A characterization of semiprojectivity for commutative C*-algebras

A CHARACTERIZA TION OF SEMIPROJECTIVITY F OR COMMUT A TIVE C ∗ -ALGEBRAS. ADAM P . W . SØRENSEN AND HANNES THIEL A B S T R A C T . Given a compact, metric space X , we show that the commutative C ∗ - algebra C ( X ) is semiproj ective if and only if X is an absolute neighbor hood retract of dimension at most one. This con firms a conjecture of Blackadar . Generalizing to the non-unital setting, we derive a chara cterization of semiproj ec- tivity for separable, commutative C ∗ -algebras. As fur ther application of our find- ings we ve rify two conjectures of Loring and Blackad ar in the commutative case, and we give a partial answer to the question, when a commutative C ∗ -algebra is weakly (semi-)projective. 1. I N T R O D U C T I O N Shape theor y is a machine ry that allows to focus on the global properties of a space by abstracting fr om its local be ha vior . This is done by approximating the space by a system of nicer spaces, and then studying this approximat ing system instead of the original space. After this idea was successfully applied to commutative spaces, it was first introduced to the noncommutative world by Effr os and Kaminker , [EK86]. Soon after , noncommutative shape theory was developed to its modern form by Blackadar , [Bla85]. In classical shape theory one appr oximates a space by absolute neighbor hoo d re- tracts (ANR s). In the noncommutative world, the role of these nice spaces is played by the semip rojective C ∗ -algebras. It is however not true that every (compact) A NR X gives a semipr ojectiv e C ∗ -algebra C ( X ) . In fact, already the two-disc D 2 is a coun- ter example (see 3 .2 and 3.3). This hints to a possible problem in noncommutative shape theory: While it easy to show that there are enough A NRs to approx imate ev- ery compact metric space, the analogue for C ∗ -algebra is not obvious at all. In fact it is still an open pr oblem whether every separable C ∗ -algebra can be written as an inductive limit of semiproj ective C ∗ -algebras. Some pr ogr ess on this pr oblem was r ecently made by Loring and Shulman, [LS10]. Hence, it is important to know which C ∗ -algebras are semiprojectiv e. And a l- though semipro jectivity was modeled on A NRs, the first large class of C ∗ -algebras Date : October 4, 2 018. 2000 Mathematics Subject Classification. Primary 46 L05, 54 C55, 55M 15, 54F5 0 ; Sec ondary 4 6 L80, 46M10 , 54F15, 54D35, 54C56, 55 P55 . Key words and ph rases. C ∗ -algebras, n on-commutative shape theory , semipro jectivity , absolute neighborho od retracts. This research was supported b y the Danish National Research Foundation throug h the Centre for Symmetry and Deformation. The second named author was partially supported by the Marie Curie Research T ra ining Network EU-NC G . 1 2 ADAM P . W . SØRENSEN AND HANNES THIEL shown to be semipro jective were the highly noncommutative Cuntz-Krieger alge- bras, see [Bla85]. Since then, these results have been extended to cover all UCT Kir chber g algebras with finitely generated K-theory and fr ee K 1 -gr oup, see [Szy02] and [Spi09], and it is conjectur ed that in fact all Kir chber g algebras with finitely gen- erated K-theory ar e semiprojectiv e. Y et, the following natural question remained unanswered: Question 1.1. Which commutative C ∗ -algebras are semiproj ective? An imp ortant partial answer was obtained by Loring, [Lor97, Propositio n 16.2. 1 , p.125], who showed that all one-dimensional CW - complexes give rise to semipr o- jective C ∗ -algebras. In [ELP98] this was extended to the class of one-dimensional NCCW -co mplexes. In a nother directio n, Chigogidze and Dranishnikov recently gave a characteriza- tion of the commutative C ∗ -algebras that are project ive: They show in [CD10, The- or em 4.3 ] that C ( X ) is projectiv e in S 1 (the category of unital, separable C ∗ -algebras with unital ∗ -homomorphisms) if and only if X is an AR and dim( X ) ≤ 1 . Inspired by their results we obtain the following answer to question 1.1: Theorem 1.2. Let X be a compact, m etric space. Th e n the following are equivalent: (I) C ( X ) is semip rojective. (II) X is an ANR and dim( X ) ≤ 1 . This confirms a conjectur e of Blackadar , [Bla06, II.8.3.8, p.163 ]. W e proceed as fol- lows: C O N T E N T S 1. Intr oduction 1 2. Pr eliminaries 4 3. One implication of the main theor em: Necessity 9 4. Str uctur e of compact, one-dimensional AN R s 13 5. The other implication of the main theorem: Suffi ciency 19 6. Applications 23 Acknowledgments 29 References 29 In section 2 (Pr eliminaries), we recall the basic concepts of commutative and noncom- mutative shape theory , in particular the notion of an ANR and of semiproject ivity . In sectio n 3 (Necessity), we show the implication ”(I) ⇒ (II)” of our main result 1. 2 . The ide a is to use the topological propert ies of higher dime nsional spaces, to show that if C ( X ) was semipr ojective and X an ANR of d ime nsion at least 2 then we could solve a lifting pro blem known to be unsolvable. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 3 In sectio n 4 we study the structur e of compact, one-dimensional ANRs. W e charac- terize when a one-dimensional Peano continuum X is an ANR, see 4.12. A s it turns out, one criterium is that X contains a finite subgraph that contains all homotopy in- formation, a (homotopy) cor e, see 4.10. This is also equivalent to K ∗ ( X ) being finitely generated, which is a recurring property in connection with semiproj ectivity . The main result of this section is theor em 4.17 which d e scribes the internal struc- tur e of a compact, one-dimensional ANR X . Starting with the homotopy cor e Y 1 ⊂ X ther e is an incr easing sequence of subgraphs Y 1 ⊂ Y 2 ⊂ . . . ⊂ X that exhaust X , and such that Y k +1 is obtained fr om Y k by simply attaching a line segment at one end to a point in Y k . This generalizes the classical structur e theor em for dendrites (which are pr ecisely the contractible , compact, one-dimensional ANRs). In section 5 (Suffi ciency) we show the implication ”(II) ⇒ (I)” of 1.2. Using the struc- tur e theor em 4.17 for X , we obtain subgr aphs Y k ⊂ X such that X ∼ = lim ← − Y k . The first graph Y 1 contains all K-theory information, and the subsequent graphs are ob- tained by attaching line segments. Dualizing, we can write C ( X ) as an inductive limit, C ( X ) = lim − → C ( Y k ) . Since the maps Y k +1 → Y k ar e r etractions, the dual bonding morphisms C ( Y k ) → C ( Y k +1 ) are accessible for lifting pr oblems. The main r esult of this section is 5.3. Given a lifting problem C ( X ) → C / S k J k and an initial lift from C ( Y 1 ) to some C /J l , there exists a lifting from any C ( Y k ) to the same height, and finally a lift from the inductive limit C ( X ) to C /J l . This idea is central in [CD10], but it has also bee n used be for e, for instance by Blackadar in order to prov e that the Cuntz algebra O ∞ is semipro jective. W e note that some form of inductive limit argument seems necessary for lifting a n infinite number of generators. W e also wish to point out that Chigogidze and Dranishnikov only needed semiprojectiv ity , and not pro jectivity , in many steps of their proofs . The pr oof ”(II) ⇒ (I)” follows fr om 5.3 if we can find an initial lift from C ( Y 1 ) . For this we use Loring’s deep r esult, [Lor97], which says that C ( Y ) is semipro jective for every finite graph Y . W e also need Loring’s r esult to write the algebras C ( Y k ) as universal C ∗ -algebras. T o summarize, the pr oof pro ceeds in two steps. First, we constr uct an initial lift C ( Y 1 ) → C /J l fr om the homotopy cor e. This will lift all K-theory information of X . But once the K-theory information is lifted, we do not need to ”sink to a lower level”. In section 6 we give applications of our main result 1.2. First, we analyze the structur e of non-compact, one-dime nsional ANRs. W e give a characterization when the one- point compactific ation of such spaces is again an ANR, see 6. 1 . This is motivat ed by the fact that a C ∗ -algebra A is semipr ojective i f a nd only if its minimal unital- ization e A is semiprojectiv e. For commutative C ∗ -algebras, the minimal unitalization corr esponds to taking the one-point compactificat ion of the underlying commutative space. U sing the characterizatio n of semiprojectiv ity for unital, separable, commu- tative C ∗ -algebras given in 1.2, we derive a characterizatio n of semiprojectivit y for non-unital, separable, commutative C ∗ -algebras, see 6. 2 . 4 ADAM P . W . SØRENSEN AND HANNES THIEL In 6.1 we also note that the one-point compactification of the considered spaces is an ANR if and only e very finite-point compactification is a n A NR. This allows us to study short exact sequences 0 / / I / / A / / F / / 0 with F finite-dimensional. It was conjectured by Loring and also by Blackadar , [Bla04, Conjectur e 4.5], that in this situation A is semiproject ive if and only if I is. One implication was r e ce ntly pr oven by Dominic Enders, [End11], who showed that semipr ojectivity passes to ideals when the quotient is finite-dimensional. The con- verse implication is in general not even known for F = C . However , in 6.3 we verify this conjectur e under the add itional assumption that A is commutative. Then, we will study the semiprojectiv ity of C ∗ -algebras of the form C 0 ( X , M k ) . W e derive in 6.9 that for a separable, commutative C ∗ -algebra A , the algebra A ⊗ M k is semipr ojective if and only if A i s semiproj ective. Again, this q uestion can be asked in general. It is known that semiprojectivit y of A implies that A ⊗ M k is semiproject ive as well, see [Bla85, Cor ollary 2.28] and [Lor97, Thoer e m 1 4.2.2, p. 1 1 0]. For the converse, it is known that semiproj ectivity passes to full corners, [Bla85 , Proposit ion 2.27]. It was conject ur ed by Blackadar , [Bla04, Conjectur e 4.4], that the same holds for full heredit ary sub- C ∗ -algebras. Note that A always is a full her editary sub- C ∗ -algebra of A ⊗ M k . Thus, we verify the conjectur e for commutative C ∗ -algebras. As a final application, we consider the following variant of question 1.1: When is a commutative C ∗ -algebra weakly (semi-)pr ojective? In order to study this problem, we analyze the str uctur e of one-dimensional app roximative a bsolute (neighborho od) r etracts, abbreviated AA (N)R . In 6.1 5 we show that such spaces are appr oximated fr om within by finite trees (finite graphs). Since finite trees (finite graphs) give (semi- )pr ojective C ∗ -algebras, we derive in 6. 1 6 that C ( X ) is wea kly (semi-)pr ojective in S 1 if X is a one-dimensional A A(N)R. Summarizing our r esults, 1.2 and 6.1 6, and the result of Chigogidze and Dranish- nikov , [CD10, Theorem 4.3], we get: Theorem 1.3. Let X be a compact, m etric space with dim ( X ) ≤ 1 . Then: (1) C ( X ) is projective in S 1 ⇔ X is an AR (2) C ( X ) is weakly projective in S 1 ⇔ X is an AAR (3) C ( X ) is semiprojective S 1 ⇔ X is an ANR (4) C ( X ) is weakly sem iprojective S 1 ⇔ X is an AANR Moreover , C ( X ) projective or semiprojective already im p lies dim( X ) ≤ 1 . 2. P R E L I M I N A R I E S By A, B , C , D we mostly denote C ∗ -algebras, usually assumed to be separable her e, and by a morphism between C ∗ -algebras we understand a ∗ -homomorphism. By a n ideal in a C ∗ -algebra we mean a closed, two-s ided ideal. If A is a C ∗ -algebra, then we denote by e A its minimal unitalization, and by A + the forced unitalization. Thus, if A is unital, then e A = A and A + ∼ = A ⊕ C . W e use the symbol ≃ to denote homotopy equivalence. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 5 By a map between two topological spaces we mean a continuous map. Given ε > 0 and subsets F , G ⊂ X of a metric space, we say F is ε -contained in G , denoted by F ⊂ ε G , if for every x ∈ F there exists some y ∈ G such that d X ( x, y ) < ε . Given two maps ϕ, ψ : X → Y between metric spaces and a subset F ⊂ X we say ” ϕ and ψ agr ee on F ”, denoted ϕ = F ψ , if ϕ ( x ) = ψ ( x ) for all x ∈ F . If mor eover ε > 0 is given, then we say ” ϕ and ψ agr ee up to ε ”, denoted ϕ = ε ψ , if d Y ( ϕ ( x ) , ψ ( x )) < ε for all x ∈ X (for normed spaces, this is usually denoted by k ϕ − ψ k ∞ < ε ). W e say ” ϕ and ψ agree on F up to ε ” , denoted ϕ = F ε ψ , if d Y ( ϕ ( x ) , ψ ( x )) < ε for all x ∈ F . 2.1 ((Approximative) absolute (neighbor hood) retr acts) . A metric space X is an (ap- proximative) absolute retract , abbreviated by (A)AR , if for all pairs 1 ( Y , Z ) of metric spaces and maps f : Z → X (and ε > 0 ) there exists a map g : Z → X such that f = g ◦ ι (resp. f = ε g ◦ ι ), where ι : Z ֒ → Y is the inclusion map. This means that the following diagram can be completed to commute (up to ε ): Y g ~ ~ X Z f o o ?  ι O O A metric spa ce X is an (approximative) ab solute ne ighborhood r e tract , abbrevi- ated by (A)ANR , if for all pa irs ( Y , Z ) of metric spaces and maps f : Z → X (and ε > 0 ) there exists a neighbor hood V of Z and a map g : V → X such that f = g ◦ ι (r esp. f = ε g ◦ ι ) where ι : Z ֒ → V is the inclusion map. This means that the following diagram can be completed to commute (up to ε ): Y V ?  O O g ~ ~ X Z f o o ?  ι O O For details about ARs an d ANRs see [Bor67]. W e will only consider compact A ARs and AA NRs in this pa per , and the reader is referr ed to [Cla7 1] for more details. W e consider shape theory for separable C ∗ -algebras a s developed by Blackadar , [Bla85]. Let us shortly recall the main notions and results: 2.2 ((W eakly) (semi-)project ive C ∗ -algebras) . Let D be a subcatego ry of the category of C ∗ -algebras, closed under q uotients 2 . A D -morphism ϕ : A → B is called (weakly) projective in D if for any C ∗ -algebra C in D and D -morphism σ : B → C / J to some 1 A ( Y , Z ) pair of spaces is simply a spa ce Y with a c losed subspa c e Z ⊂ Y . 2 This mea ns the following: Assume B is a quotient C ∗ -algebra of A with quotient morphism π : A → B . If A ∈ D , then B ∈ D and π is a D -morphism. 6 ADAM P . W . SØRENSEN AND HANNES THIEL quotient (and finite subset F ⊂ A , ε > 0 ), there exists a D -morphism ¯ σ : A → C such that π ◦ ¯ σ = σ ◦ ϕ (resp. π ◦ ¯ σ = F ε σ ◦ ϕ ), where π : C → C / J is the quotient morphism. This means that the following diagram can be completed to commute (up to ε on F ): C π   A ϕ / / ¯ σ 7 7 B σ / / C / J A C ∗ -algebra A is called (weakly) projective in D if the identity morphism id A : A → A is (weakly) proj ective. A D -morphism ϕ : A → B is called (weakly) semiprojective in D if for a ny C ∗ - algebra C in D and increasing sequence of id e als J 1 ✁ J 2 ✁ . . . ✁ C and D -morphism σ : B → C / S k J k (and finite subset F ⊂ A , ε > 0 ), there exists a n index k and a D -morphism ¯ σ : A → C /J k such that π k ◦ ¯ σ = σ ◦ ϕ (resp. π k ◦ ¯ σ = F ε σ ◦ ϕ ), where π k : C /J k → C / S k J k is the quotient morphism. This means that the following dia- gram can be completed to commute (up to ε on F ): C   C / J k π   A ϕ / / ψ 6 6 B σ / / C / S k J k A C ∗ -algebra A is called (weakly) sem iprojective in D if the identity morphism id A : A → A is (weakly) semiproject ive. It is well known that if A is separable then A is semipr ojective in the category of all C ∗ -algebras if and only if it is in the categor y of separable C ∗ -algebras. If D is the category S of all separable C ∗ -algebras (with all ∗ -homomorphisms), then one dr ops the refer ence to D and simply speaks of (wea kly) (semi-)pr ojective C ∗ -algebras. Besides S one often considers the category S 1 of all unital separable C ∗ -algebras with unital ∗ -homomorphisms as morphisms. A projectiv e C ∗ -algebra cannot have a unit. For a (separable) C ∗ -algebras A we get fr om [Bla85, Pr oposition 2.5], see also [Lor97, Theorem 10.1 . 9 , p.75], that the follow- ing are equivalent: (1) A is pro jective (2) e A is pro jective in S 1 The situation for semipro jectivity is even easier . A unital C ∗ -algebra is semipro - jective if and only if it is semip rojective in S 1 . Further , for a separable C ∗ -algebra A we get from [Bla8 5, Corollary 2.16], see also [Lor97, Theorem 14.1. 7 , p. 1 08], that the following are equivalent: (1) A is semiproj ective (2) e A is semiproj ective SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 7 (3) e A is semiproj ective in S 1 2.3 (Connection between (approx imative) a bsolute (neighbor hood) retract s and (weakly) (semi-)pr ojective C ∗ -algebras) . Let S C be the full subcategory of S consisting of (sep- arable) commutative C ∗ -algebras, and similarly let S C 1 be the full subcategory of S 1 consisting of (separable, unital) commutative C ∗ -algebras. In general, for a C ∗ -algebra it is easier to be (weakly) (semi-)pr ojective in a smaller full subcategory , since there are fewer quotients to map into. I n particular , if a com- mutative C ∗ -algebra is (weakly) (semi-)proj ective, then it will be (weakly) (semi- )pr ojective with respect to S C . If one compar es the definitions carefully , then one gets the following equivalences for a c ompact , metric space X (see [Bla85, Proposi- tion 2.11]): (1) C ( X ) is pro jective in S C 1 ⇔ X is an AR (2) C ( X ) is weakly p rojective in S C 1 ⇔ X is an AAR (3) C ( X ) is semipro jective in S C 1 ⇔ X is an ANR (4) C ( X ) is weakly semip r ojective in S C 1 ⇔ X is an AANR Thus, the notion of (weak) (semi-)pr ojectively is a translation of the concept of an (appr oximate) absolute (neighbor hood) retract to the world of noncommutative spaces. Let us clearly state a point which is used in the pro of of the main theorem: If C ( X ) is (weakly) (semi-)pr ojective in S C 1 , then X is an (ap pr oximate) absolute (neighbor hood) retract. As we will see, the converse is not tr ue in general. W e need an assumption on the dimension of X . 2.4 (Covering dime nsion) . 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 of X such that ord( V ) ≤ n + 1 . Her e ord( V ) is the largest number k such that there exists some point x ∈ X that is contained in k differ e nt elements of V . T o an open cover V one can naturally assign an abstract simplicial complex 3 N ( V ) , called the ne rve of the covering. I t 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 ) corr esponds to a choice of n dif fer ent el e ments in the cover that have non-empty intersection. Given an abstract simplicial complex C , one can naturally a ssociate to it a space | C | , calle d the geometric realization of C . The space | C | is a polyhedron, in particular it is a CW -complex. 3 An a b stra ct simplicial complex over a set S is a fa mily C of finite subsets of S such that X ⊂ Y ∈ C implies X ∈ C . A n element X ∈ C with n + 1 elements is ca lled an n -simplex (of the a bstract simplicial complex). 8 ADAM P . W . SØRENSEN AND HANNES THIEL Note that ord( V ) ≤ n + 1 if and only if the nerve N ( V ) of the covering V is an abstract simplicial set of dimension 4 ≤ n , or equivale n tly the geometric realization of |N ( V ) | is a polyhedr on of covering dimension 5 ≤ n . Let U be a finite open covering of a space X , and { e u : U ∈ U } a partition of unity that is subordinate to U . This naturally defines a map α : X → |N ( U ) | sending a point x ∈ X to the (unique) point α ( x ) ∈ |N ( U ) | that has ”coordinat es” e U ( x ) . By lo cdim( X ) we de note the local covering dime nsion of a space X . By definition lo cdim( X ) ≤ n if every point x ∈ X has a closed neighbor hood D such that dim ( D ) ≤ n . If X is paracompact (e . g. if it is compact, or locally compact and σ -compact), then lo cdim( X ) = dim( X ) . See [Nag70] for mor e details on nerves, polyhedra and the (local) covering dimen- sion of a space. A particularly nice class of one-dimensional 6 spaces are the so-called dendrites. Be- for e we look at them, let us recall some notions from continuum theory . A good r eference is Nadler ’s book, [Nad92]. A continuum is a compact, connected, metric space, and a generalized continuum is a locally compact, connected, metric space. A Peano continuum is a locally con- nected continuum, and a gener a lize d Pea n o continuum is a locally connected gen- eralized continuum. By a finite graph we mean a graph with finitely many vertices and edges, or equivalently a compact, one-dimensional C W -complex. By a finite tre e we mean a contractible finite graph. 2.5 (Dendrites) . A dendrite is a Peano continuum that does not contains a simple closed curve (i.e., ther e is no embedding of the cir cle S 1 into it). There are many other characterizations of a d endrite. W e collect a few and we will use them without further mentioning. Let X be a Peano continuum. Then X is a dendrite if and only if one (or equiva- lently all) of the following conditions holds: (1) X is one-dimensional and contractible (2) X is tree-like 7 . (3) X is dendritic 8 4 The dimension of an a bstract simplicial set is the largest integer k such that it contains a k -simplex. 5 The covering dimension of polyhedra, or more generally CW -complexes, is easily understood. These space s are successively build b y attaching cells of higher and higher dimension. The ( covering) dimension of a CW - complex is simply the highest dimension of a cell that was a ttached when building the complex. 6 W e say a space is one-dimensional if dim( X ) ≤ 1 . So, a lthough it sounds weird, a one-dimensional space can a lso be zero-dimensional. It would probably be more precise to spea k of ”at most one- dimensional” spa ce, however the usage of the term ”one-dimensional space” is well established. 7 A (compact, metric) space X is tree-like, if for every ε > 0 there exists a finite tree T and a map f : X → T onto T such that diam( f − 1 ( y )) < ε for a ll y ∈ T . 8 A space X is called dendritic, if any two points of X can be separated by the omission of a third point SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 9 (4) X is hereditarily unicoherent 9 . For more information about dendrites see [Na d92, Chapter 10 ], [Lel76], [CC6 0]. 3. O N E I M P L I C AT I O N O F T H E M A I N T H E O R E M : N E C E S S I T Y Proposition 3.1. Let C ( X ) be a unital, separable C ∗ -algebra that i s sem iprojective. T h en X is a compact ANR with dim( X ) ≤ 1 . Proo f. A ssume such a C ( X ) is given. The n X is a compact, metric space. As noted in 2.3, semipr ojectivity (in S 1 ) implie s semipro jectivity in the full subcategory S C 1 and this means exactly that X is a (compact) ANR. W e are le ft with showing dim( X ) ≤ 1 . Assume otherwise, i.e., assume dim( X ) ≥ 2 . Since X is paracompact, we have lo cdim( X ) = dim( X ) ≥ 2 . This means ther e exists x 0 ∈ X such that dim( D ) ≥ 2 for each closed neighbor hood D of x 0 . For each k consider D k := { y ∈ X : d ( y , x 0 ) ≤ 1 /k } . This defines a d e cr easing sequence of closed neighborho ods around x 0 with dim( D k ) ≥ 2 . It was noted in [CD10, Pr oposition 3.1] that a Pea n o space of dimension at least 2 ad mits a topological embedding 10 of S 1 . Indeed, a Peano space that contains no simple ar c (i.e . in which S 1 cannot be embedded) is a dendrite, and therefor e at most one-dimensional. It follows that ther e are embeddings ϕ k : S 1 ֒ → D k ⊂ X . Putting these together we get a map (not necessarily an embedding) ϕ : Y → X where Y is the space of ”smaller and smaller circles”: Y = { (0 , 0) } ∪ [ k ≥ 1 S ((1 / 2 k , 0) , 1 / ( 4 · 2 k )) ⊂ R 2 , wher e S ( x, r ) is the circle of radius r ar ound the point x . W e define ϕ as ϕ k on the cir cle S ((1 /k , 0) , 1 / 3 k ) . The map ϕ : Y → X induces a morphism ϕ ∗ : C ( X ) → C ( Y ) . Next we construc t a C ∗ -algebra B with a nested sequence of ideals J k ✁ B , such that C ( Y ) = B / S k J k and ϕ ∗ : C ( X ) → C ( Y ) cannot be lifted to some B /J k . Let T be the T oeplitz algebra a nd let T 1 , T 2 , . . . be a seque nce of copies of the T oeplitz a lgebra, and set: B := ( M k ∈ N T k ) + = { ( b 1 , b 2 , . . . ) ∈ Y k ≥ 1 T such that ( b k ) k conver ges to a scalar multiple of 1 T } . 9 A continuum X is ca lled unicoherent if f or ea c h two subcontinua Y 1 , Y 2 ⊂ X with X = Y 1 ∪ Y 2 the intersection Y 1 ∩ Y 2 is a continuum ( i.e. connected). A continuum is called hereditarily unicoherent if all its subcontinua are unicoherent. 10 If 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 . W e usually c onsider a topologically e mbedded space as a subset with the subset topology . 10 ADAM P . W . SØRENSEN AND HANNES THIEL The algebras T k come with ideals K k ✁ T k (each K k a copy of the algebra of compact operators K ). Define ideals J k ✁ B a s follows: J k := K 1 ⊕ . . . ⊕ K k ⊕ 0 ⊕ 0 ⊕ . . . = { ( b 1 , . . . , b k , 0 , 0 , . . . ) ∈ B : b i ∈ K i ✁ T i } . Note B /J k = C ( S 1 ) ⊕ . . . ( k ) ⊕ C ( S 1 ) ⊕ ( L l ≥ k +1 T l ) + ( k summands of C ( S 1 ) ). Also J k ⊂ J k +1 and J := S k J k = L k ∈ N K k and B /J = ( L l ≥ 1 C ( S 1 )) + ∼ = C ( Y ) . The semipro jectivity of C ( X ) gives a lift of ϕ ∗ : C ( X ) → C ( Y ) = B /J to some B /J k . Consider the pr ojection ρ k +1 : B / J k → T k +1 onto the (k+1)-th coordinate, and similarly  k +1 : B / J → C ( S 1 ) . The composition C ( X ) → C ( Y ) ∼ = B /J → C ( S 1 ) is ϕ ∗ k +1 , the morphism in d uced by the inclusion ϕ k +1 : S 1 ֒ → X . Note that ϕ ∗ k +1 is surjec- tive since ϕ k +1 is an inclusion. The situation is viewed in the following commutative diagram: B /J k   ρ k +1 / / T k +1   C ( X ) ϕ ∗ k +1 2 2 ϕ ∗ / / 5 5 k k k k k k k k k k k k k k k k k k C ( Y ) ∼ = / / B /J  k +1 / / C ( S 1 ) The unitary id S 1 ∈ C ( S 1 ) lifts under ϕ ∗ k +1 to a normal eleme nt in C ( X ) , but it does not lift to a normal ele ment in T k +1 . This is a contradiction, a nd our assumption dim( X ) ≥ 2 must be wrong.  It is well known that C ( D 2 ) , the C ∗ -algebra of continuous functions on the two- dimensional disc D 2 = { ( x, y ) ∈ R 2 : x 2 + y 2 ≤ 1 } , is not wea kly semiproject ive. For completeness we include the argument which is essentially taken from Loring [Lor97, 17. 1 , p.131], see also [Lor95]. Proposition 3.2. C ( D 2 ) is not weakly semiprojective. Proo f. The ∗ -homomorphisms fr om C ( D 2 ) to a C ∗ -algebra A are in natural one-one corr espondence with normal contract ions in A . Thus, statements about (weak) (semi- )pr ojectivity of C ( D 2 ) corr e spond to statements about the (approx imate) liftability of normal elements. For example, that C ( D 2 ) is projectiv e would corr espond to the (wr ong) statement that normal elem e nts lift fr om quotient C ∗ -algebras. T o disprove weak semip rojectivit y of C ( D 2 ) one uses a constr uction of operators that are approx- imately normal but do not lift in the requir e d way due to a n index obstruct ion. Mor e precisely , d e fine weighted shift operators t n on the separable Hilbe rt space l 2 (with basis ξ 1 , ξ 2 , . . . ) as follows: t n ( ξ k ) = ( (( r + 1) / 2 n − 1 ) ξ k +1 if k = r 2 n +1 + s, 0 ≤ s < 2 n +1 ξ k +1 if k ≥ 4 n . SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 11 Each t n is a finite-rank perturbation of the unilateral shift. Ther efor e the t n lie in the T oeplitz algebra T and have index − 1 . The const ruct ion of t n is made so that k t ∗ n t n − t n t ∗ n k = 1 / 2 n − 1 . Consider the C ∗ -algebra B = Q N T / L N T . The sequence ( t 1 , t 2 , . . . ) defines an element in Q N T . Let x = [( t 1 , t 2 , . . . )] ∈ B be the equivalence class in B . Then x is a normal eleme nt of B , and we let ϕ : C ( D 2 ) → B be the corresponding morphism. W e have the following lifting problem: Q k ≥ N T k π   C ( D 2 ) ϕ / / ¯ ϕ 8 8 Q N T / L N T Assume C ( D 2 ) is weakly semipr ojective. Then the lifting prob lem can be solved, and ¯ ϕ defines a normal ele ment y = ( y N , y N +1 , . . . ) in Q k ≥ N T k . But the inde x of each y l is zero, while the index of each t l is − 1 , so that the norm-distance between y l and t l is at least one. Therefor e the distance of π ( y ) and x is at least one, a contradiction. Thus, C ( D 2 ) is not weakly semipro jective.  Remark 3.3 (Spaces cont aining a two-dimensional disc) . W e have seen above that C ( D 2 ) is not weakly semiprojectiv e. Even mor e is tr ue: Whenever a (compact, met- ric) space X contains a two-dimensional disc, then C ( X ) is not wea kly semiprojec- tive. This was noted by Loring, [Lor09a]. For completeness we include the a r gument: Let D 2 ⊂ X be a two-dimensional disc with inclusion ma p i : D 2 → X . Since D 2 is an absolute r etract, ther e exists a retraction r : X → D , i.e., r ◦ i = id : D 2 → D 2 . Pass- ing to C ∗ -algebras, we get induced momorphisms i ∗ : C ( X ) → C ( D 2 ) , r ∗ : C ( D 2 ) → C ( X ) such that i ∗ ◦ r ∗ is the identity on C ( D 2 ) . Assume C ( X ) is weakly semiproj ec- tive. Then any lifting p roblem for C ( D 2 ) could be solved as follows: Using the weak semipr ojectivity of C ( X ) , the morphism ϕ ◦ i ∗ can be lifted. Then σ ◦ r ∗ is a lift for ϕ = ϕ ◦ i ∗ ◦ r ∗ . The situation is viewe d in the following commutative dia gram: Q k ≥ N B k π   C ( D 2 ) r ∗ / / C ( X ) i ∗ / / σ 4 4 C ( D 2 ) ϕ / / Q k ≥ 1 B k / L k ≥ 1 B k This gives a contradiction, as we have shown above that C ( D 2 ) is not weakly semipro- jective. However , that a space does not contain a two-dimensional disc is no guarantee that it has dimension at most one. These kind of que stions are studied in continuum theory , and Bing, [Bin51], gave examples of spaces of arbitrarily high dimension that 12 ADAM P . W . SØRENSEN AND HANNES THIEL ar e hereditar ily indecomposable 11 , in pa rticular they d o not contain an arc or a copy of D 2 . These pathologies cannot occur if we restrict to ”nicer ” spaces. For example, if a CW -complex d oes not contain a two-dimensional disc, then it has dimension at most one. What about ANRs? Bing and Borsuk, [BB64], gave an example of a thr ee-dimensional A R that does not contain a copy of D 2 . The question for four- dimensional AR’s is still open, i.e., it is unknown whether there exist high-dimensional AR’s (or just ANRs) that do not contain a copy of D 2 . The point we want to ma ke clear is the following: T o prove that an ANR is one- dimensional it is not enough to prove that it d oes not contain a copy of D 2 . Remark 3 .4 (Spaces contained in ANRs of dimension ≥ 2 ) . Although an ANR X of with dim( X ) ≥ 2 might not contain a disc, one can show that it must contain (a copy of) one of the following thr ee spaces: Space 1: The space Y 1 of distinct ”smaller and smaller cir cles” as consider ed in the pr oof of 3.1, i.e ., Y 1 = { (0 , 0) } ∪ S k ≥ 1 S ((1 / 2 k , 0) , 1 / ( 4 · 2 k )) ⊂ R 2 . Space 2: The Hawaiian earrings, i.e., Y 2 = S k ≥ 1 S ((1 / 2 k , 0) , 1 / 2 k ) ⊂ R 2 . Space 3: A variant of the Hawaiian earrings, where the cir cles do not just intersect in one point, but have a segment in common. I t is homeomorphic to: Y 3 = { ( x, x ) , ( x, − x ) : x ∈ [0 , 1 ] } ∪ S k ≥ 1 { 1 /k } × [ − 1 /k , 1 /k ] ⊂ R 2 . T o prove this, one uses the same idea as in the proo f of 3.1: If dim( X ) ≥ 2 , then ther e exists a point x 0 wher e the local dime nsion is at least two. Then one can embed into X a sequence of circles that get smaller and smaller and conver ge to x 0 . Note that the cir cles may intersect or overlap. B y passing to subspaces, we can get rid of ”unnecessary” intersections and overlappings, and finally there are only three qual- itatively differ ent ways a bunch of ”smaller and smaller ” can look like. W e skip the details. Note that none of the three spaces Y 1 , Y 2 , Y 3 ar e semiproject ive. Further , no (com- pact, metric) space X that contains a copy of Y 1 , Y 2 or Y 3 can be semipr ojective. One uses a similar ar gument as for an e mbe dded D 2 . Assume for some k ther e is an in- clusion i : Y k ֒ → X . Since Y k is not an AR, there will in general be no r etraction onto it. Instead, choose an embedding f : Y k ֒ → D 2 . This map can be extended a map ˜ f : X → D 2 on all of X since D 2 is an A R. 11 A continuum (i.e. compact, connected, metric space) is called decomposable if it can be written as the union of two proper subcontinua. Note that the union is not assumed to be disjoint. For example the interva l [0 , 1] is decomposable as it can be written as the union of [0 , 1 / 2] and [1 / 2 , 1] . A continuum is called hereditar ily indecomposable if none of its subcontinua is decomposable. See [Nad92] for f urther information. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 13 (a) Space Y 1 (b) Space Y 2 (c) Space Y 3 F I G U R E 1 . Spaces contained in high-dimensional AN Rs D 2 Y k i / / f O O X ˜ f ` ` If C ( X ) is semiprojective, then any lifting pro blem as shown in the diagram be low can be solved. However , using T oeplitz algebras as in 3 .1 we see that the morphism f ∗ = i ∗ ◦ ˜ f ∗ : C ( D 2 ) → C ( Y k ) is not semiproject ive. B /J N π   C ( D 2 ) ˜ f ∗ / / C ( X ) i ∗ / / σ 4 4 C ( Y k ) ϕ / / B / S k ≥ 1 J k Finally let us note that the C ∗ -algebras C ( Y 1 ) , C ( Y 2 ) and C ( Y 3 ) are weakly semipro- jective. 4. S T R U C T U R E O F C O M PA C T , O N E - D I M E N S I O N A L A N R S In this se ction we prove structural theor ems about compact, one-dimensional abso- lute neighborhoo d r etracts (ANR s). The r esults are used in the next section to show 14 ADAM P . W . SØRENSEN AND HANNES THIEL that the C ∗ -algebra of continuous functions on such a space is semip rojective. In sec- tion 6 we will study the structur e on non-compact, one-dimensional ANR s. W e start with some preparatory lemmas. By π ( X , x 0 ) we denote the fundamental group of X based at x 0 ∈ X . Statements about the fundame ntal gr oup often do not depend on the basepoint, and then we will simply write π ( X ) to mean that any (fixed) basepoint may be chosen. Lemma 4.1. Let X be a Hausdorff space . Assume X has a sim p ly connected c overing space. Then ever y path in X is homotopic (relative endpoints) to a path that is pie cewise arc. Proo f. Le t p : e X → X be a simply connected, Hausdorf f covering space. Let α : [0 , 1] → X be a p a th, and let e α : [0 , 1] → e X be a lift. Then the image of e α is a Pea no continuum (i.e., a compact, connected, locally connected, metric space), and is therefor e arcwise connected. C hoose any arc β : [0 , 1] → e X fro m e α (0) to e α (1) . The arc may of course be chosen within the image of e α . Since e X is simply connected, the paths e α and β are homotopic (relative endpoints). Then α = p ◦ e α a nd p ◦ β are homotopic paths in X . Since p is locally a homeomorphism, p ◦ β is piecewise arc, i.e., there exists a finite subdevision 0 = t 0 < t 1 < . . . < t N = 1 such that each r estrictio n p ◦ β | [ t j ,t j +1 ] is an ar c.  Lemma 4.2. Let X be a Hausdorff space, and x 0 ∈ X . Assume X has a sim ply connected covering spac e , and π ( X , x 0 ) is fi nitel y generated. Th en there e xists a finite graph Y ⊂ X with x 0 ∈ Y such that π ( Y , x 0 ) → π ( X , x 0 ) is surjective . Proo f. C hoose a set of generators g 1 , . . . , g k for π ( X , x 0 ) , repr esented by loops α 1 , . . . , α k : S 1 → X . Fr om the above lemma we can homotope e a ch α j to a loop β j that is piecewise ar c. Then the image of each β j in X is a finite graph. Consequently , also the union Y := S j im( β j ) is a finite graph (containing x 0 ). By construct ion e ach g j lies in the image of the natural map π ( Y , x 0 ) → π ( X , y 0 ) . Therefor e this map is surjective.  Remark 4.3. Let X be a connected, locally pathwise connected space. Then X has a simply connected covering space (also called universal cover) if and only if X is semilocally simply connected 12 (s.l.s.c.), see [Bre93, Theorem III.8.4, p.155]. Proposition 4.4. Let X be a s .l .s.c. Peano continuum and x 0 ∈ X . Then ther e exists a finite graph Y ⊂ X with x 0 ∈ Y such that π ( Y , x 0 ) → π ( X , x 0 ) is surj e ctive. Proo f. Peano continua are connected and locally pathwise connected. Therefor e, by the above remark 4 .3, X has a simply connected covering space. By [CC06, Lemma 7.7], π ( X , x 0 ) is finitely generated (even finitely pr esented). Now we may apply the above lemma 4.2.  12 A space X is called semilocally simply connected (sometimes a lso called locally relatively simply connected) if for each x 0 ∈ X there exists a neighbo rhood U of x 0 such that π ( U, x o ) → π ( X , x 0 ) is zero. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 15 Remark 4.5. The fundamental gr oup of a finite graph is finitely generated (f.g.), free and abelian. Thus, the above map π ( Y , x 0 ) → π ( X , x 0 ) will in general not be injective. Even if π ( X , x 0 ) is f.g., free and abelia n, the construct ed map might not be injective. The r eason is simply that the constru cted graph could contain ”unnecessary” loops (e.g. consider a circle embedded into a disc). However , by rest ricting to a subgraph one can get π ( Y , x 0 ) → π ( X , x 0 ) to be an isomorphism. Thus, if X is a Hausdorff space that has a simply connected covering space, and π ( X , x 0 ) is finitely generated, fr ee and abelian, then ther e exists a finite graph Y ⊂ X such that π ( Y , x 0 ) → π ( X , x 0 ) is an isomorphism. Let us consider a one-dimensional space X . This situation is special, since Cannon and Conner , [C C 06, C or ollary 3. 3 ], have shown that an inclusion Y ⊂ X of one- dimensional spaces induces an injective map on the fundamental gro up. Thus, we get the following: Proposition 4.6. Let X be a one-dimensional, Hausdorff spac e, and x 0 ∈ X . Assume X has a simply connected covering space, and π ( X , x 0 ) i s finitely generated. Then there exists a finite graph Y ⊂ X wi th x 0 ∈ Y such that π ( Y , x 0 ) → π ( X , x 0 ) is an isomorph i sm. Above we have studied, when there is a finite subgraph containing (up to homotopy) all loops of a space. W e now turn to the question, when there is canonical such subgraph. It is clear that we can only hope for this to happen if the space is one- dimensional. W e will use results fro m the master thesis of Meilstr up, [Mei05], where also the fol- lowing concept is intro duced: A one-dimensional Peano continuum is called a core continuum if it contains n o pro per deformation retr acts. Proposition 4.7 (see [Mei05, Corollary 2.6]) . Let X be a one-dimensional Peano contin- uum. Then the foll owing are equivalent: (1) X is a core (2) X has no attached de ndrites (an attached dendrite is a dendrite C ⊂ X such that for some y ∈ C there is a stron g deform ation retract r : X → ( X \ C ) ∪ { y } ) (3) every point of X is on an ess ential loop that cannot be hom otoped off it (4) whenever Y ⊂ X is a subset with π ( Y ) → π ( X ) surjective (h e nce bijecti v e), then Y = X Proo f. The e quivalence of (1),(2) and (3) is proved in [Mei05, Cor ollary 2.6]. ”(3) ⇒ (4)”: Let Y ⊂ X be a subset with π ( Y ) → π ( X ) surjective. Let x ∈ X be any point. The n x is on an essential loop, say α , which cannot be homoto ped off it. Since [ α ] ∈ π ( Y , x ) ther e is a loop β with image in Y that is homotopic to α . The r efor e x ∈ Y . ”(3) ⇒ (4)”: For any subset Y that is a deformation retract of X the map π ( Y ) → π ( X ) surjective.  16 ADAM P . W . SØRENSEN AND HANNES THIEL T o proc eed further and p rove that every one-dimensional Peano continuum contains a cor e we ne e d the notion of reduced loop fr om [CC06, Definition 3.8]. In fact, we will slighty generalize this to the notion of r educed path. This will help to simplify some proofs below . Definition 4.8 (see [CC06, Definition 3.8]) . A non-constant path α : [0 , 1] → X is called red ucible , if there is an open arc I = ( s, t ) ⊂ [0 , 1] such that f ( s ) = f ( t ) and the loop α | [ s,t ] based at f ( s ) is nullhomotpic. A path is c al l ed r educed if it is not reducible. A constant path is also c al l ed reduced. By [CC 0 6, Theor em 3 . 9 ] every loop is homotopic to a r educed loop, and if the space is one-dimensional, then this r educed loop is even unique (up to r eparame trization of S 1 ). The analogue for paths is proved in the same way . Proposition 4.9 (see [CC06, Theor em 3.9]) . Let X be a spac e, and α : [0 , 1] → X a path. Then α is homotopic (relative endpoints) to a reduced path β : [0 , 1] → X and we m ay assume the homotopy takes place inside the image of α , so that also the image of β l ies inside the image of α . If X is one-dimensional, then the r educed path is unique up to reparametrizing of [0 , 1] . Proposition 4.10 (see [Mei05, Theor em 2.4]) . Let X be a non-contractible, one-dimensional Peano c ontinuum. Then there exists a unique strong d eformation retract C ⊂ X that is a core continuum. We call it the c ore of X and de note it by core ( X ) . Further: (1) core( X ) is the sm allest strong deformation retract of X (2) core( X ) is the sm allest subset Y ⊂ X such that the map π ( Y ) → π ( X ) is surjectiv e Proo f. Le t core ( X ) ⊂ X be the union of all essential, reduced loops in X . In the pro of of [Mei05, Theor em 2 . 4 ] it is shown that core( X ) is a core continuum and a str ong deformation retract of X . For every stro ng deformation retr act Y ⊂ X the map π ( Y ) → π ( X ) is surjectiv e. Thus, to pr ove the two statements it is enough to show that core( X ) is contained in every subset Y ⊂ X such that the map π ( Y ) → π ( X ) is surjective. Let Y ⊂ X be any subset such that the map π ( Y ) → π ( X ) is surjective, and let α be an essential, reduced loop in X . Then α is homotopic to a loop α ′ in Y . By the above remark the image of α ′ contains the image of α . Thus, Y contains a ll essential, r educed loops in X , and therefor e core( X ) ⊂ Y .  Remark 4.11. If X is a contractible, one-dimensional Peano continuum (i.e. a den- drite), then it can be contracted to any of its points. That is why core( X ) is not defined in this situation. However , to simplify the following statements we will consider the cor e of a dendrite to be just any fixed point. If X is a finite graph, then the core is obtained by successively remov ing all ”loose” edges, i.e., vertices that ar e endpoints and the edge connecting the endpoint to the r est of the graph. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 17 Next, we combine a bunch of known facts with some of our results to obtain a list of equivalent characterizations when a one-dimensional Peano continuum is an A NR. Theorem 4.12. Let X be a one-dimensional Peano continuum. Then the following are equiv- alent: (1) X is an absolute neighborhood retract (ANR) (2) X is locall y contractible (3) X has a simply connected covering space (4) π ( X ) is finitely generated (5) there exists a finite g raph Y ⊂ X such that π ( Y ) → π ( X ) is an isomorphis m (6) core( X ) is a finite graph Proo f. ”(1) ⇒ (2)”: Every ANR is locally contractible, see [Bor67, V .2.3, p.101]. ”(2) ⇒ (3)”: By the above remark 4.3. ”(3) ⇒ (4)”: By [CC06, Le mma 7.7]. ”(4) ⇒ (1)”: This follows from [Bor67, V .13.6, p.138]. ”(3)+(4) ⇒ (5)”: Follows from 4.6. ”(5) ⇒ (6)”: B y 4.10 (2), core( X ) ⊂ Y . Then π (core( X )) → π ( Y ) is an isomorphism, and ther efor e core( X ) = core( Y ) . By the above remark 4.1 1 the cor e of a finite graph is again a finite graph. ”(6) ⇒ (4)”: Follows since π (cor e ( X )) → π ( X ) is bijective and the fundamental gr oup of a finite graph is finitely generated.  Remark 4.13. Let X be a one-dimensional Peano continuum. In the same way as the above theor em 4.12 one obtains that the following are equivalent: (1) X is an absolute retract (AR) (2) X is contractible (3) X is simply connected (4) π ( X , x 0 ) is z e r o (5) ther e exists a finite tr ee Y ⊂ X such that π ( Y , x 0 ) → π ( X , x 0 ) is a n isomor - phism (for any x 0 ∈ Y ) (6) core( X ) is a p oint Note that X is a dendrite if and only if it is a one-dimensional Peano continuum that satisfies one (or equivalently all) of the above conditions. Let us proceed with the study of the internal structur e of compact, one-dimensional ANRs. W e will give a struc tur e theorem which says that these spaces can be ap- pr oximated by finite graphs in a nice way , namely from within. This generalzes a theor em from Nadler ’s book, [Nad 9 2], about the struc tur e of dendrites (which ar e exactly the contractible one-dimensional, compact ANR s). The point is that compact, one-dimensional ANRs can be approximat ed fr om within by finite graphs in exactly the same way as dendrites can be appr oximated by finite tr ees (which are e xactly the contractible finite graphs). 18 ADAM P . W . SØRENSEN AND HANNES THIEL Lemma 4.14. Let X be a one-dimensional Peano continuum, and Y a subcontinuum wi th core( X ) ⊂ Y . For each x ∈ X \ Y there is a unique point r ( x ) ∈ Y such that r ( x ) is a point of an arc in X from x to any point of Y . Proo f. This is the analogue of [Nad 9 2, Lemma 1 0.24, p.175]. W e use ideas from the pr oof of [Mei05, Theorem 2.4]. Let X , Y be given, and x ∈ X \ Y . Pick some point y ∈ Y . Since X is arc-co nnected, ther e exists an ar c α : [0 , 1] → X starting at α (0) = x and ending at α (1) = y . Let y 0 = α ( min α − 1 ( Y )) , which is the first point in Y of the arc (starting from x ). Note that y 0 ∈ Y since Y i s closed. Assume ther e ar e two ar cs α 1 , α 2 : [0 , 1] → X fr om x to dif fer ent points y 1 , y 2 ∈ Y such that α i ([0 , 1)) ⊂ X \ Y . W e show that this leads to a contradictio n. Let β be a r educed path in Y fr om y 1 to y 2 . Define t 1 := sup { t ∈ [0 , 1] : α 1 ( t ) ∈ im( α 2 ) } t 2 := sup { t ∈ [0 , 1] : α 2 ( t ) ∈ im( α 1 ) } , so that x 0 = α 1 ( t 1 ) = α 2 ( t 2 ) is the first point where the ar cs α 1 , α 2 meet (looking from y 1 and y 2 ). Connecting ( α 1 ) | [ t 1 , 1] (fr om x 0 to y 1 ) with β (from y 1 to y 2 ) and the inverse of ( α 1 ) | [ t 2 , 1] (fr om y 2 to x 0 ), we get a reduced loop containing x 0 which cont radicts x 0 / ∈ core( X ) ⊂ Y . It follows that ther e exists a uniq ue point y ∈ Y with the desir ed pr operties.  Definition 4 .15 (see [Nad92, Definition 10.26, p.1 7 6]) . Let X be a one-dimensional Peano continuum, and Y a subcontinuum with core( X ) ⊂ Y . Define a m ap r : X → Y by le tting r ( x ) as in the lemm a 4. 1 4 above i f x ∈ X \ Y , and r ( x ) = x if x ∈ Y . Thi s map is called the first point map . The first point map is continuou s, and thus a retr action of X onto Y . This is the analogue of [Nad92, Lemma 10.2 5, p.176] and pr oved the same way . But mor e is true: As in the pr oof of [Mei05, Theorem 2.4], one can show that Y is a str ong d e formation retr act of X . Proposition 4.16. Let X be a one-dimensional Peano continuum, and Y a subcontinuum with core( X ) ⊂ Y . Then the fi rst point map i s c ontinuous. Further , there is a strong deformation retraction to the fi r st point m ap. Proo f. Le t X , Y be given. As in the proo f of [Mei05, Theor em 2.4], the complement X \ Y consist of a collection of attached d e ndrites { C i } . That me a ns each C i ⊂ X is a dendrite such that C i ∩ Y consists of exactly one point y i and such that there is a str ong deformation r etract r i : X → ( X \ C i ) ∪ { y i } . Meilstrup shows that these str ong deformation retr acts can be assembled to give a strong deformation retr act to the first point map r .  Theorem 4.17. Let X be a one-dimensional Peano continuum. The n ther e is a sequence { Y k } ∞ k =1 such that: SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 19 (1) each Y k is a subcontinuum of X (2) Y k ⊂ Y k +1 (3) lim k Y k = X (4) Y 1 = cor e( X ) and for eac h k , Y k +1 is obtained from Y k by attaching a li ne segment at a point, i.e ., Y k +1 \ Y k is an arc with an end point p k such that Y k +1 \ Y k ∩ Y k = { p k } (5) letting r k : X → Y k be the fi rst p oint m ap for Y k we have that { r k } ∞ k =1 converges uniformly to the identity m ap on X If X is also ANR, then all Y k are finite graphs. If X is even contractible (i .e. i s an AR, or equivalently a dendrite), then core( X ) is just some point, and all Y k are finite trees. Proo f. This is the analogue of [Nad92, Le mma 1 0 . 2 4, p.1 7 5], and the proof goes thr ough if we use our a naloguous lemmas 4.14 and 4.16.  5. T H E O T H E R I M P L I C AT I O N O F T H E M A I N T H E O R E M : S U FFI C I E N C Y For this implication we aim to mirror the approach of Chigogidze and Dranishnikov , [CD10]. However we first show how to go fro m C ( X ) being a universal C ∗ -algebra to C ( Y ) being one, where Y is obtained from X by attaching a line segment at one point. This step is not needed in [CD10], since they are able to give a general descrip- tion of the generators and r elations of the relevant spaces. W e have n ot been able to find such generators and r elations, and doing so might be of inde pendent interest . Lemma 5. 1 . Suppose X is a space, that C ( X ) = C ∗ hG | Ri and that { ˆ g | g ∈ G } is a generating set of C ( X ) that fulfills R . Let Y be the space f ormed from X by attaching a li ne segment at a point v , and let λ g = ˆ g ( v ) . Then C ( Y ) = C ∗ hG ∪ { h } | R ′ i , where R ′ = R ∪ { g h = λ g h and g h = hg | g ∈ G } ∪ { 0 ≤ h ≤ 1 } . Proo f. Extending the ˆ g to Y by letting them be constant on the added line segment and letting ˆ h be the function that is zero on X and grows linearly to one on the line segment (identifying it with [0 , 1] ), shows that that there is a generating family in C ( Y ) that fulfills R ′ . W e will use [Lor97, Lemma 3.2.2, p.26] to show that C ( Y ) i s universal for R . B y this lemma, it suf fices to show , that whenever we have a family { T g | g ∈ G ∪ { h }} of operators, on some Hilbert space H , that fulfills R and { T g | g ∈ G } ′ = C I , then we can find a morphism fr om C ( Y ) to B ( H ) taking ˆ g to T g for all g ∈ G ∪ { h } . Suppose we have such operators. Since C ( X ) is commutative and R ′ for ces h to commute with all the other generators, we have that T g = µ g I for some µ g ∈ C , for all g ∈ G ∪ { h } . W e need to find a morphism fro m C ( Y ) to C . There are two cases. • Case 1: µ h = 0 : In this case we can find a morphism φ : C ( X ) → C such that φ ( ˆ g ) = µ g for all g ∈ G , since C ( X ) = C ∗ hG | Ri . Then φ = ev u for some point u ∈ X . The morphism ev u : C ( Y ) → C maps ˆ h = 0 and ˆ g = µ g , and thus is the r equired morphism. • Case 2: µ h 6 = 0 Since 0 ≤ T h ≤ 1 , we have 0 < µ h ≤ 1 . For g ∈ G we have µ g µ h I = T g T h = λ g µ h I . 20 ADAM P . W . SØRENSEN AND HANNES THIEL So since µ h 6 = 0 , we have µ g = λ g for all g ∈ G . Let us now identify the added line segment with [0 , 1] . The m orphism ev µ h : C ( Y ) → C , takes ˆ h to µ h and ˆ g to λ g = µ g . Hence it is the requir ed morphism.  W e now p rovide a slightly alter ed (in both pr oof and statement) version of [CD10 , Pr oposition 4 .1]. Lemma 5. 2 . Suppose X i s a one-dimensional fi nite graph, that C ( X ) = C ∗ hG | Ri , th at { ˆ g | g ∈ G } is a generating set of C ( X ) that f ulfills R , and that G is finite. Let Y be the space formed from X by attaching a line segment at a p oint v . Suppose we have a commutative squar e C ( X ) ι   ψ / / C π   C ( Y ) φ / / C / J where J i s an ideal in the unital C ∗ -algebra C , π is the quotient m orphism, ψ and φ ar e unital morphisms , and ι is i nduce d by the retraction from Y onto X , i.e., ι takes a function in C ( X ) to the f unction in C ( Y ) gi ven by ι ( f )( x ) =  f ( x ) , x ∈ X , f ( v ) , x is i n the adde d line segme nt . Then for every ε > 0 we can find a morphism χ : C ( Y ) → C such that π ◦ χ = φ and k ( χ ◦ ι )( ˆ g ) − ψ ( ˆ g ) k ≤ ε for ev e ry g ∈ G . Proo f. Throughout the proof we use the notation of Lemma 5.1. Let δ > 0 be given. W e will construct a δ -repr esentation { d g | g ∈ G ∪ { h } } of R ′ in C such that π ( d g ) = φ ( ι ( ˆ g )) for g ∈ G and π ( d h ) = φ ( ˆ h ) . Let q κ : X → X be the map that collapses the ball B κ/ 2 ( v ) , fixes X \ B κ ( v ) , and extends linearly in between. Since there are only finitely many ˆ g , we can find κ 0 such that k q ∗ κ 0 ( ˆ g ) − ˆ g k ≤ δ / 2 , where q ∗ κ is the morphism on C ( X ) induced by q κ . For simpler notation we let q = q κ 0 , and put w g = q ∗ ( ˆ g ) for all g ∈ G . Let f 0 be a positive function in C ( X ) of norm 1 that is zero on X \ B κ 0 / 2 ( v ) and 1 at v . Observe that if f ∈ q ∗ ( C 0 ( X \ { v } )) , then f f 0 = 0 . Since ˆ h ≤ ι ( f 0 ) and ψ ( f 0 ) is a lift of φ ( ι ( f 0 )) , we can, by [Lor97, C or ollary 8.2.2, p.63], find a lift ¯ h of φ ( ˆ h ) such that 0 ≤ ¯ h ≤ ψ ( f 0 ) . W e now claim that { ψ ( ˆ g ) | g ∈ G } ∪ { ¯ h } , is a δ -repr esentation of R . Since the ¯ g fulfill the relations R and ¯ h is a positive contraction, we only nee d to check that ψ ( ˆ g ) and ¯ h almost commute, and that ψ ( ˆ g ) ¯ h is almost λ g ¯ h . First we note that since 0 ≤ ¯ h ≤ ψ ( f 0 ) for any f ∈ q ∗ ( C 0 ( X \ { v } )) we have k ψ ( f ) ¯ h 1 / 2 k 2 = k ψ ( f ) ¯ hψ ( f ) ∗ k ≤ k ψ ( f ) ψ ( f 0 ) ψ ( f ) ∗ k = 0 . Thus ψ ( f ) ¯ h = 0 . In particular we have ψ ( w g − λ g ) ¯ h = 0 . SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 21 Now we ha ve k ψ ( ˆ g ) ¯ h − ¯ hψ ( ˆ g ) k = k ψ ( ˆ g ) ¯ h − ψ ( w g − λ g ) ¯ h − ¯ hψ ( ˆ g ) + ¯ hψ ( w g − λ g ) k = k ψ ( ˆ g − w g ) ¯ h + λ g ¯ h − ¯ h ( ψ ( ˆ g − w g )) − λ g ¯ h k ≤ k ¯ h k ( k ψ ( ˆ g − w g ) k + k ψ ( ˆ g − w g ) k ) ≤ 2 k ˆ g − w g k ≤ 2 · δ / 2 = δ, for all g ∈ G . Likewise we have k ψ ( ˆ g ) ¯ h − λ g ¯ h k = k ψ ( ˆ g ) ¯ h − λ g ¯ h − ψ ( w g − λ g ) ¯ h k = k ψ ( ˆ g − w g ) ¯ h + λ g ¯ h − λ g ¯ h k = k ψ ( ˆ g − w g ) ¯ h k ≤ k ˆ g − w g k ≤ δ / 2 ≤ δ, for al l g ∈ G . So { ψ ( g ) | g ∈ G } ∪ { ¯ h } is indeed a δ -repr esentation of R ′ . Further we have that π ( ψ ( ˆ g )) = φ ( ι ( ˆ g )) and that π ( ¯ h ) = φ ( h ) . Since X is a one-dimensional finite graph, Y is also a one-dimensional finite graph, so C ( Y ) is semiproject ive by [Lor97, Propositio n 16.2.1, p.125]. By [Lor97, Theo- r em 14.1.4, p.1 06] the r elations R ′ ar e then stable. So the fact that we can find a δ -repr esentation for all δ implies that we can find a morphism χ : C ( Y ) → C such that π ◦ χ = φ and k χ ( ι ( ˆ g ) ) − ψ ( ˆ g ) k ≤ ε for all g ∈ G .  W e ar e now ready to show that some inductive limits have good lifting properties. In particular if we have an initial lift then we can lift all that follows. Proposition 5.3. Suppose that X is a compact space such that C ( X ) can be written as an inductive limit lim − → n C ( Y n ) = C ( X ) , where each Y n is a finite graph, Y n +1 is just Y n with a line segments attached at a point (as in Lemma 5.2), and the bonding morphisms ι n,n +1 : C ( Y n ) → C ( Y n +1 ) are as the morp hism in Lemma 5.2, i.e., induced by retracting the attached interval to the attaching point. If there is a unital morphism φ : C ( X ) → C /J , where J i s an ide al in a unital C ∗ -algebra C , and a unital morphism ψ 1 : C ( Y 1 ) → C such that π ◦ ψ 1 = φ ◦ ι 1 , ∞ , then there is a unital morphism ¯ ψ : C ( X ) → C such that π ◦ ¯ ψ = φ . Proo f. W e have the following situation: C π   C ( Y 1 ) ι 1 , ∞ / / ψ 1 5 5 j j j j j j j j j j j j j j j j j j j j C ( X ) φ / / ¯ ψ ; ; C / J As Y 1 is a finite graph, C ( Y 1 ) is finitely generated. Thus C ( Y 1 ) is a universal C ∗ - algebra for some finite set of generators and relations, C ( Y 1 ) = C ∗ hG 1 | R 1 i , say . In view of Lemma 5 . 1 we can now a ssume that C ( Y n ) = C ∗ hG n | R n i , wher e G 1 ⊆ G 2 · · · , and likewise for the R n . W e also get fr om Lemma 5.1 that all the G n and R n ar e finite. 22 ADAM P . W . SØRENSEN AND HANNES THIEL Since we are given ψ 1 , we can, using Lemma 5.2 inductively , for any sequence of positive numbers ( ε n ) find morphisms ψ n : C ( Y n ) → C for each n > 1 such that π ◦ ψ n = φ ◦ ι n, ∞ and such that k ψ n ( ˆ g ) − ψ n − 1 ( ˆ g ) k ≤ ε n for the gene rators ˆ g of C ( Y n ) . W e now wish to define new morphisms χ n : C ( Y n ) → C such that π ◦ χ n = φ ◦ ι n, ∞ and χ n +1 extends χ n . T o this end we define, for each n ∈ N , e le ments { ¯ g n | g ∈ G n } , by ¯ g n = lim k ψ n + k ( ˆ g ) . The sequence ( ψ n + k ( ˆ g )) is Cauchy if P ε n < ∞ , so we will assume that. W e claim that for any n ∈ N the elements { ¯ g n | g ∈ G n } in C fulfill R n . By [Lor97, Lemma 13.2.3, p.103] the set { ¯ g n | g ∈ G n } is an ε -repr esentation of R n for all ε > 0 since { ψ n + k ( ˆ g ) | g ∈ G n } is a r epresentat ion of R n for all k . Thus { ¯ g n | g ∈ G n } is a r epresent ation of R n . Observe that if m ≥ n , then ¯ g m = ¯ g n , since ¯ g m is the limit of a tail of the sequence ¯ g n is the limit of. Thus, we will drop the subscript s, and simply say that we have eleme nts { ¯ g | g ∈ ∪G n } such that for any n ∈ N the set { ¯ g | g ∈ G n } fulfills R n . Now we can define the χ n . W e put χ n ( ˆ g ) = ¯ g , for g ∈ G n , and this extends to a morphism since C ( Y n ) ∼ = C ∗ hG n | R n i . W e get χ n 1 ◦ ι n,n +1 = χ n and π ◦ χ n = φ ◦ ι by universality , since it holds on generators. By the universal property of an inductive limit we get a morphism χ : C ( X ) → C such that π ◦ χ = φ .  Remark 5.4. U sing the structur e theorem for dendrites, [Nad92, Theorem 10.27, p.17 6], see 4.17, and the above Proposit ion 5.3 we may deduce that for a d e ndrite X the C ∗ - algebra C ( X ) is projective in S 1 (the categor y of unital C ∗ -algebras, see 2.2). Thus, we recover the implication ”(1) ⇒ (2)” of [CD10, Theor em 4.3]. T o elaborate: Each dendrite X can be approximat ed fro m within by finite tr ees, i.e., C ( X ) ∼ = lim − → C ( Y k ) where Y 1 is just a single point and the trees Y k ar e obtained by successive attaching of line segments. Since C ( Y 1 ) = C is p rojective in S 1 , we obtain fr om 5.3 that morphisms from C ( X ) into a quotients can be lifted, i.e., C ( X ) is pr o- jective in S 1 . W e ar e now r eady to pro ve our main theorem: Proo f of theorem 1.2. The implication ” ( I ) ⇒ ( I I ) ” is Propo sition 3.1. Let us prove ” ( I I ) ⇒ ( I ) ”: So assume X is a compact ANR with dim( X ) ≤ 1 . Note that X can have at most finitely many components X i . If we can show that each C ( X i ) is semiprojective, then C ( X ) = L i C ( X i ) will be semipr ojective (since semiproject iv- ity is preserved by finite direct sums, see [Lor97, Theorem 14.2.1, p.110]). So we may assume X is connected. Then theorem 4.17 applies, and we may find a n increasing sequence Y 1 ⊂ Y 2 ⊂ . . . ⊂ X of finite subgraphs such that: (1) lim k Y k = X , i.e., S k Y k = X (2) Y k +1 is obtained from Y k by attaching a line segment at a p oint SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 23 Then C ( X ) = lim − → k C ( Y k ) where each bonding morphism ι k ,k +1 : C ( Y k ) → C ( Y k +1 ) is induced by the retract ion from Y k +1 to Y k that contracts Y k +1 \ Y k to the point Y k +1 \ Y k ∩ Y k . Suppose now that we are given a unital C ∗ -algebra C and an incr eas- ing sequence of idea ls J 1 ✁ J 2 ✁ . . . ✁ C and a unital morphism σ : C ( X ) → C / S k J k . W e need to find a lift ¯ σ : C ( X ) → C / J l for some l . Consider the unital morphism σ ◦ ι 1 , ∞ : C ( Y 1 ) → C / S k J k . By [Lor97, Pr oposition 16.2.1, p.125], the in itial C ∗ -algebra C ( Y 1 ) is semipr ojective. Ther efor e, we can find an index l and a unital morphism α : C ( Y 1 ) → C / J l such that π l ◦ α = σ ◦ ι 1 , ∞ . This is viewed in the following commutative diagram: C   C / J l π l   C ( Y 1 ) ι 1 , 2 / / α 2 2 C ( Y 2 ) / / . . . / / C ( X ) σ / / C / S k J k Now we can apply 5.3 to find a unital morphism ¯ σ : C ( X ) → C /J l such that π l ◦ ¯ σ = σ . This shows that C ( X ) is semiprojective.  6. A P P L I C AT I O N S In this section we give applications of our findings. First, we characterize semipr o- jectivity of non-unital, separable commutative C ∗ -algebras. Building on this, we ar e able to confirm a conjectur e of Loring in the particular case of commutative C ∗ -algebras. Then, we will study the semipr ojectivit y of C ∗ -algebras of the form C 0 ( X , M k ) . Finally , we will give a partial solution to the pro blem when a commuta- tive C ∗ -algebra is weakly (semi-)pr ojective. T o keep this article short, we will omit most of the proof s in this sections. T o charact erize semipro jectivity of non-unital commutative C ∗ -algebras we have to study the struct ur e of non-compact, one-dimensional A N Rs. W e are particularly inter ested in the one-point compactifications of such spaces. The motivation ar e the following r esults: If X is a locally compact, Hausdorf f space, then naturally ^ C 0 ( X ) ∼ = C ( α X ) , wher e α X is the one-point comapctifi cation of X . Further , a C ∗ -algebra A is semipr ojective if and only if e A is semipr ojective. Thus, C 0 ( X ) is semiproject ive if and only if C ( α X ) is semipr ojective. By our main result 1.2 this happens precisely if α X is a one-dime nsional ANR. The following result gives a topological characterization of such spaces. W e de- rive a characterization of semiprojectivit y for non-unital, sep a rable commutative C ∗ - algebras, see coro llary 6.2. W e al so show that α X is a one-dimensional ANR if a nd 24 ADAM P . W . SØRENSEN AND HANNES THIEL only if e very finite-point compactification 13 of X is a one-dimensional ANR. Using this, we can confirm a conjectur e about the semipr ojective of extensions in the com- mutative case, see 6.3 a nd 6.4. Theorem 6.1. Let X be a one-dimensional, locally compact, separable, me tr i c ANR. Then the followi ng are equivalent: (1) The one-point compactifi c ation α X is an A N R (2) X has only finitely many compact components and also only finitel y m any compo- nents C ⊂ X such that α C is not a de ndrite (3) Every finite-point c ompactification of X is an AN R (4) Some finite-point c ompactification of X is an AN R Corollary 6.2. Let X be a locally compact, s e parable, metric space. T h en the following are equivalent: (1) C 0 ( X ) is semip rojective. (2) X is a one-dime nsional ANR that has only finitely many compact components, and X has als o only finitel y many comp onents C ⊂ X such that α C is not a dendrite Corollary 6.3. Let A be a separable, c om mutative C ∗ -algebra, and I ✁ A an ideal. Assume A/I is finite-dim ensional, i.e.. A/I ∼ = C k for some k . Then A is semiprojective if and only if I is sem i projective. Proo f. Le t A = C 0 ( X ) for a locally compact, sep arable, metric space X . Then I = C 0 ( Y ) for a n open subset Y ⊂ X . Since A/I is finite-dimensional, X \ Y is finite. It follows that also α X \ Y is finite, and so the closure Y ⊂ α X is a finite-point compacti- fication of Y . Set F := α X \ Y (which is also finite). Note that Y ⊂ α X is a component, so that α X = Y ⊔ F . It follows that α X is an ANR if and only Y is. Then we argue as follows: A = C 0 ( X ) is semipr ojective ⇔ e A = C ( α X ) is semiproject ive ⇔ α X is a one-dimensional A NR [ by theorem 1.2 ] ⇔ Y ⊂ α X is a one-dimensional ANR [ since α X = Y ⊔ F ] ⇔ α Y is a one-dimensional ANR [by theor em 6 .1 since Y is a finite-point compactification of Y ] ⇔ e I = C ( α Y ) is semip rojective [ by theorem 1.2 ] ⇔ I = C 0 ( Y ) is semiprojectiv e  13 A compactification of a space X is a pa ir ( Y , ι Y ) where Y is a compact space, ι : X → Y is an embedding and ι ( X ) is dense in Y . Usually the embed ding is understood and one denotes a com- pactification just by the spa ce Y . A compactification γ ( X ) of X is called a finite-point compactification if the remainder γ ( X ) \ X is finite. SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 25 Remark 6.4. Let A be a se p a rable C ∗ -algebra, and I ✁ A an ideal so that the quotient is finite-dimensional. W e get a short exact sequence: 0 / / I / / A / / F / / 0 It was conjectur e d by Loring and also by Blackadar , [Bla04, Conjecture 4.5], that in this situation A is semiprojective if and only if I is semipr ojective. One implication was r ecently pro ven by Dominic Enders, [End11], who showed that semipr ojectivity passes to id e als when the quotient is finite-dimensional. The converse implication is in general not even known for F = C . Our above r e- sult 6.3 confirms this conjectur e in the case that A is commutative. Let us now study the semiproj ectivity of C ∗ -algebras of the form C 0 ( X , M k ) . Lemma 6.5. Let X be a locally compact metric space and le t k ∈ N . If φ : C 0 ( X , M k ) → M k is a morphism then the re is a unitary u ∈ M k and a unique point x ∈ α X such that φ = Ad u ◦ ev x . Proposition 6. 6 . Let X be a locally compac t, separable, me tric space and let k ∈ N . If C 0 ( X , M k ) is projective, then α X is an AR. Proo f. Suppose we are given a compact metric space Y with an embedding ι : α X → Y . Dualiz ing and e mbedding C 0 ( X ) into C ( α X ) , we get the following d ia gram C 0 ( Y ) ι ∗   C 0 ( X ) / / C ( α X ) T e nsoring everything by the k by k matrices M k , we get C 0 ( Y , M k ) ( ι ∗ ) k   C 0 ( X , M k ) / / C ( α X , M k ) Since C 0 ( X , M k ) is projective, ther e is a morphism ψ : C 0 ( X , M k ) → C 0 ( Y , M k ) such that ( ι ∗ ) k ◦ ψ is the inclusion of C 0 ( X , M k ) into C ( α X , M k ) . For each y ∈ Y lemma 6.5 tells us that the morphism ev y ◦ ψ , has the form Ad u y ◦ ev x y for some unitary u y ∈ M k and some uniq ue x y ∈ α X . Hence we can define a function λ : Y → α X such that ev y ◦ ψ = Ad u y ◦ ev λ ( y ) . This map λ is continuous. For each x ∈ α X we have the following commutative diagram 26 ADAM P . W . SØRENSEN AND HANNES THIEL C 0 ( Y , M k ) ( ι ∗ ) k   ev ι ( x ) / / M k C 0 ( X , M k ) / / ψ 7 7 o o o o o o o o o o o C ( α X , M k ) ev x / / M k Fr om this diagram, it follows that if x ∈ α X then Ad u ι ( x ) ◦ ev λ ( ι ( x )) = ev ι ( x ) ◦ ψ = ev x ◦ ( ι ∗ ) k ◦ ψ = ev x . So for any function g ∈ C 0 ( X , M k ) we get ev λ ( ι ( x ))   g . . . g   = ( Ad u ι ( x ) ◦ ev λ ( ι ( x )) )   g . . . g   = ev x   g . . . g   . Hence we must have λ ( ι ( x )) = x . All in all, we have found a continuous map λ : Y → α X such that λ ◦ ι = id , i.e., the embedded space α X ⊂ Y is a retract. As the e mbe dding was arbitrary , α X is an AR.  The proof can be modified to show: Proposition 6. 7 . Let X be a locally compac t, separable, me tric space and let k ∈ N . If C 0 ( X , M k ) is se miprojective, then α X is an ANR. Using the ide a of the pr oof of 3.1 one can show the following: Proposition 6. 8 . Let X be a loc all y compact, separable, metric s pace, and k ∈ N . If C 0 ( X , M k ) is se miprojective, then dim( X ) ≤ 1 . Corollary 6.9 . Let A be a separable, commutative C ∗ -algebra, and k ∈ N . If A ⊗ M k is projective, then so is A . Analogously , if A ⊗ M k is sem iprojective, then so is A . Proo f. Le t A = C 0 ( X ) for a locally compact, separable, metric space X . First, a ssume A ⊗ M k is semip rojective. By pr oposition 6.8, dim( X ) ≤ 1 . This implies that the dimension of α X is at most one. By pro position 6.7, α X is an ANR . Then our main theor em 1.2 shows that C ( α X ) is semiproject ive. Since C ( α X ) is the unitization of C 0 ( X ) , we also have that C 0 ( X ) is semiproject ive. Assume now that A ⊗ M k is proj ective. I t follows that A cannot be unital, for oth- erwise A ⊗ M k would be unital and that is impossible for pro jective C ∗ -algebras. As in the semipr ojective case we deduce dim( α X ) ≤ 1 . By 6.6, α X is an AR. It follows fr om [CD10, Theor em 4.3], see also 1.3, the C ( α X ) is pr ojective in S 1 . It follows that C 0 ( X ) is pr ojective, see 2.2.  SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 27 W e will now turn to the q ue stion, whe n a unital, commutative C ∗ -algebra is weakly (semi-)pr ojective in S 1 . The analogue of a weakly (semi-)project ive C ∗ -algebra in the commutative world is an ap p roximative absolute (neighbor hood) r etract (abbr e vi- ated by AAR and AANR). A s mentioned in 2.3, if C ( X ) is weakly (semi-)pr ojective, then X is A A(N)R. W e will show below , that for one-dimensional spaces the converse is also true. 6.10. Let X be a compact, metric space. Consider the following conditions: (1) for each ε > 0 there exists a map f : X → Y ⊂ X such that Y is an AR (an ANR), and d ( f ) ≤ ε (2) X is an AAR (an A ANR) Here, by d ( f ) < ε we mean that the distance of x and f ( x ) is less than ε for all x ∈ X , i.e., d ( x, f ( x )) < ε for all x ∈ X . The first condition means that X can be appro ximated fr om within by ARs (by ANRs). As shown by Clapp, [Cla71, Theorem 2.3], see also [CP05, Proposit ion 2.2(a)], the implication ”(1) ⇒ (2)” holds in general. It was asked by Cha ratonik and Prajs, [CP05, Question 5.3], whether the converse also holds (at least for continua). They showed that this is indeed the case for heredi- tarily unicoher ent continua, [CP05, Observation 5.4]. In theorem 6.15 below we show that the two conditions are also equivalent for one-dimensional, compact, metric spaces. The following is a standard result fr om continuum theory: Proposition 6.11. Let X be a one-dimensional Peano continuum, and ε > 0 . Then there exists a fi nite subgraph Y ⊂ X and a surj ective map f : X → Y ⊂ X such that d ( f ) < ε . Corollary 6.12. Every one-dimensional Peano continuum is an AANR. Proo f. Le t X be a one-dimensional Peano continuum. By 6.11, X can be approxi- mated fro m within by finite subgraphs. A finite graph is an ANR. It follows fr om [Cla71, Theorem 2.3], see 6.10, that X is an A A NR.  The following Le mma is a direct translation of [Lor09b, Lemma 5.5] to the commuta- tive setting. Lemma 6.13 (see [Lor09b, Lemma 5.5]) . Let X be an compact AAR, and D any ANR. Then ever y map f : X → D is inessential, i.e., homotopic to a constant map. Corollary 6.14. Every one-dimensional, comp act AAR is tree-like. Proo f. Le t X be a one-dimensional, compact AA R. The n X is connected and thus a continuum. In [CC 60, Theorem 1] tr ee-like continua a r e characterized as one- dimensional continua such that every map into a finite graph is inessential. Thus, 28 ADAM P . W . SØRENSEN AND HANNES THIEL we need to show that every map fr om X into a finite graph is ine ssential. This fol- lows from the above L e mma since every finite graph is an AN R.  Theorem 6.15. Let X be a one-dim e nsional, com pact, metri c space. Th e n the following are equivalent: (1) for each ε > 0 there ex i sts a map f : X → Y ⊂ X such that Y i s a finite tree (a finite graph), and d ( f ) ≤ ε (2) for each ε > 0 the re exists a map f : X → Y ⊂ X such that Y is an AR (an ANR), and d ( f ) ≤ ε (3) X is an AAR (an AANR) Moreover , in (1) and (2) the map f may be assumed to be surjec tive. Proo f. ”(1) ⇒ (2)” is clear , and ”(2) ⇒ (3)” follows fr om [Cla71, Theor em 2.3], see 6.1 0. ”(3) ⇒ (1)”: It was shown by Clapp, [Cla71, Theor em 4.5], that for e ach embedding of a compact AANR X in the Hilbert cube Q a nd δ > 0 there exists a compact poly- hedro n P ⊂ Q with ma ps f : X → P and g : P → X such that d ( f ) < δ and d ( g ) < δ . Note that g maps each component of P onto a Peano subcontinuum of X . Thus, the image Y := g ( P ) ⊂ X is a finite union of Peano subcontinua. Moreov er , the map g ◦ f : X → Y ⊂ X satisfies d ( f ) < 2 δ . Assume X is a one-dimensional, compact AANR and fix some ε > 0 . W e app ly the r esult of Clapp for δ = ε/ 4 and obtain a compact subspace Y ⊂ X that is the (disjoint) union of finitely many Peano continua, together with a surjective ma p f : X → Y such that d ( f ) < ε/ 2 . Since Y ⊂ X is closed, dim( Y ) ≤ dim( X ) ≤ 1 . App lying 6.11 to each component of Y and ε / 2 we obtain a finite subgraph Z ⊂ Y and a surjective map g : Y → Z such that d ( g ) < ε/ 2 . W e may consider Z as a finite subgraph of X . The map h := g ◦ f : X → Z ⊂ X is surjective and satisfies d ( h ) < ε . So we have shown the imp lication for the case that X is AA NR. Assume additionally that X is an A AR. W e have already shown that X can be appro ximated fr om within by finite subgraphs. W e need to show that the same is true with finite tr ees. By 6.1 4, X is tree-like. By [Le l7 6, 2.2 and 2.3 ], e very tree-like continuum is he r edi- tarily unicoherent. A coherent finite graph is a finite tree. It follo ws that ever y finite subgraph Z ⊂ X is a finite tree, and so X can be approximated fr om within by finite subgraphs which automatically are finite trees.  Corollary 6.16. Let X be a compact, metri c space. The n the following im plications hold: (1) If X is an AAN R and dim( X ) ≤ 1 , then C ( X ) is weakly sem iprojective S 1 . (2) If X is an AAR and dim( X ) ≤ 1 , then C ( X ) is we akly projective in S 1 . Proo f. Le t X be a one-dimensional, compact AA R (A ANR). By 6.15, X can be appro x- imated fr om within by finite tr e e s (finite graphs), i.e., for each n ≥ 1 ther e exists a finite tree (graph) Y n ⊂ X and a surjective map f n : X → Y n with d ( f n ) < 1 /n . W e want to use [Lor09b, Theor em 4.7] to show C ( X ) is we a kly (semi-)pr ojective in S 1 . SEMIPROJECTIVITY OF COMMUT A TIVE C ∗ -ALGEBRAS. 29 The surjective maps f n induce injective morphisms f ∗ n : C ( Y n ) → C ( X ) . Consider also the inclusion map ι n : Y n ֒ → X and the dual morphism ι ∗ n : C ( X ) → C ( Y n ) . Set θ n := f ∗ n ◦ ι ∗ n : C ( X ) → C ( X ) . Since d ( f n ) tends to zero, the morphisms θ n conver ge (pointwise) to the identity morphism. Further , the image of θ n is eq ual to the im a ge of f ∗ n , a nd ther efore isomor - phic to C ( Y n ) . As shown by Loring, [Lor97, Proposit ion 16.2.1, p.125], C ( Y ) is semiproject ive (in S 1 ) if Y is a finite graph. Similarly , C ( Y ) is projective in S 1 if Y is a finite tree Y (see also [CD10]). N ow , it follows from [Lor09b, Theorem 4.7] (and the analogous r esult for weakly semiproj ective C ∗ -algebras) that C ( X ) is we a kly (semi-)pr ojectiv e in S 1 .  Remark 6.17. W e remark that the converse implications of 6.16 also hold. As ex- plained in 2.3, if C ( X ) is wea kly (semi-)proj ective in S 1 , then X is ne cessarily an ap- pr oximative a bsolute (ne ighborhood) retract . 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