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A CHARACTERIZATION OF SEMIPROJECTIVITY FOR COMMUTATIVE C∗-ALGEBRAS. 1 1 ADAMP.W.SØRENSENANDHANNESTHIEL 0 2 ∗ ABSTRACT. Given a compact, metric space X, we show that the commutative C - n algebra C(X)is semiprojective if and only if X is anabsolute neighborhood retract a J ofdimensionatmostone. ThisconfirmsaconjectureofBlackadar. 6 Generalizingtothenon-unitalsetting,wederiveacharacterizationofsemiprojec- ∗ 2 tivity for separable, commutative C -algebras. As further application of our find- ingsweverifytwoconjecturesofLoringandBlackadarinthecommutativecase,and ] ∗ A wegiveapartialanswertothequestion, whenacommutativeC -algebraisweakly (semi-)projective. O . h t a m 1. INTRODUCTION [ Shape theory is a machinery that allows to focus on the global properties of a space 2 by abstracting from its local behavior. This is done by approximating the space by a v system of nicer spaces, and then studying this approximating system instead of the 6 5 originalspace. Afterthisideawassuccessfullyappliedtocommutativespaces,itwas 8 firstintroducedtothenoncommutative worldbyEffrosandKaminker,[EK86]. Soon 1 . after,noncommutativeshapetheorywasdevelopedtoitsmodernformbyBlackadar, 1 0 [Bla85]. 1 In classical shape theory one approximates a space by absolute neighborhood re- 1 tracts (ANRs). In the noncommutative world, the role of these nice spaces is played : v by the semiprojective C∗-algebras. It is however not true that every (compact) ANR i X X givesasemiprojective C∗-algebraC(X). Infact,alreadythetwo-discD2 isacoun- r terexample (see 3.2 and 3.3). This hints to a possible problem in noncommutative a shape theory: While it easy to show that there are enough ANRsto approximate every compact metric space, the analogue for C∗-algebra is not obvious at all. In fact it is still an open problem whether every separable C∗-algebra can be written as an inductive limit of semiprojective C∗-algebras. Some progress on this problem was recently made byLoring andShulman, [LS10]. Hence, it is important to know which C∗-algebras are semiprojective. And al- though semiprojectivity was modeled on ANRs, the first large class of C∗-algebras Date:January27,2011. 2000 Mathematics Subject Classification. Primary 46L05, 54C55, 55M15, 54F50 ; Secondary 46L80, 46M10,54F15,54D35,54C56,55P55. ∗ Key words and phrases. C -algebras, non-commutative shape theory, semiprojectivity, absolute neighborhoodretracts. ThisresearchwassupportedbytheDanishNationalResearchFoundationthroughtheCentrefor SymmetryandDeformation. Thesecond namedauthorwaspartiallysupportedbytheMarieCurie ResearchTrainingNetworkEU-NCG. 1 2 ADAMP.W.SØRENSENANDHANNESTHIEL shown to be semiprojective were the highly noncommutative Cuntz-Krieger algebras, see [Bla85]. Since then, these results have been extended to cover all UCT Kirchberg algebras with finitely generated K-theory and free K -group, see [Szy02] 1 and[Spi09], anditisconjectured that infactall Kirchberg algebraswith finitelygen- erated K-theory are semiprojective. Yet, the following natural question remainedunanswered: Question 1.1. Whichcommutative C∗-algebras are semiprojective? An important partial answer was obtained by Loring, [Lor97, Proposition 16.2.1, p.125], who showed that all one-dimensional CW-complexes give rise to semiprojective C∗-algebras. In [ELP98] this was extended to the class of one-dimensional NCCW-complexes. In another direction, Chigogidze and Dranishnikov recently gave a characterization of the commutative C∗-algebras that are projective: They show in [CD10, The- orem 4.3] that C(X) is projective in S (the category of unital, separable C∗-algebras 1 with unital ∗-homomorphisms) if and only if X is an AR and dim(X) ≤ 1. Inspired by theirresults we obtain the following answerto question 1.1: Theorem 1.2. LetX be acompact,metricspace. Thenthe followingare equivalent: (I) C(X) issemiprojective. (II) X isan ANRand dim(X) ≤ 1. This confirms a conjecture of Blackadar, [Bla06, II.8.3.8, p.163]. We proceed as follows: CONTENTS 1. Introduction 1 2. Preliminaries 4 3. One implication ofthe main theorem: Necessity 9 4. Structure ofcompact, one-dimensional ANRs 13 5. The other implication of the main theorem: Sufficiency 19 6. Applications 23 Acknowledgments 29 References 29 Insection2(Preliminaries),werecallthebasicconceptsofcommutativeandnoncom- mutative shape theory, inparticular the notion ofan ANRandof semiprojectivity. In section 3 (Necessity), we show the implication ”(I) ⇒ (II)” of our main result 1.2. The idea is to use the topological properties of higher dimensional spaces, to show thatifC(X)wassemiprojectiveandX anANRofdimensionatleast2thenwecould solve a liftingproblem known tobe unsolvable. SEMIPROJECTIVITYOFCOMMUTATIVEC∗-ALGEBRAS. 3 In section 4 we study the structure of compact, one-dimensional ANRs. We charac- terize when a one-dimensional Peano continuum X is an ANR, see 4.12. As it turns out, onecriterium isthatX containsafinitesubgraph thatcontainsallhomotopy information,a(homotopy)core,see4.10. ThisisalsoequivalenttoK∗(X)beingfinitely generated, which isa recurring property in connection with semiprojectivity. The main result of this section is theorem 4.17 which describes the internal struc- tureofacompact,one-dimensionalANRX. StartingwiththehomotopycoreY ⊂ X 1 there is an increasing sequence of subgraphs Y ⊂ Y ⊂ ... ⊂ X that exhaust X, and 1 2 such thatY isobtainedfrom Y bysimplyattachingalinesegmentatoneendtoa k+1 k point in Y . This generalizes the classical structure theorem for dendrites (which are k precisely the contractible,compact, one-dimensional ANRs). Insection 5(Sufficiency) weshowtheimplication ”(II)⇒(I)”of1.2. Usingthestruc- ∼ ture theorem 4.17 for X, we obtain subgraphs Y ⊂ X such that X = limY . The k ←− k first graph Y contains all K-theory information, and the subsequent graphs are ob- 1 tained by attaching line segments. Dualizing, we can write C(X) as an inductive limit, C(X) = limC(Y ). Since the maps Y → Y are retractions, the dual bonding −→ k k+1 k morphisms C(Y ) → C(Y )are accessible forlifting problems. k k+1 Themainresultofthissectionis5.3. GivenaliftingproblemC(X) → C/ J and k k aninitialliftfromC(Y )tosomeC/J ,thereexistsaliftingfromanyC(Y )tothesame 1 l k S height, and finallyaliftfrom the inductive limitC(X)toC/J . Thisideaiscentral in l [CD10], but it has also been used before, for instance by Blackadar in order to prove that the Cuntz algebra O is semiprojective. We note that some form of inductive ∞ limit argument seems necessary for lifting an infinite numberof generators. Wealso wish to point out that Chigogidze and Dranishnikov only needed semiprojectivity, and not projectivity, in manystepsof theirproofs. The proof ”(II)⇒ (I)” follows from 5.3 if we can find an initial lift from C(Y ). For 1 this we use Loring’s deep result, [Lor97], which says that C(Y) is semiprojective for every finite graph Y. We also need Loring’s result to write the algebras C(Y ) as k universal C∗-algebras. To summarize, the proof proceeds in two steps. First, we construct an initial lift C(Y ) → C/J from the homotopy core. This will lift all K-theory information of X. 1 l ButoncetheK-theoryinformationislifted,wedonotneedto”sinktoalowerlevel”. Insection6wegiveapplicationsofourmainresult1.2. First,weanalyzethestructure of non-compact, one-dimensional ANRs. We give a characterization when the one- point compactification of such spaces is again an ANR, see 6.1. This is motivated by the fact that a C∗-algebra A is semiprojective if and only if its minimal unitalization A is semiprojective. For commutative C∗-algebras, the minimal unitalization correspondstotakingtheone-pointcompactificationoftheunderlyingcommutative space. Uesing the characterization of semiprojectivity for unital, separable, commutative C∗-algebras given in 1.2, we derive a characterization of semiprojectivity for non-unital, separable,commutative C∗-algebras, see 6.2. 4 ADAMP.W.SØRENSENANDHANNESTHIEL In 6.1 we also note that the one-point compactification of the considered spaces is an ANR if and only every finite-point compactification is an ANR. This allows us to study short exactsequences 0 // I // A // F // 0 with F finite-dimensional. It was conjectured by Loring and also by Blackadar, [Bla04, Conjecture 4.5], that in this situation A is semiprojective if and only if I is. OneimplicationwasrecentlyprovenbyDominicEnders,[End11],whoshowedthat semiprojectivity passes to ideals when the quotient is finite-dimensional. The converse implication isingeneral notevenknown forF = C. However,in 6.3weverify this conjecture underthe additional assumption thatAiscommutative. Then,wewillstudythesemiprojectivity ofC∗-algebrasoftheform C (X,M ). We 0 k derive in 6.9 that for a separable, commutative C∗-algebra A, the algebra A ⊗M is k semiprojectiveifandonlyifAissemiprojective. Again,thisquestioncanbeaskedin general. ItisknownthatsemiprojectivityofAimpliesthatA⊗M issemiprojectiveas k well,see[Bla85,Corollary2.28]and[Lor97,Thoerem14.2.2,p.110]. Fortheconverse, it is known that semiprojectivity passes to full corners, [Bla85, Proposition 2.27]. It was conjectured by Blackadar, [Bla04, Conjecture 4.4], that the same holds for full hereditary sub-C∗-algebras. Note that A always is a full hereditary sub-C∗-algebra of A⊗M . Thus, weverify the conjecture for commutative C∗-algebras. k As a final application, we consider the following variant of question 1.1: When is a commutative C∗-algebra weakly(semi-)projective? In order to study this problem, weanalyzethestructureofone-dimensionalapproximativeabsolute(neighborhood) retracts, abbreviated AA(N)R. In 6.15 we show that such spaces are approximated fromwithinbyfinitetrees(finitegraphs). Sincefinitetrees(finitegraphs)give(semi- )projective C∗-algebras,wederivein6.16thatC(X)isweakly(semi-)projective inS 1 ifX isa one-dimensional AA(N)R. Summarizing our results, 1.2 and 6.16, and the result of Chigogidze and Dranish- nikov, [CD10,Theorem 4.3],we get: Theorem 1.3. LetX be acompact,metricspacewithdim(X) ≤ 1. Then: (1) C(X)isprojectiveinS ⇔X isan AR 1 (2) C(X)isweakly projectiveinS ⇔X isan AAR 1 (3) C(X)issemiprojectiveS ⇔X isan ANR 1 (4) C(X)isweakly semiprojectiveS ⇔X isan AANR 1 Moreover,C(X) projectiveor semiprojectivealreadyimpliesdim(X) ≤ 1. 2. PRELIMINARIES By A,B,C,D we mostly denote C∗-algebras, usually assumed to be separable here, and by a morphism between C∗-algebras we understand a ∗-homomorphism. By an idealinaC∗-algebrawemeanaclosed,two-sidedideal. IfAisaC∗-algebra,thenwe denote byAits minimal unitalization, and byA+ the forced unitalization. Thus, if A is unital, then A = A and A+ ∼= A ⊕ C. We use the symbol ≃ to denote homotopy equivalencee. e SEMIPROJECTIVITYOFCOMMUTATIVEC∗-ALGEBRAS. 5 Byamapbetweentwotopologicalspaceswemeanacontinuousmap. Givenε > 0 and subsets F,G ⊂ X of a metric space, we say F is ε-contained in G, denoted by F ⊂ G, if for every x ∈ F there exists some y ∈ G such that d (x,y) < ε. Given ε X two maps ϕ,ψ: X → Y between metric spaces and a subset F ⊂ X we say ”ϕ and ψ agree on F”, denoted ϕ =F ψ, if ϕ(x) = ψ(x) for all x ∈ F. If moreover ε > 0 is given, thenwesay”ϕandψ agreeuptoε”,denotedϕ = ψ,ifd (ϕ(x),ψ(x)) < εfor ε Y all x ∈ X (for normed spaces, this is usually denoted by kϕ− ψk < ε). We say ”ϕ ∞ and ψ agree on F up toε”, denoted ϕ =F ψ, ifd (ϕ(x),ψ(x)) < εfor all x ∈ F. ε Y 2.1 ((Approximative) absolute (neighborhood) retracts). A metric space X is an (approximative)absoluteretract,abbreviated by(A)AR,ifforallpairs1(Y,Z)ofmetric spaces and maps f: Z → X (and ε > 0) there exists a map g: Z → X such that f = g◦ι(resp. f = g◦ι), where ι: Z ֒→ Y isthe inclusion map. Thismeansthat the ε following diagram can becompleted tocommute (up to ε): Y OO g ι ~~ (cid:31)? X oo Z f A metric space X is an (approximative) absolute neighborhood retract, abbreviated by (A)ANR, if for all pairs (Y,Z) of metric spaces and maps f: Z → X (and ε > 0) there exists a neighborhood V of Z and a map g: V → X such that f = g ◦ ι (resp. f = g◦ι)whereι: Z ֒→ V istheinclusionmap. Thismeansthatthefollowing ε diagram can be completed to commute (up to ε): Y OO (cid:31)? V OO g ι ~~ (cid:31)? X oo Z f FordetailsaboutARsandANRssee[Bor67]. WewillonlyconsidercompactAARs and AANRsin this paper,andthe readerisreferred to [Cla71]for more details. WeconsidershapetheoryforseparableC∗-algebrasasdevelopedbyBlackadar,[Bla85]. Letusshortly recall the mainnotions andresults: 2.2 ((Weakly) (semi-)projective C∗-algebras). Let D be a subcategory of the category ofC∗-algebras,closedunderquotients2. AD-morphismϕ: A → B iscalled(weakly) projective in D if for any C∗-algebra C in D and D-morphism σ: B → C/J to some 1A(Y,Z)pairofspacesissimplyaspaceY withaclosedsubspaceZ ⊂Y. 2This means the following: Assume B is a quotient C∗-algebra of A with quotient morphism π: A→B. IfA∈D,thenB ∈DandπisaD-morphism. 6 ADAMP.W.SØRENSENANDHANNESTHIEL quotient (and finite subset F ⊂ A, ε > 0), there exists a D-morphism σ¯: A → C such thatπ◦σ¯ = σ◦ϕ(resp. π◦σ¯ =F σ◦ϕ),whereπ: C → C/J isthequotientmorphism. ε Thismeansthatthefollowingdiagram canbecompletedtocommute (uptoεonF): C 77 σ¯ π (cid:15)(cid:15) A // B // C/J ϕ σ AC∗-algebraAiscalled(weakly)projective inD iftheidentitymorphism id : A → A Ais(weakly) projective. A D-morphism ϕ: A → B is called (weakly) semiprojective in D if for any C∗- algebra C in D andincreasing sequence ofidealsJ (cid:1)J (cid:1)...(cid:1)C andD-morphism 1 2 σ: B → C/ J (and finite subset F ⊂ A, ε > 0), there exists an index k and a k k D-morphism σ¯: A → C/J such that π ◦ σ¯ = σ ◦ ϕ (resp. π ◦ σ¯ =F σ ◦ ϕ), where k k k ε S π : C/J → C/ J is the quotient morphism. This means that the following dia- k k k k gram can becompleted tocommute (up to εon F): S C (cid:15)(cid:15) C/J k 66 ψ π (cid:15)(cid:15) A // B // C/ J ϕ σ k k A C∗-algebra A is called (weakly) semiprojectSive in D if the identity morphism id : A → Ais(weakly) semiprojective. A It is well known that if A is separable then A is semiprojective in the category of all C∗-algebras if and only if it is in the category of separable C∗-algebras. If D is the category S of all separable C∗-algebras (with all ∗-homomorphisms), then one dropsthereferencetoDandsimplyspeaksof(weakly)(semi-)projectiveC∗-algebras. BesidesS oneoftenconsidersthecategoryS ofallunitalseparableC∗-algebraswith 1 unital ∗-homomorphisms asmorphisms. Aprojective C∗-algebracannothaveaunit. Fora(separable)C∗-algebrasAweget from [Bla85, Proposition 2.5], see also [Lor97, Theorem 10.1.9, p.75], that the following are equivalent: (1) Ais projective (2) Ais projective in S 1 The situation for semiprojectivity is even easier. A unital C∗-algebra is semiprojective ief and only if it is semiprojective in S . Further, for a separable C∗-algebra A 1 we get from [Bla85, Corollary 2.16], see also [Lor97, Theorem 14.1.7, p.108], that the following are equivalent: (1) Ais semiprojective (2) Ais semiprojective e SEMIPROJECTIVITYOFCOMMUTATIVEC∗-ALGEBRAS. 7 (3) Ais semiprojective in S 1 e 2.3(Connectionbetween(approximative)absolute(neighborhood)retractsand(weakly) (semi-)projective C∗-algebras). LetSC bethefullsubcategoryofS consistingof(separable) commutative C∗-algebras, and similarly let SC be the full subcategory of S 1 1 consisting of (separable,unital) commutative C∗-algebras. Ingeneral, foraC∗-algebraitiseasiertobe(weakly)(semi-)projective inasmaller full subcategory, since there are fewer quotients to map into. In particular, if a commutative C∗-algebra is (weakly) (semi-)projective, then it will be (weakly) (semi- )projective with respect to SC. If one compares the definitions carefully, then one gets the following equivalences for a compact, metric space X (see [Bla85, Proposi- tion 2.11]): (1) C(X)isprojective in SC ⇔ X isanAR 1 (2) C(X)isweakly projective in SC ⇔ X isanAAR 1 (3) C(X)issemiprojective in SC ⇔ X isanANR 1 (4) C(X)isweakly semiprojective in SC ⇔ X isanAANR 1 Thus, the notion of (weak) (semi-)projectively is a translation of the concept of an (approximate) absolute (neighborhood) retract to the world of noncommutative spaces. Let us clearly state a point which is used in the proof of the main theorem: If C(X) is (weakly) (semi-)projective in SC , then X is an (approximate) absolute 1 (neighborhood) retract. As we will see, the converse is not true in general. We need an assumption on the dimension of X. 2.4 (Covering dimension). By dim(X) we denote the covering dimension of a space X. By definition, dim(X) ≤ n if every finite open cover U of X can be refined by a finite open cover V ofX such that ord(V) ≤ n+1. Here ord(V) is the largest number k such that there exists some point x ∈ X that is contained in k different elementsof V. ToanopencoverV onecannaturallyassignanabstractsimplicialcomplex3N(V), called the nerve of the covering. It is is defined as the family of finite subsets V′ ⊂ V with non-empty intersection, in symbols: N(V) := {V′ ⊂ V finite : V′ 6= ∅}. \ A n-simplex of N(V) corresponds to a choice of n different elements in the cover that have non-empty intersection. Given an abstract simplicial complex C, one can naturally associate to it a space |C|, called the geometric realization of C. The space |C|is apolyhedron, in particular itis aCW-complex. 3AnabstractsimplicialcomplexoverasetSisafamilyCoffinitesubsetsofSsuchthatX ⊂Y ∈C impliesX ∈C.AnelementX ∈Cwithn+1elementsiscalledann-simplex(oftheabstractsimplicial complex). 8 ADAMP.W.SØRENSENANDHANNESTHIEL Note that ord(V) ≤ n + 1 if and only if the nerve N(V) of the covering V is an abstractsimplicialsetofdimension4≤ n,orequivalentlythegeometricrealizationof |N(V)|isa polyhedron ofcovering dimension5≤ n. Let U be a finite open covering of a space X, and {e : U ∈ U} a partition of u unity that is subordinate to U. This naturally definesa map α: X → |N(U)| sending a point x ∈ X tothe (unique)point α(x) ∈ |N(U)|that has”coordinates” e (x). U By locdim(X) we denote the local covering dimension of a space X. By definition locdim(X) ≤ nifeverypointx ∈ X hasaclosedneighborhoodDsuchthatdim(D) ≤ n. IfX is paracompact (e.g. if itis compact, or locally compact and σ-compact), then locdim(X) = dim(X). See[Nag70]formore detailsonnerves,polyhedra andthe(local)coveringdimen- sion ofa space. A particularly nice class of one-dimensional6 spaces are the so-called dendrites. Be- fore we look at them, let us recall some notions from continuum theory. A good reference isNadler’sbook, [Nad92]. Acontinuumisacompact,connected,metricspace,andageneralizedcontinuum is a locally compact, connected, metric space. A Peano continuum is a locally connected continuum, and a generalized Peano continuum is a locally connected generalized continuum. By a finite graph we mean a graph with finitely many vertices andedges,orequivalentlyacompact,one-dimensionalCW-complex. Byafinitetree we meana contractible finite graph. 2.5 (Dendrites). A dendrite is a Peano continuum that does not contains a simple closed curve (i.e., there is no embedding of the circle S1 into it). There are many other characterizations ofadendrite. Wecollectafewandwewill use themwithout further mentioning. Let X be a Peano continuum. Then X is a dendrite if and only if one (or equiva- lently all)ofthe following conditions holds: (1) X isone-dimensional and contractible (2) X istree-like7. (3) X isdendritic8 4Thedimensionofanabstractsimplicialsetisthelargestintegerksuchthatitcontainsak-simplex. 5The covering dimension of polyhedra, or more generally CW-complexes, is easily understood. Thesespacesaresuccessivelybuildbyattachingcellsofhigherandhigherdimension. The(covering) dimensionofaCW-complexissimplythehighestdimensionofacellthatwasattachedwhenbuilding thecomplex. 6Wesayaspaceisone-dimensionalifdim(X)≤1.So,althoughitsoundsweird,aone-dimensional space can also be zero-dimensional. It would probably be more precise to speak of ”at most one- dimensional”space,howevertheusageoftheterm”one-dimensionalspace”iswellestablished. 7A (compact, metric) spaceX is tree-like, if for everyε > 0 thereexists a finite treeT and a map −1 f: X →T ontoT suchthatdiam(f (y))<εforally ∈T. 8AspaceX iscalleddendritic,ifanytwopointsofX canbeseparatedbytheomission ofathird point SEMIPROJECTIVITYOFCOMMUTATIVEC∗-ALGEBRAS. 9 (4) X ishereditarily unicoherent9. For more information about dendritessee [Nad92, Chapter10],[Lel76], [CC60]. 3. ONE IMPLICATION OF THE MAIN THEOREM: NECESSITY Proposition 3.1. LetC(X)beaunital,separableC∗-algebrathatissemiprojective. ThenX is acompactANRwithdim(X) ≤ 1. Proof. Assumesuch aC(X)isgiven. ThenX isacompact, metricspace. Asnoted in 2.3, semiprojectivity (in S ) implies semiprojectivity in the full subcategory SC and 1 1 thismeansexactlythatX isa(compact) ANR.Weareleftwithshowingdim(X) ≤ 1. Assume otherwise, i.e., assume dim(X) ≥ 2. Since X is paracompact, we have locdim(X) = dim(X) ≥ 2. This means there exists x ∈ X such that dim(D) ≥ 2 for 0 each closed neighborhood D of x . For each k consider D := {y ∈ X : d(y,x ) ≤ 0 k 0 1/k}. This defines a decreasing sequence of closed neighborhoods around x with 0 dim(D ) ≥ 2. k It was noted in [CD10, Proposition 3.1] that a Peano space of dimension at least 2 admits a topological embedding10 of S1. Indeed, a Peano space that contains no simplearc(i.e. inwhichS1 cannotbeembedded)isadendrite,andtherefore atmost one-dimensional. It follows that there are embeddings ϕ : S1 ֒→ D ⊂ X. Putting k k these together we get a map (not necessarily an embedding) ϕ: Y → X where Y is the space of ”smallerand smallercircles”: Y = {(0,0)}∪ S((1/2k,0),1/(4·2k)) ⊂ R2, k≥1 [ where S(x,r) is the circle of radius r around the point x. We define ϕ as ϕ on the k circle S((1/k,0),1/3k). The map ϕ: Y → X inducesa morphism ϕ∗: C(X) → C(Y). Next we construct a C∗-algebra B with a nested sequence of ideals J (cid:1) B, such k that C(Y) = B/ J andϕ∗: C(X) → C(Y)cannot belifted tosome B/J . LetT be k k k the Toeplitz algebra and let T ,T ,... be a sequence of copies of the Toeplitz algebra, 1 2 S and set: B := ( T )+ k k∈N M = {(b ,b ,...) ∈ T such that (b ) converges to ascalar multiple of1 }. 1 2 k k T k≥1 Y 9AcontinuumX iscalledunicoherentifforeachtwosubcontinuaY1,Y2 ⊂X withX =Y1∪Y2the intersectionY1∩Y2isacontinuum(i.e. connected). Acontinuumiscalledhereditarilyunicoherentif allitssubcontinuaareunicoherent. 10If X,Y are spaces, then an injective map i: X → Y is called a topological embedding if the original topology of X is the same as initial topology induced by the map i. We usually consider a topologicallyembeddedspaceasasubsetwiththesubsettopology. 10 ADAMP.W.SØRENSENANDHANNESTHIEL The algebras T come with ideals K (cid:1)T (each K a copy of the algebra of compact k k k k operators K). Define idealsJ (cid:1)B asfollows: k J := K ⊕...⊕K ⊕0⊕0⊕... k 1 k = {(b ,...,b ,0,0,...) ∈ B : b ∈ K (cid:1)T }. 1 k i i i Note B/J = C(S1) ⊕ ... ⊕ C(S1) ⊕ ( T )+ (k summands of C(S1)). Also k (k) l≥k+1 l J ⊂ J and J := J = K andB/J = ( C(S1))+ ∼= C(Y). k k+1 k k k∈N k L l≥1 The semiprojectivity of C(X) gives a lift of ϕ∗: C(X) → C(Y) = B/J to some S L L B/J . Consider the projection ρ : B/J → T onto the (k+1)-th coordinate, and k k+1 k k+1 similarly ̺ : B/J → C(S1). The composition C(X) → C(Y) ∼= B/J → C(S1) is k+1 ϕ∗ ,themorphisminducedbytheinclusionϕ : S1 ֒→ X. Notethatϕ∗ issurjec- k+1 k+1 k+1 tive since ϕ isan inclusion. Thesituation isviewedinthe followingcommutative k+1 diagram: C(X) kkkkϕk∗k// kCk(kYkkk)kkk∼=kkk55//BB//(cid:15)(cid:15)JJk ̺ρkk++11// C// T(kS+(cid:15)(cid:15) 11) 22 ϕ∗ k+1 The unitary id ∈ C(S1) lifts under ϕ∗ to a normal element in C(X), but it does S1 k+1 not lift to a normal element in T . This is a contradiction, and our assumption k+1 dim(X) ≥ 2 must bewrong. (cid:3) It is well known that C(D2), the C∗-algebra of continuous functions on the two- dimensional disc D2 = {(x,y) ∈ R2 : x2 + y2 ≤ 1}, is not weakly semiprojective. For completeness we include the argument which is essentially taken from Loring [Lor97, 17.1,p.131],seealso [Lor95]. Proposition 3.2. C(D2) is notweakly semiprojective. Proof. The ∗-homomorphisms from C(D2) to a C∗-algebra A are in natural one-one correspondencewithnormalcontractionsinA. Thus,statementsabout(weak)(semi- )projectivity ofC(D2)correspond tostatementsaboutthe(approximate)liftabilityof normal elements. For example, that C(D2) is projective would correspond to the (wrong) statement that normal elements lift from quotient C∗-algebras. To disprove weaksemiprojectivity ofC(D2)oneusesaconstruction ofoperatorsthatareapprox- imately normal butdo notlift in the required waydue to anindexobstruction. Moreprecisely,defineweightedshiftoperatorst ontheseparableHilbertspacel2 n (with basisξ ,ξ ,...)asfollows: 1 2 ((r +1)/2n−1)ξ ifk = r2n+1+s,0 ≤ s < 2n+1 k+1 t (ξ ) = n k (ξk+1 ifk ≥ 4n.