(B ⊗K)/B ⊗K.We may drop the condition on the structure space, but it appears thatthen we can only prove the existence of stable full subalgebras rather thanfull copies of the compact operators, K. This is nevertheless sufficient:Theorem 3.5. Separable stable real rank zero C∗-algebras have the propertythat full multiplier projections are properly infinite.Proof. Under this hypothesis, the primitive ideal space is a second-countablezero-dimensional locally compact topological space. It may therefore bewritten as a countable union of disjoint compact open sets. By the Dauns-Hoffmann theorem, the support functions of these sets can be regarded ascentral projections in the multiplier algebra M(A). Let us denote theseprojections by qi.By Lemma 2.3, it is sufficient to prove that, given a positive full elementc of the multiplier algebraM(A), the hereditary subalgebra cAc contains astable full subalgebra. Since Prim(qiA) is compact, we may use Theorem 3.2followed by Lemma 2.3 again, and assume that qicAc ⊆ A contains a stablefull subalgebra, say Si. We may moreover suppose that the Si are separable(since A is).REAL RANK ZERO AND THE CORONA FACTORIZATION PROPERTY 9Define S ⊆ cAc to beS := C0(Si) = {(s1, s2, s3, · · · ) : si ∈ Si, ‖si‖ → 0 as i→∞} .We claim first that S is full. Let pi be an arbitrary nonzero representationof A. We may extend it to a representation of M(A) by taking closuresin the strict topology. Since the qi converge strictly to 1M(A), at least oneof the qi is not in the kernel of pi. But then since the corresponding Si isfull in qiA, it follows that pi(Si) 6= {0}. This implies that S is not properlycontained in the kernel of any nonzero representation.Lastly, since S can be written as an inductive limit of stable separablesubalgebras⊕N1 Si, it is stable [7]. ¤References[1] O. Bratteli, G.A. Elliott, Structure spaces of approximately finite-dimensional C∗-algebras. II, J. Funct. Anal. 30 (1978), pp. 74–82.[2] L. G. Brown, Stable isomorphism of hereditary subalgebras of C∗-algebras, PacificJ. Math, 71 (1977), pp. 335–384.[3] L. G. Brown, G. K. Pedersen, C∗-algebras of real rank zero, J. Funct. Anal. 99(1991), pp. 131–149.[4] J. Dixmier, C∗-algebras, North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977).[5] E. Kirchberg, W. Winter, Covering dimension and quasidiagonality, Internat. J.Math.,15 (2004), pp.63–85.[6] G. A. Elliott, D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption the-orem, Pacific J. Math., 198 (2001), pp. 1–25.[7] J. Hjelmborg and M. Rordam, On stability of C∗-algebras, J. Funct. Anal., 155(1998) pp. 153–170.[8] D. Kucerovsky, Large Fredholm Triples, J. Funct. Anal., 236 (2006), pp. 395-408.[9] D. Kucerovsky, Extensions contained in ideals, Trans. Amer. Math. Soc., 356 (2004),pp. 1025-1043.[10] D. Kucerovsky When are Fredholm Triples Operator Homotopic?, Proc. Amer.Math. Soc., posted on August 21, 2006, PII S 0002-9939(06)08481-4 (to appearin print).[11] D. Kucerovsky, Extensions of C∗-algebras with the corona factorization property,Int. J. Pure and Applied Math., 16 (2004), pp. 181–191.[12] D. Kucerovsky, P. W. Ng, Decomposition rank and absorbing extensions of type Ialgebras, J. Funct. Anal., 221 (2005), pp. 25–36.10 DAN KUCEROVSKY AND FRANCESC PERERA[13] D. Kucerovsky, P. W. Ng, The corona factorization property and approximate uni-tary equivalence, Houston J. Math., 32 (2006), pp. 531–550.[14] D. Kucerovsky, P. W. Ng, AF-skeletons and real rank zero algebras with the coronafactorization property, Can. Math. Bull., to appear.[15] D. Kucerovsky, P. W. Ng, The corona factorization property and stability, Preprint,2005.[16] D. Kucerovsky, P. W. Ng, A simple C∗-algebra with perforation and the coronafactorization property, Preprint, 2005.[17] P. W. Ng, The corona factorization property, arXiv preprint math.OA/0510248 v2(2006).[18] H. Lin, Full extensions and approximate unitary equivalence, arXiv preprintmath.OA/0401242 v1 (2006).[19] E. Pardo Finite projections in multiplier algebras, Rocky Mountain J. Math. 31(2001), pp. 581–594.[20] G.K. Pedersen, C∗-algebras and their automorphism groups. London MathematicalSociety Monographs, 14. Academic Press, Inc., London-New York, 1979.[21] F. Perera, M. Rørdam, AF-embeddings into C∗-algebras of real rank zero, J. Funct.Anal., 217 (2004), pp. 142–170.[22] M. A. Rieffel, Morita equivalence for operator algebras, in Operator algebras andapplications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math. 38, pp. 285–298.[23] W. Winter, Covering dimension for nuclear C∗-algebras, J. Funct. Anal., 199(2003), pp. 535–556.Department of Mathematics, University of New Brunswick at Fredericton,NB, Canada E3B 5A3E-mail address: dkucerov@unb.caDepartament de Matema`tiques, Universitat Auto`noma de Barcelona, 08193Bellaterra, Barcelona, SpainE-mail address: perera@mat.uab.cat

Real rank zero algebras have the corona factorization property

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Real rank zero algebras have the corona factorization property

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