The Jiang-Su algebra revisited

The Jiang–Su algebra revisited Mik ael Rørdam ∗ and Wilhelm Win ter † Octob er 22, 2018 Abstract W e giv e a number of new characteri zations of the Jiang–Su algebra Z , both intrinsic and extrinsic, in terms of C ∗ -algebraic, dynamical, top olog ical and K -t heoretic conditions. Along the wa y w e study divisibility p roperties of C ∗ -algebras, we give a precise c haracterization of those unital C ∗ -algebras of stable rank one that ad mit a unital embedd ing of th e dimension- drop C ∗ -algebra Z n,n +1 , and w e prove a ca ncellation t heorem for the Cun tz semigroup o f C ∗ -algebras of stable rank one. 1 In tro duction In Elliott’s progra m to classify nuclear C ∗ -algebra s b y K -theory data (see [14] for an in tro duction), the systematic use of stro ngly self-abs o rbing C ∗ -algebra s play a c en tral role. The term “strongly self-absor bing C ∗ -algebra s” was formally coined in the pap er [18] to deno te the class of C ∗ -alge- bras D 6 = C for whic h there is an iso morphism fro m D to D ⊗ D which is appr o ximately unitarily equiv alent to the embedding d 7→ d ⊗ 1. Strongly self-a bsorbing C ∗ -algebra s ar e automatically simple, nuclear and hav e at most one tracial state. The Cuntz algebras O 2 and O ∞ and the Jiang–Su a lgebra Z are s tr ongly self-abso r bing. Most classific a tion results obtained s o far ca n be interpreted a s classific ation up to D -stability , where D is o ne of the (few) known strongly self-abso rbing examples (cf. [16]). The cla s sification of Kir c hberg algebra s can th us b e view ed a s cla ssification up to O ∞ -stability . There is at present m uch interest in classificatio n up to Z -sta bilit y , which appear s to b e the lar gest possible cla ss of “ D -stable” C ∗ -algebra s. One may view Z as being the stably finite analog ue o f O ∞ . The original construction of the Jiang–Su algebra in [8] is as an inductive limit of a sequence of C ∗ -algebra s with sp e cified connecting mappings. Whereas everything in this co nstruction in principle is concrete, the presentation is not canonica l, and it depends on infinitely many choices. Since the Jiang–Su algebra ha s b ecome to pla y suc h a cen tral role in the classific a tion pro gram it is desirable to have a more concr e te and “finite” pr esen tation of this algebra, o r to b e a ble to characterize it in a more streamlined w ay . W e r efer to the r ecen t pap er b y Da da rlat and T oms, [3], for a very nice such c hara cterization. In this pap er w e pr esen t o ther characterizations and presentations o f the Jiang–Su a lgebra. The many alternative descriptions av ailable for the Cuntz algebra O ∞ provide a guideline o f what kind of characterizations one migh t expect for Z . They in v olve ( C ∗ -)algebra ic, dynamica l and K -theor etic conditio ns ; in the present pa per we shall employ similar co nditions to characterize the Jiang–Su algebra in v arious manners. W e also give a top ologica l characterization of Z which currently has no known analog ue for O ∞ . Besides its orig inal pr esen tation as a universal C ∗ -algebra with g enerators and relations, O ∞ may b e written as a cro ssed pro duct of a canonical subalgebra by an endo morphism (see [2]). Both these des c r iptions a re concr ete, and entirely intrinsic. Kirch berg has obtained a completely different ch ara cterization o f O ∞ , as the uniquely deter mined purely infinite, separa ble, unital, ∗ Supported by the Danish Natural Science Researc h Council and the Fields Institute † Supported by the Deutsc he F orsch ungsgemeinsc haft through the SFB 478 1 nu clear C ∗ -algebra which is K K -equiv a len t to the complex num b ers (cf. [9]; see also [10]). Note that this descr iption is not intrinsic, since it compares O ∞ with the complex nu mbers (at le ast on the le vel of K K -theory). Using the well known facts that str ongly self-absor bing C ∗ -algebra s are nu clear and either stably finite with a uniq ue tracia l sta te or purely infinite, it is then immediate that O ∞ is the unique strongly self-abs o rbing C ∗ -algebra that has no tracial state and is K K - equiv alent to C . Moreov er, one might rephra se the co ndition o f b eing pur ely infinite in terms of the Cun tz s emigroup: a simple C ∗ -algebra is pur ely infinite if and only if it is infinite and has almo s t unper forated Cuntz semigr oup, in whic h case its Cun tz semigr oup coincides with the semigroup { 0 , ∞} . Regarding the Cuntz semigroup as a K -theoretic inv a r ian t in the bro adest s ense, one arrives at an abstract (but extrinsic) c haracteriz a tion of O ∞ among strongly self-absorbing C ∗ - algebras in terms of K -theor y da ta . Let us compar e the characterizations of O ∞ and of Z in more detail. Cun tz’s orig inal descrip- tion of O ∞ uses (infinitely many) generators and relations . While Jia ng and Su’s construction is not quite of this type, the building blocks of their inductiv e limit ar e given by (finitely many) generator s and rela tions—and for many purp oses this has proven to b e just as useful as if the whole a lgebra was presented as a universal C ∗ -algebra . Cun tz’s description of O ∞ as a crossed pro duct uses the dynamics of a certain ca nonical subalgebra . It is not so easy to write the Jiang–Su algebra as a crossed product by a single endomorphism, since such algebras tend to hav e nontrivial K 1 -groups, but we can nonetheless use dynamical prope r ties of certain canonica l subalgebra s to write Z a s a stationary inductive limit of s uc h a subalgebra; the co nnecting map is not ea sy to describe explicitly (its existence follo ws from a result of I. Hir s h be r g and the authors), but its p ertinent prop erty ca n b e stated in a very elegant ma nner. Mo re precisely , we show that the J iang–Su algebra is a statio nary inductiv e limit of a generalize d prime dimensio n drop C ∗ -algebra a nd a tra ce-collapsing endomorphism; any s uc h limit is is omorphic to Z . Although the connecting maps of the inductive system are not given explicitly , this is still an entirely in trinsic descr iption of the Jiang –Su algebra. W e wish to p oint out that this picture ha s already pro ven highly useful in [22]. The largest part of the pap er w ill b e devoted to finite v ersions (for Z ) of K irch b erg’s ch ara c- terization of O ∞ . The gener al pattern o f s uc h c harac ter izations go es as follows: one states v arious conditions ( C ∗ -algebra ic, K -theor etic and/or topologic a l), a nd shows that, if met by a stro ngly self-absor bing C ∗ -algebra D , then D is isomorphic to Z . Then one obser v es that Z itself satisfies the conditions in question. The latter will follow mostly from known r esults by Jiang and Su, the first na med a uthor, and the s econd named a utho r and E. Kir chberg. T o es tablish an iso morphism betw een D and Z , it will suffice to c o nstruct unital em bedding s in both directions, as we w ork within the class of stro ng ly self-abso rbing C ∗ -algebra s. Our first characteriza tio n singles out Z a s the uniquely determined s trongly self-a bsorbing C ∗ -algebra of stable ra nk one, for which the unit can be approximately divided in the C untz semigroup, and whic h is absorb ed b y an y UHF algebra. The latter condition will guarantee that the algebra in question is absorb ed b y the Jia ng–Su alge br a, using a joint result by I. Hirshberg and the authors. That the alge bra abso rbs Z follo ws from stable rank one toge ther with a ca ncellation theorem for the Cunt z semigroup es tablished in Section 4, and from the divisibility c o ndition. The key technical to ol here will be P rop osition 5.1, which provides criteria for embeddability of certa in dimension dro p int erv als into a unital C ∗ -algebra . Essen tially , this is done by analyzing a set of generator s and r elations quite differen t from those us ed to describ e dimension drop in terv als in [8]. Along similar lines, we then obtain ano ther characteriza tion of Z , as the uniquely determined strongly self-abso rbing finite C ∗ -algebra which has almost unp erforated Cuntz semigro up a nd which is absor bed by any UHF alg bra. W e p oint out that the latter condition in par ticular e ntails that the algebra in q ue s tion is K K -equiv alen t to the complex num b ers, whence this characteri- zation indeed may be view ed a s a finite analogue of Kirc hberg’s character iz a tion o f O ∞ . Again, the pro of uses ideas fro m P r opo sition 5.1 in a cr ucial wa y , along with a further careful a nalysis of divisibility prop erties of strongly self-absorbing C ∗ -algebra s. Our last c hara cterization of the Jia ng–Su algebra in volv es the decomposition r ank, a notion of co vering dimension for n uclear C ∗ -algebra s intro duced by E. Kirch b erg and the second named 2 author in [11]. Our r esult says that Z is the uniquely deter mined strongly self-a bsorbing C ∗ - algebra with finite deco mposition rank which is K K - equiv alent to the complex num b e rs. The pro of uses the fact that finite decompos ition ra nk entails sufficient regularity on the level of the Cun tz semigro up; tog ether with Pr opo sition 5.1 this shows that finite deco mposition ra nk and strongly s e lf- a bsorbing imply Z -stability . That Z is the o nly s uc h alg e br a then follows fro m a recent classifica tion theor em of the second named a uthor ([2 2]). W e note that decomp osition rank is of a very top olog ic a l flavour, and that there is current ly no analogous characterizatio n for O ∞ . The paper is org anized as follows. In Section 2, we reca ll some background r esults about strongly s elf-absorbing C ∗ -algebra s, the Jiang– Su alg ebra, and order zer o maps. In Section 3, w e characterize the Jiang–Su algebra as a sta tionary inductiv e limit of generaliz e d dimension drop algebras . Section 4 pr o vides a cancellatio n theor e m for the Cuntz semig roup of C ∗ -algebra s with stable rank one. In Section 5 we derive a n abstract c haracteriza tion of the Jiang– Su algebr a among strongly self-a bsorbing C ∗ -algebra s of s table rank one; in the subsequent s e ction w e obtain a v aria tio n of this result, asking the Cuntz semigroup to b e almost unper forated. Fina lly , in Section 7, we c haracter ize the Jiang–Su algebra amo ng strongly self-abso rbing C ∗ -algebra s of finite dec o mpositio n rank. The author s thank The Fields Institute and George Ellio tt for hos pitalit y during our stay in the fall of 2007, and we thank Georg e Elliott and E berha rd Kirch berg for a num b er of inspiring conv ersations on the question of ho w to c hara cterize the Jiang–Su algebr a a bstractly . 2 Some b ackg round results In this section we reca ll some well-known results abo ut stro ngly se lf-absorbing C ∗ -algebra s in gen- eral and ab out the Jia ng–Su alg e bra, Z , in particular. (The reader is referred to the introduction and to [18] for a definition and pro perties of s trongly se lf-absorbing C ∗ -algebra s.) W e als o rec all some fac ts ab out co mpletely p ositive contractiv e (c.p.c.) order zero maps. W e quote b elo w a result by Andre w T oms and the second named author ab out the hierarch y of strongly self-abs o rbing C ∗ -algebra s: Prop osition 2.1 (T oms –Win ter, [1 8]) L et D and E b e str ongly self-absorbing C ∗ -algebr as. Then: (i) D emb e ds unital ly into E if and only if D ⊗ E is isomorphic to E . (ii) D and E ar e isomorphic if D emb e ds u nital ly into E and E emb e ds unital ly into D . F or each sup ernatural n umber p le t M p denote the UH F algebr a of t ype p . W e say that p is of infinite typ e if p ∞ = p , in which case M p is stro ngly self-absorbing. (If p is a natura l n umber , then M p will denote the C ∗ -algebra of p × p matrices over the co mplex num be r s.) If p and q are natural or sup ernatural n um b ers, then we set Z p,q = { f ∈ C ([0 , 1] , M p ⊗ M q ) | f (0) ∈ M p ⊗ C , f (1) ∈ C ⊗ M q } . If p and q are natura l num b ers, then Z p,q is a so- called dimension-dr op C ∗ -algebr a . If p and q are relatively pr ime, then Z p,q is sa id to be prime . It is worth while noting that Z p,q has no non-triv ial pro jections (other than 0 and 1) if and only if p and q are relatively prime (natura l or sup ernatural num ber s ), and that its K -theory in that case is given by K 0 ( Z p,q ) ∼ = Z , K 1 ( Z p,q ) = 0 . Prime dimension-drop C ∗ -algebra s play a crucial role in the definition of the Jiang– Su algebra : 3 Theorem 2.2 (Jiang–Su, [8]) The induct ive limit of the se quenc e A 1 → A 2 → A 3 → · · · , wher e e ach A j is a prime dimension-dr op C ∗ -algebr a and wher e the c onne cting mappings ar e unital, is isomorphic to the Jiang–Su algebr a Z if and only if it is simple and has a unique tr acial state. Let p b e a natural num b er. Recall fro m [20] that a c.p.c. map ϕ : M p → A is said to hav e or der zer o if it preserves or thogonality . W e collect below some w ell known facts ab out order zero maps (see [20, Prop osition 3.2(a)] and [21, 1.2] for Pr opo s ition 2.3, and [1 9, 1 .2.3] for Pro positio n 2.4). W e let e ij , or sometimes e ( p ) ij , denote the canonical ( i, j )th matrix unit in M p . Prop osition 2.3 (Wi n ter, [20, 21]) L et A b e a C ∗ -algebr a, let p ∈ N , and let ϕ : M p → A b e a c.p.c. or der zer o m ap. (i) Ther e is a unique ∗ -homomorphi sm ˜ ϕ : C 0 ((0 , 1]) ⊗ M p → A such that ϕ ( x ) = ˜ ϕ ( ι ⊗ x ) for al l x ∈ M p , wher e ι ( t ) = t . (ii) Ther e is a un ique ∗ -homomorphi sm ¯ ϕ : M p → A ∗∗ given by s en ding the matrix u nit e ij in M p to the p artial isometry in A ∗∗ in the p olar de c omp osition of ϕ ( e ij ) . We have ϕ ( x ) = ¯ ϕ ( x ) ϕ (1 p ) = ϕ (1 p ) ¯ ϕ ( x ) for al l x ∈ M p ; and ¯ ϕ (1 p ) is the s u pp ort pr oje ction of ϕ (1 p ) . (iii) If, for some h ∈ A ∗∗ with k h k ≤ 1 , the element h ∗ h c ommutes with ¯ ϕ ( M p ) and satisfies h ∗ h ¯ ϕ ( M p ) ⊆ A , then the map ϕ h : M p → A given by ϕ h ( x ) = h ¯ ϕ ( x ) h ∗ , for x ∈ M p , is a wel l define d c.p.c. or der zer o map. The map ¯ ϕ in (ii) ab ov e will be called the supp orting ∗ -homomorphi sm o f ϕ . Prop osition 2.4 (Wi n ter, [19]) Supp ose x 1 , x 2 , . . . , x p ∈ A satisfy the r elations k x i k ≤ 1 , x 1 ≥ 0 , x i x ∗ i = x ∗ 1 x 1 , x ∗ j x j ⊥ x ∗ i x i , ( R p ) for al l i, j = 1 , . . . , n with i 6 = j . Then t he line ar map ψ : M p → A given by ψ ( e ij ) = x ∗ i x j is a c.p.c. or der zer o m ap. Note that the original version of the ab ov e r e s ult was phr ased in terms of elemen ts of the form e i 1 , i = 2 , . . . , p . Ho w ever, it is straightforward to check that the t wo versions ar e in fact equiv alent. The next pr o positio n contains a re cipe for finding a unital ∗ -homomorphis m from a dimensio n drop C ∗ -algebra Z p,q int o a unital C ∗ -algebra A . Prop osition 2.5 L et A b e a unital C ∗ -algebr a. F or r elatively prime natu r al numb ers p and q , supp ose t hat α : M p → A and β : M q → A ar e c.p.c. or der zer o maps satisfying α (1 p ) + β (1 q ) = 1 A , [ α ( M p ) , β ( M q )] = 0 . (2.1) Then ther e is a (unique) un ital ∗ -homomorphi sm ϕ : Z p,q → A , which makes the diagr am Z p,q ϕ   C 0 ([0 , 1) , M p ) 8 8 r r r r r r r r r r ˜ α ′ & & M M M M M M M M M M M C 0 ((0 , 1] , M q ) f f L L L L L L L L L L ˜ β x x q q q q q q q q q q q A c ommut ative, wher e the upwar ds maps ar e the obvious ones, wher e ˜ α and ˜ β a r e as in Pr op osi- tion 2.3(i), and whe r e ˜ α ′ is obtaine d fr om ˜ α by r eversing the orientation of the interval [0 , 1 ] . 4 Pro of: By [8, Prop osition 7.3], Z p,q is the universal C ∗ -algebra with genera tors a 1 , a 2 , . . . , a p , b 1 , b 2 , . . . , b q and rela tions ( R p ) from Prop osition 2.4 (with the x i ’s repla ced b y the a i ’s), ( R q ) (with the x i ’s replaced by the b i ’s), and [ a i , b j ] = 0 , [ a i , b ∗ j ] = 0 , p X k =1 a ∗ k a k + q X l =1 b ∗ l b l = 1 , for i = 1 , . . . , p and j = 1 , . . . , q . Iden tifying Z p,q with a sub- C ∗ -algebra of C ([0 , 1]) ⊗ M p ⊗ M q in the ca nonical wa y , and letting ι ∈ C ([0 , 1]) denote the function ι ( t ) = t , we can take the ge nerators in Z p,q to b e a i = (1 − ι ) 1 / 2 ⊗ e ( p ) 1 i ⊗ 1 q , b j = ι 1 / 2 ⊗ 1 p ⊗ e ( q ) 1 j . It is straightforward to c heck that the elemen ts ¯ a i = α (1 p ) 1 / 2 ¯ α ( e ( p ) 1 i ) = ˜ α ′ ((1 − ι ) 1 / 2 ⊗ e ( p ) 1 i ) , ¯ b j = β (1 q ) 1 / 2 ¯ β ( e ( q ) 1 j ) = ˜ β ( ι 1 / 2 ⊗ e ( q ) 1 j ) in A satisfy the relations a bov e, where ¯ α and ¯ β are the suppo rting ∗ -homomorphis ms fo r α and β , resp ectiv ely . By the universal pr oper t y of Z p,q there is (precisely) o ne unital ∗ -homomorphis m ϕ : Z p,q → A such that ϕ ( a i ) = ¯ a i and ϕ ( b j ) = ¯ b j for all i and j ; and one chec ks (on elements o f the form (1 − ι ) 1 / 2 ⊗ e ( p ) 1 i ∈ C 0 ([0 , 1) , M p ) and ι 1 / 2 ⊗ e ( q ) 1 j ∈ C 0 ((0 , 1] , M q )) that the diagr am in the prop osition is commutativ e.  3 The Jiang–Su algebra and the C ∗ -algebras Z p,q In this sectio n we characterize the Jiang–Su algebra using dynamical pro perties of the C ∗ -algebra s Z p,q (defined in the pr evious section, and with p and q sup e r natural num bers ). The fir st result is an immediate consequence of one of the main result fro m [7]: Prop osition 3.1 L et D b e a str ongly self-absorbing C ∗ -algebr a which tensorial ly is absorb e d by every UHF-algebr a B , i.e., D ⊗ B ∼ = B . Then D ⊗ Z p,q ∼ = Z p,q whenever p and q ar e infinite sup ern atur al numb ers. Pro of: The C ∗ -algebra Z p,q is in a canonica l wa y a C ([0 , 1])- algebra with fibres b eing UHF- algebras of type p at the left end-p oint, o f type pq a t (0 , 1), and of type q at the rig h t end-p oint. Each fibre is accordingly a UHF-algebra and so a bs orbs D tensorially . As the in terv a l [0 , 1] has finite dimensio n it follows fro m [7] that Z p,q also abs o rbs D .  The Jiang –Su alg ebra is strong ly self-a bsorbing ([18]) and it is b eing abs orbe d by all UHF-alg ebras ([8]), and so w e get: Corollary 3.2 L et p and q b e infinite su p ernatur al nu m b ers. Then Z p,q absorbs t he Jiang–Su algebr a: Z ⊗ Z p,q ∼ = Z p,q . The prop osition b elow is proved in [15, Prop osition 2.2] in the case where p = n ∞ and q = m ∞ , and where n and m are na tural nu mbers, that are relatively prime. W e shall need this r esult in the slightly more general case where p and q are arbitrary supernatur al num b ers that are r elativ ely prime. Assume tha t such p and q are given. Then write M p and M q as inductive limits M p 1 → M p 2 → M p 3 → · · · → M p , M q 1 → M q 2 → M q 3 → · · · → M q (with unital connecting mappings) for suitable seque nc e s of natural num b ers { p j } and { q j } . As p j | p and q j | q , it is automatic that p j and q j are relatively pr ime for a ll j . Let σ j : M p j ⊗ M q j → M p j +1 ⊗ M q j +1 be a unital ∗ -homomorphis m such that σ j ( M p j ⊗ C ) ⊆ M p j +1 ⊗ C a nd σ j ( C ⊗ M q j ) ⊆ C ⊗ M q j +1 . Then Z p,q is the limit of the inductive system Z p 1 ,q 1 ρ 1 / / Z p 2 ,q 2 ρ 2 / / Z p 3 ,q 3 ρ 3 / / · · · / / Z p,q , 5 where ρ j is given by ρ j ( f ) = σ j ◦ f . Pro ceeding as in the pro of of [15, Prop osition 2.2] one o btains the fo llo wing: Prop osition 3.3 L et p and q b e sup ernatu r al nu mb ers t hat ar e r elatively prime. Th en Z p,q emb e ds unital ly into Z . Combining Pro p osition 3.3 and Corollary 3 .2 we g et unita l embeddings Z p,q → Z → Z p,q whenever p and q are infinite s uperna tur al num b ers that a r e rela tiv ely pr ime. As w e shall see below, this characterizes Z a mong s trongly self-abso rbing C ∗ -algebra s. Firs t we note a related result. A unital e ndo morphism ϕ on a unital C ∗ -algebra A is said to b e tr ac e-c ol lapsing if τ ◦ ϕ = τ ′ ◦ ϕ for a n y pair of tra cial states τ and τ ′ on A . Theorem 3.4 L et p a nd q b e infinite su p ernatur al numb ers t hat ar e r elatively prime. (i) Ther e exists a tr ac e-c ol lapsing unital endomorphism on Z p,q . (ii) L et ϕ b e any tr ac e-c ol lapsing unital en domorphism on Z p,q . Then the Jiang–Su algebr a Z is isomorphi c to the inductive limit of the stationary inductive se quenc e: Z p,q ϕ / / Z p,q ϕ / / Z p,q ϕ / / · · · . Pro of: (i). T ake the compos itio n o f a n y unital em b eddings Z p,q → Z → Z p,q (cf. the remarks ab o ve) and re call (eg. from [8]) that Z has a unique trace. (ii). W e note first that the inductive limit, call it A , of the sequence ab ov e is a n inductiv e limit of prime dimension-drop C ∗ -algebra s, i.e., of C ∗ -algebra s of the form Z n,m with n and m na tur al nu mbers that are r elativ ely prime. Indeed, ea c h Z p,q is suc h an inductive limit, cf. the rema rks ab o ve. Hence A ca n lo cally b e approximated by prime dimension-dro p C ∗ -algebra s. Each (prime) dimension-drop C ∗ -algebra is w eakly stable by [8, Prop osition 7.3], whence any C ∗ -algebra that lo cally can b e approximated by pr ime dimension- drop C ∗ -algebra s is an actual inductiv e limit o f them, cf. [12]. It now follows from Jiang a nd Su, [8], c f. Theorem 2.2, that A is isomorphic to the J iang–Su algebra Z if a nd only if A is simple and has unique trace. Uniqueness o f the trace of A follows easily from the assumption that ϕ is trace-collapsing . The endomor phism ϕ is necessa rily injectiv e. Indeed, if it w ere not and I is the kernel of ϕ , then ϕ would induce an embedding o f Z p,q /I into Z p,q . But any non-trivial quotient of Z p,q has non-trivial pro jections (i.e., pro jections other than 0 and 1), wher eas Z p,q only contains the trivial pro jections, cf. the remar ks in Section 2. That A is s imple now follows from the fact that ϕ ( a ) is full in Z p,q for all no n-zero a ∈ Z p,q . T o see this, let π t : Z p,q → M pq denote the fibre ma p (for t ∈ [0 , 1 ]). L e t τ be the (unique) tracial state o n M pq . Then t 7→ ( τ ◦ π t ◦ ϕ )( a ∗ a ) is co nstan t by the ass umption that ϕ is trace-c o llapsing, and this function is non-z ero (b ecause a is non-zero and ϕ is injective). Hence π t ( ϕ ( a )) 6 = 0 for all t ∈ [0 , 1], whic h entails that ϕ ( a ) is full in Z p,q .  Prop osition 3.5 The Jiang–Su algebr a Z is the only str ongly self-absorbing C ∗ -algebr a for which ther e ar e r elatively prime infinite sup ernatur al numb ers p and q and unital emb e ddings Z p,q → Z → Z p,q . Pro of: Suppo s e that p and q are infinite sup ernatural num b ers that are re la tiv ely prime a nd tha t A is a str ongly self-absor bing C ∗ -algebra for which there are unital ∗ -homomorphis ms λ : Z p,q → A and µ : A → Z p,q . Co nsider the inductiv e s ystem A µ / / Z p,q λ / / A µ / / Z p,q λ / / A µ / / Z p,q λ / / · · · . 6 The inductiv e limit of this system coincides w ith the inductive limits of the tw o subsy stems b elo w: A λ ◦ µ / / A λ ◦ µ / / A λ ◦ µ / / · · · , Z p,q µ ◦ λ / / Z p,q µ ◦ λ / / Z p,q µ ◦ λ / / · · · . An y unital endomorphism on a stro ng ly self-absor bing C ∗ -algebra is approximately unitar ily equiv alent to the identit y by [1 8 , Coro llary 1.12]. It thus follows from a n inductiv e limit a rgument (after Elliott — see fo r ex ample [1 4, Co rollary 2 .3.3]) that the former inductiv e system above has inductive limit is omorphic to A . As A ha s unique tra ce (cf. [18, Theo r em 1.7]) the unital endomo rphism µ ◦ λ is trac e -collapsing. Hence the latter of the tw o inductive systems above has limit isomorphic to Z by Theorem 3.4. This proves that A is isomorphic to Z .  4 A cancellation theorem for the Cun tz semigroup In this section we prov e a cancellation theorem for the Cun tz semig roup for C ∗ -algebra s of stable rank one. This res ult, which might b e of independent interest, and which e x tends a recent result of Elliott, [5], is needed for the next section. W e r efer the reader to [15] and [13] for notation and background ma terial on Cun tz compariso n of positive elements and on the Cunt z semigroup. Recall the following fa c t, proved in [13]: Prop osition 4.1 L et A b e a unital C ∗ -algebr a of stable r ank one, let a, b b e p ositive elements in A such that a - b , and let ε > 0 . It fol lows that ther e is a unitary element u ∈ A such that u ∗ ( a − ε ) + u ∈ bAb. The tw o res ults below show that the Cun tz semig roup W ( A ) of a C ∗ -algebra of stable ra nk o ne has a lmost cancellatio n: Prop osition 4.2 L et A b e a C ∗ -algebr a of stable r ank one, let a, b b e p ositive elements in M ∞ ( A ) , and let p b e a pr oje ction in M ∞ ( A ) such tha t a ⊕ p - b ⊕ p. Then a - b . Pro of: Up on replacing A by a s uitable matrix algebr a ov er A w e ca n assume tha t a, b, p all b elong to A and that a ⊥ p and b ⊥ p . Let 0 < ε < 1. As ( p − ε ) + = (1 − ε ) p , we ca n use Pro positio n 4.1 to find a unitary u in the unitization of A such that u  ( a − ε ) + + p  u ∗ ∈ ( b + p ) A ( b + p ) def = B . Being a he r editary sub- C ∗ -algebra of A , B and hence also its unitiza tion are of stable rank one. Now, u pu ∗ and p are equiv alent pro jections in B , and so there is a unitary v in the unitization of B (that we may reg ard as b eing a sub- C ∗ -algebra of the unitization of A ) such tha t upu ∗ = v pv ∗ . Note that v ∗ u ( a − ε ) + u ∗ v ∈ B , v ∗ u ( a − ε ) + u ∗ v ⊥ v ∗ upu ∗ v = p, which entails that v ∗ u ( a − ε ) + u ∗ v b elongs to (1 − p ) B (1 − p ) = bAb . This proves that ( a − ε ) + - b ; and as ε > 0 was ar bitrary , we conclude that a - b .  Theorem 4.3 (Cancellatio n) L et A b e a C ∗ -algebr a of stable r ank one, and let x, y b e elements in the Cunt z semigr oup W ( A ) such that x + h c i ≤ y + h ( c − ε ) + i . for some c ∈ M ∞ ( A ) + and for some ε > 0 . Then x ≤ y . 7 Pro of: Up on replacing A by a matrix algebra ov er A we can ass ume that c belo ngs to A , and that x = h a i , y = h b i for some p ositive elements a, b in A with a ⊥ c and b ⊥ c . Next, upon adjoining a unit to A we may assume that A is unital (this w ill not affect the compariso n of the elements a, b, c ). Let h ε : R + → R + be given by h ε ( t ) = ( ε − 1 ( ε − t ) , 0 ≤ t ≤ ε, 0 , t ≥ ε . (4.1) Then ( c − ε ) + ⊥ h ε ( c ) a nd c + h ε ( c ) is inv ertible. Hence a ⊕ 1 A - a ⊕ ( c + h ε ( c )) - a ⊕ c ⊕ h ε ( c ) - b ⊕ ( c − ε ) + ⊕ h ε ( c ) - b ⊕ (( c − ε ) + + h ε ( c )) - b ⊕ 1 A . The claim now follows from Prop osition 4.2.  One ca nnot strengthen Theore m 4.3 to the more intuitiv e statement: x + z ≤ y + z implies x ≤ y , when x, y , z are elements in the Cuntz semig roup, W ( A ), of a n arbitrar y C ∗ -algebra A o f stable rank one. Indeed, if one takes A to b e a UHF algebra with trace τ , p to b e a pro jection, a nd a, b to b e positive elemen ts in A s uc h that τ ( p ) = d τ ( a ) (= lim n →∞ τ ( a 1 /n )) , and such that 0 is an a ccum ulation p oint of sp( a ) \ { 0 } and of sp( b ) \ { 0 } , then p - | a but p ⊕ b - a ⊕ b (see [1] for more details ). W e shall als o need the lemma b elow for the next section. Firs t we fix s o me nota tio n to b e used here a nd in the sequel. Notation 4.4 F or positive n um b ers 0 ≤ η < ε ≤ 1 define con tin uous functions f ε , g η, ε : [0 , 1] → R + by g η, ε ( t ) =      0 , t ≤ η , 1 , ε ≤ t ≤ 1 , linear, else, f ε = g 0 ,ε . Lemma 4.5 L et A b e a unital C ∗ -algebr a of stable r ank one, and let a, b ∈ A + b e such that h a i + h b i ≥ h 1 A i . Then 1 A − f ε ( a ) - ( b − ε ) + for some ε > 0 . Pro of: As h (1 A − ε ) + i = h 1 A i for all ε ∈ [0 , 1) one can conclude fro m [13] that here exists δ > 0 such that h ( a − δ ) + i + h ( b − δ ) + i ≥ h 1 A i . T ake ε suc h that 0 < ε < δ . Observe that 1 A − f ε ( a ) ⊥ ( a − ε ) + . It follows that h 1 A − f ε ( a ) i + h ( a − ε ) + i ≤ h 1 A i ≤ h ( a − δ ) + i + h ( b − δ ) + i ≤ h ( b − ε ) + i + h ( a − δ ) + i = h ( b − ε ) + i +  ( a − ε ) + − ( δ − ε )  +  . By Theo rem 4.3 this implies that 1 A − f ε ( a ) - ( b − ε ) + .  5 An axiomatic description of the Jiang–Su algebra The main r esult of this section is Theorem 5 .5 b elow in which a new c hara cterization of the J iang– Su algebr a is giv en. The pro of uses facts about the Cun tz semigr oup and comparison theory for po sitiv e ele ments deriv ed in the pr evious sectio n. Two pos itiv e e lemen ts a and b in a C ∗ -algebra A ar e said to b e e quivalent , written a ∼ b , if there is x ∈ A such that a = x ∗ x and b = xx ∗ . It is ea sy to see that a ∼ b implies h a i = h b i in W ( A ). Recall the definition of the dimension-dro p C ∗ -algebra Z p,q from Sectio n 2. 8 Prop osition 5.1 L et A b e a u nital C ∗ -algebr a of stable r ank one, and let n b e a natur al numb er. The fol lowing four c onditions a r e e quivalent: (i) Ther e exists x ∈ W ( A ) such that nx ≤ h 1 A i ≤ ( n + 1) x . (ii) Ther e exist ε > 0 and mut u al ly e quivalent and ortho gonal p ositive elements b 1 , b 2 , . . . , b n in A such that 1 A − ( b 1 + b 2 + · · · + b n ) - ( b 1 − ε ) + . (iii) Ther e ar e elements v , s 1 , s 2 , . . . , s n ∈ A of norm 1 such t hat s ∗ 1 s 1 = s i s ∗ i , s ∗ i s i ⊥ s ∗ j s j , v ∗ v = 1 A − n X k =1 s ∗ k s k , vv ∗ s ∗ 1 s 1 = v v ∗ , (5.1) for al l i and j with i 6 = j . (iv) Ther e is a un ital ∗ -homomorphi sm fr om the C ∗ -algebr a Z n,n +1 into A . The hyp othesis of stable r ank one is only ne e de d fo r the imp lic ation (i) ⇒ (ii) . Pro of: (i) ⇒ (ii). Find a p ositive element d in s ome matrix algebra M k ( A ) over A such that x = h d i . There is δ > 0 suc h that ( n + 1) h ( d − δ ) + i ≥ h 1 A i (cf. [13]). As n h d i ≤ h 1 A i there is a row matrix t ∈ M 1 ,nk ( A ) suc h that t ∗ t = t ∗ 1 A t = ( d − δ ) + ⊕ ( d − δ ) + ⊕ · · · ⊕ ( d − δ ) + . W rite t = ( t 1 , t 2 , . . . , t n ) with t i ∈ M 1 ,k ( A ). T he n t ∗ i t j = ( ( d − δ ) + , i = j, 0 , i 6 = j. Put e j = t j t ∗ j . Then e 1 , e 2 , . . . , e n are pair wise orthog onal p ositive elements in A each of whic h is equiv alent to ( d − δ ) + . It follows in particular that h 1 A i ≤ ( n + 1 ) h ( d − δ ) + i = ( n + 1) h e 1 i = h e 1 + e 2 + · · · + e n i + h e 1 i . Now use Lemma 4.5 (and r ecall fro m 4.4 the definition of the function f η ) to see that there exists η > 0 such that 1 A − f η ( e 1 + e 2 + · · · + e n ) - ( e 1 − η ) + . Put b j = f η ( e j ). Note that e 1 - f η ( e 1 ) = b 1 (and a lso b 1 - e 1 ), so there ex ists ε > 0 such that ( e 1 − η ) + - ( b 1 − ε ) + (see [13]). It now follows that the elements b 1 , b 2 , . . . , b n are as desired, bec ause f η ( e 1 + e 2 + · · · + e n ) = b 1 + b 2 + · · · + b n . (ii) ⇒ (iii). W e may ass ume that ε < 1. Since each b i is equiv alent to b 1 , there are x 2 , . . . , x n ∈ A suc h that x i x ∗ i = b 1 and x ∗ i x i = b i . Let x i = v i | x i | = | x ∗ i | v i be the po lar decomp osition with v i a partial iso metry in A ∗∗ . Put s 1 = f ε ( b 1 ) 1 / 2 and put s i = v i f ε ( b i ) 1 / 2 = f ε ( b 1 ) 1 / 2 v i ∈ A for i = 2 , 3 , . . . , n (cf. 4.4). Then s i s ∗ i = f ε ( b 1 ) = s ∗ 1 s 1 for all i , and s ∗ i s i = f ε ( b i ) whenc e s ∗ i s i ⊥ s ∗ j s j when i 6 = j . Note that 1 − ( f ε ( b 1 ) + · · · + f ε ( b n )) belo ngs to the hereditary sub- C ∗ -algebra genera ted by (1 − ( b 1 + · · · + b n ) − (1 − ε )) + . Cho ose 0 < η < 1 − ε and note that g η, 1 − ε (1 − ( b 1 + · · · + b n )) is a unit for (1 − ( b 1 + · · · + b n ) − (1 − ε )) + and hence also for 1 − ( f ε ( b 1 ) + · · · + f ε ( b n )) (cf. 4.4). It follows from the hypothesis and [13, Prop osition 2.4 ] that there is x ∈ A such tha t x ∗ x = g η, 1 − ε (1 − ( b 1 + · · · + b n )) , xx ∗ ∈ ( b 1 − ε ) + A ( b 1 − ε ) + . Set v = x  1 A − ( f ε ( b 1 ) + · · · + f ε ( b n ))  1 / 2 . 9 Then v ∗ v = 1 A − ( f ε ( b 1 ) + · · · + f ε ( b n )) = 1 A − n X k =1 s ∗ k s k . Since vv ∗ belo ngs to ( b 1 − ε ) + A ( b 1 − ε ) + and ( b 1 − ε ) + f ε ( b 1 ) = ( b 1 − ε ) + , w e get that v v ∗ s ∗ 1 s 1 = v v ∗ f ε ( b 1 ) = v v ∗ . (iii) ⇒ (iv). In the light of P rop osition 2.5, it suffices to construct or der zero c.p.c. maps α : M n +1 → A and β : M n → A with comm uting ima ges such that α (1 n +1 ) + β (1 n ) = 1 A . The construction of α a nd β (and the verification that they hav e the desired prop erties) is rather long and tedious. It may be constructive to note that one q uite easily can write down order zero c.p.c. maps µ : M n +1 → A and ρ : M n → A suc h that µ (1 n +1 ) + ρ (1 n ) ≥ 1 A . Indeed, put t 1 = ( v ∗ v ) 1 / 2 , t j +1 = v ∗ s j ( j = 1 , 2 , . . . , n ) . One ca n easily verify that the elements s 1 , s 2 , . . . , s n satisfy the r elations ( R n ) of Prop osition 2 .4, and that t 1 , t 2 , . . . , t n +1 satisfy the rela tio ns ( R n +1 ). It therefor e follows from Pro positio n 2 .4 that there a r e order zero c.p.c. maps µ : M n +1 → A, µ ( e ( n +1) ij ) = t ∗ i t j , ρ : M n → A, ρ ( e ( n ) ij ) = s ∗ i s j . These maps fail to ha v e commuting images, and µ (1 n +1 ) + ρ (1 n ) is larg er tha n but not equal to 1 A . W e shall in the follo wing modify these maps so that they g et the des ir ed prop erties. In the pro cess w e shall make muc h us e of the map ρ , but we shall mak e no further explicit use of the map µ . Upo n repla c ing s 1 by ( s ∗ 1 s 1 ) 1 / 2 we may assume that s 1 ≥ 0. Let ¯ v | v | b e the p olar decomp osition of v with ¯ v a partia l isometry in A ∗∗ and | v | = ( v ∗ v ) 1 / 2 . Let us no te some r e lations satisfied by the e le men ts v , ¯ v , s 1 , . . . , s n to be used later in the pro of: (a) s 1 v = v , s 1 ¯ v = ¯ v . (b) s j v = s j ¯ v = 0 for j = 2 , 3 , . . . , n . (c) vv ∗ ⊥ v ∗ v , (d) s i s j = 0 for all i = 2 , 3 , . . . , n , j = 1 , 2 , . . . , n . (e) [ s i , v ∗ v ] = [ s ∗ i , v ∗ v ] = 0 for a ll i = 1 , . . . , n , (f ) c ¯ v ∗ ¯ v = c for all c ∈ v ∗ v Av ∗ v . (g) ¯ v c ∈ A for all c ∈ v ∗ v Av ∗ v . The first par t of (a) follows b y the hypo thesis that v v ∗ = s ∗ 1 s 1 v v ∗ , a nd the s e cond part of (a) follows from the first pa rt and standard prop erties of the p o lar decomp o sition. T o see (b) use that s ∗ j s j v v ∗ = s ∗ j s j s ∗ 1 s 1 v v ∗ = 0. Next, v ∗ v · v v ∗ = (1 A − n X j =1 s ∗ j s j ) v v ∗ (b) = (1 A − s ∗ 1 s 1 ) v v ∗ (5.1) = 0 , whence (c) holds. F or i 6 = 1 we have s ∗ i s i s j s ∗ j = s ∗ i s i s ∗ 1 s 1 = 0, so (d) holds. F or i = 1 , 2 , . . . , n one has v ∗ v s i = (1 A − n X j =1 s ∗ j s j ) s i (d) = s i − s ∗ 1 s 1 s i (5.1) = s i − s i s ∗ i s i (d),(5.1) = s i (1 A − n X j =1 s ∗ j s j ) = s i v ∗ v . This prov es (e). (f ) a nd (g) are well-known prop erties of the pola r decomp osition. Recall the definition of the order zero c.p.c. map ρ fro m a bov e, and as socia te to it the sup- po rting ∗ -homomorphis m ¯ ρ : M n → A ∗∗ defined in P ropo sition 2 .3 (ii). Note (from (5.1) and Prop osition 2.3 (ii)) that 10 (h) ρ (1 n ) = 1 A − v ∗ v , (i) ρ (1 n ) ¯ ρ ( x ) = ρ ( x ) ∈ A for all x ∈ M n . Define a map ϕ : v ∗ v Av ∗ v → A b y ϕ ( c ) = n X i =1 s ∗ i ¯ v c ¯ v ∗ s i , c ∈ v ∗ v Av ∗ v , cf. (g). The ma p ϕ is cle arly linear and hermitian, a nd, a s shown be low, it is actually a ∗ -homo- morphism. T ake c 1 , c 2 ∈ v ∗ v Av ∗ v a nd calculate: ϕ ( c 1 ) ϕ ( c 2 ) = n X i,j =1 s ∗ i ¯ v c 1 ¯ v ∗ s i s ∗ j ¯ v c 2 ¯ v ∗ s j = n X i =1 s ∗ i ¯ v c 1 ¯ v ∗ s i s ∗ i ¯ v c 2 ¯ v ∗ s i = n X i =1 s ∗ i ¯ v c 1 ¯ v ∗ s ∗ 1 s 1 ¯ v c 2 ¯ v ∗ s i (a) = n X i =1 s ∗ i ¯ v c 1 ¯ v ∗ ¯ v c 2 ¯ v ∗ s i (f ) = n X i =1 s ∗ i ¯ v c 1 c 2 ¯ v ∗ s i = ϕ ( c 1 c 2 ) , where we in the second and third equation hav e used the relatio ns for the s i ’s from (5.1). W e note the fo llo wing r elations concerning the ∗ -homomorphis m ϕ : (j) ϕ ( c )¯ v = ¯ v c for all c ∈ v ∗ v Av ∗ v , (k) [ ϕ ( c ) , s i ] = [ ϕ ( c ) , s ∗ i ] = 0 for all c ∈ v ∗ v Av ∗ v a nd i = 1 , 2 , . . . , n , (l) [ ϕ ( c ) , ¯ ρ ( x )] = 0 for all c ∈ v ∗ v Av ∗ v a nd x ∈ M n , (m) ϕ ( v ∗ v Av ∗ v ) ⊥ v ∗ v Av ∗ v . W e first prove (j): ϕ ( c ) ¯ v (b) = s ∗ 1 ¯ v c ¯ v ∗ s 1 ¯ v (a) = ¯ v c ¯ v ∗ ¯ v (f ) = ¯ v c. Next, fo r i = 1 , 2 , . . . , n w e hav e ϕ ( c ) s i (d) = s ∗ 1 ¯ v c ¯ v ∗ s 1 s i (a) = ¯ v c ¯ v ∗ s i (a) = s ∗ 1 s 1 ¯ v c ¯ v ∗ s i (5.1) = s i s ∗ i ¯ v c ¯ v ∗ s i (5.1) = s i ϕ ( c ) , hence (k) ho lds. The image of ¯ ρ is contained in the weak closure (in A ∗∗ ) of the C ∗ -algebra generated by the s i ’s, so (l) follows fr o m (k). The calcula tio n: ϕ ( v ∗ v ) v ∗ v = n X j =1 s ∗ i ¯ v v ∗ v ¯ v ∗ s i v ∗ v = n X j =1 s ∗ i v v ∗ s i v ∗ v (e) = n X j =1 s ∗ i v v ∗ v ∗ v s i (c) = 0 , shows that (m) holds. Consider the tw o C ∗ -algebra s: D 1 = { f ∈ C 0 ([0 , 1) , M n ) | f (0) ∈ C · 1 n } , D 2 = { f ∈ C 0 ([0 , 1) , M n ⊗ M n ) | f (0) ∈ C · 1 n ⊗ 1 n } . Note (b y (e)) that v ∗ v comm utes with ρ ( M n ) a nd (hence) with ¯ ρ ( M n ). The ∗ -homomorphis m ¯ λ : C ([0 , 1] , M n ) = C ([0 , 1]) ⊗ M n → A ∗∗ given b y ¯ λ ( f ⊗ x ) = f (1 − v ∗ v ) ¯ ρ ( x ) , f ∈ C ([0 , 1]) , x ∈ M n , 11 restricts to a ∗ -homomorphis m λ : D 1 → v ∗ v Av ∗ v . Indeed, D 1 is generated as a C ∗ -algebra by the elements (1 − ι ) ⊗ 1 n and ι (1 − ι ) ⊗ x , x ∈ M n , (where ι ( t ) = t ), a nd ¯ λ ((1 − ι ) ⊗ 1 n ) = v ∗ v , ¯ λ ( ι (1 − ι ) ⊗ x ) = (1 − v ∗ v ) v ∗ v ¯ ρ ( x ) (h) , (i) = v ∗ v ρ ( x ) ∈ v ∗ v Av ∗ v . By (l) we can define a ∗ -homomorphis m γ : D 2 → A by γ ( f ⊗ x ⊗ y ) = ( ϕ ◦ λ )( f ⊗ x ) ¯ ρ ( y ) , f ∈ C 0 ([0 , 1)) , x, y ∈ M n . T o see tha t the imag e of γ is contained in A (ra ther than in A ∗∗ ) observe that the imag e of ϕ is contained the hereditary sub- C ∗ -algebra o f A gener a ted b y P n i =1 s ∗ i s i = 1 A − v ∗ v = ρ (1 n ), and so by (i), ϕ ( c ) ¯ ρ ( y ) b elongs to A for all c ∈ v ∗ v Av ∗ v a nd y ∈ M n . Let u ∈ C ([0 , 1] , M n ⊗ M n ) ∩ M ( D 2 ) b e a self-a djoin t unitary such that u ( t ) = 1 n ⊗ 1 n (0 ≤ t ≤ 1 / 3) , u ( t )( x ⊗ y ) u ( t ) = y ⊗ x (2 / 3 ≤ t ≤ 1) , for a ll x, y ∈ M n . P ut w = γ ∗∗ ( u ) ∈ A ∗∗ (where γ ∗∗ : D ∗∗ 2 → A ∗∗ is the canonical extension of γ ). Let g ∈ C 0 ([0 , 1)) b e given by g ( t ) =      1 , 0 ≤ t ≤ 2 / 3 , 0 , t = 1 , linear , 2 / 3 < t < 1 . W e list some easily verified identities inv olving u , w , a nd g . (n) u · ( g ⊗ 1 n ⊗ 1 n ) ∈ D 2 , (o) w · ( ϕ ◦ λ )( g ⊗ 1 n ) = ( ϕ ◦ λ )( g ⊗ 1 n ) · w = γ ( u · ( g ⊗ 1 n ⊗ 1 n )) ∈ A . (p) ((1 − g ) ⊗ 1 n ⊗ 1 n ) · u · (1 ⊗ x ⊗ y ) · u = (1 − g ) ⊗ y ⊗ x for all x, y ∈ M n . Put x i = λ ( g ⊗ 1 n ) 1 / 2 ¯ v ∗ s i w ( i = 1 , 2 , . . . , n ) , x n +1 = λ ( g ⊗ 1 n ) 1 / 2 . The x i ’s sa tisfy the follo wing relations: (q) x ∗ i x n +1 = w ( ϕ ◦ λ )( g ⊗ 1 n ) ρ ( e ( n ) i 1 ) ¯ v for i = 1 , 2 , . . . , n . (r) x ∗ i x j ∈ A for all i, j = 1 , 2 , . . . , n + 1 . (s) x j x ∗ j = λ ( g ⊗ 1 n ) = x ∗ n +1 x n +1 for j = 1 , 2 , . . . , n + 1. (t) x ∗ i x i ⊥ x ∗ j x j for i 6 = j . (u) P n i =1 s ∗ i ¯ v ¯ v ∗ s i ϕ ( c ) = ϕ ( c ) for all c ∈ v ∗ v Av ∗ v . (v) P n +1 j =1 x ∗ j x j = λ ( g ⊗ 1 n ) + ( ϕ ◦ λ )( g ⊗ 1 n ). Let us verify these identities. F or i = 1 , 2 , . . . , n we ha ve x ∗ i x n +1 = ws ∗ i ¯ v λ ( g ⊗ 1 n ) (recall that w = w ∗ ) a nd s ∗ i ¯ v λ ( g ⊗ 1 n ) (j) = s ∗ i ( ϕ ◦ λ )( g ⊗ 1 n ) ¯ v (k) = ( ϕ ◦ λ )( g ⊗ 1 n ) s ∗ i ¯ v (a) = ( ϕ ◦ λ )( g ⊗ 1 n ) s ∗ i s 1 ¯ v = ( ϕ ◦ λ )( g ⊗ 1 n ) ρ ( e ( n ) i 1 ) ¯ v . This prov es that (q) holds. As s ∗ i ¯ v λ ( g ⊗ 1 n ) belo ngs to A (b y (g)), we can use (o) to conclude that x ∗ i x n +1 belo ngs to A . When i, j = 1 , 2 , . . . , n we hav e x ∗ i x j = w ∗ s ∗ i ¯ v λ ( g ⊗ 1 n ) ¯ v ∗ s j w ; and ¯ v λ ( g ⊗ 1 n ) ¯ v ∗ belo ngs to A by (g). Moreov er, s ∗ i ¯ v λ ( g ⊗ 1 n ) ¯ v ∗ s j (j),(k) = ( ϕ ◦ λ )( g ⊗ 1 n ) 1 / 2 s ∗ i ¯ v ¯ v ∗ s j ( ϕ ◦ λ )( g ⊗ 1 n ) 1 / 2 . 12 W e can now use (o) to s ee that x ∗ i x j belo ngs to A . T o see that (s) holds, note that ¯ v ∗ s j ww ∗ s ∗ j ¯ v = ¯ v ∗ s ∗ 1 s 1 ¯ v = ¯ v ∗ ¯ v and us e (f ). Use (5.1) to see that x ∗ i x i ⊥ x ∗ j x j when i 6 = j and i, j ≤ n . W e pr oceed to esta blis h (u): n X i =1 s ∗ i ¯ v ¯ v ∗ s i ϕ ( c ) (5.1) = n X i =1 s ∗ i ¯ v ¯ v ∗ s i s ∗ i ¯ v c ¯ v ∗ s i (5.1) = n X i =1 s ∗ i ¯ v ¯ v ∗ s ∗ 1 s 1 ¯ v c ¯ v ∗ s i (a) = n X i =1 s ∗ i ¯ v c ¯ v ∗ s i = ϕ ( c ) . Next, n X i =1 x ∗ i x i = n X i =1 w ∗ s ∗ i ¯ v λ ( g ⊗ 1 n ) ¯ v ∗ s i w (j),(k) = n X i =1 w ∗ ( ϕ ◦ λ )( g ⊗ 1 n ) s ∗ i ¯ v ¯ v ∗ s i w (u) = w ∗ ( ϕ ◦ λ )( g ⊗ 1 n ) w (o) = ( ϕ ◦ λ )( g ⊗ 1 n ) . ¿F rom this we see that (v) holds, and we a lso see that x ∗ i x i ⊥ x ∗ n +1 x n +1 (cf. (m)). It follows from (r), (s), (t) and Pr opo s ition 2 .4 that there is an o rder zero c.p.c. map α : M n +1 → A, given b y α ( e ( n +1) ij ) = x ∗ i x j ( i, j = 1 , 2 , . . . , n + 1) . W e list some prop erties of α : (w) α (1 n +1 ) = λ ( g ⊗ 1 n ) + ( ϕ ◦ λ )( g ⊗ 1 n ) (x) [ λ ( g ⊗ 1 n ) , ¯ ρ ( M n )] = 0 , [ α ( 1 n +1 ) , ¯ ρ ( M n )] = 0 . (y) 1 A − α (1 n +1 ) ∈ ρ (1 n ) Aρ (1 n ). (z) (1 A − α (1 n +1 )) ¯ ρ ( M n ) ⊆ A . (w) is just a reformulation of (v). The fir s t pa rt o f (x) follows from (e) when we note that g is a function of 1 − ι , whence λ ( g ⊗ 1 n ) belo ngs to the C ∗ -algebra ge nerated by λ ((1 − ι ) ⊗ 1 n ) = v ∗ v , and that the image of ¯ ρ is con tained in the w eak closure (in A ∗∗ ) o f the C ∗ -algebra generated by the s i ’s. The seco nd part of (x) follows fro m the first part together with (w) a nd (l). As g ≥ 1 − ι we ge t λ ( g ⊗ 1 n ) ≥ λ ((1 − ι ) ⊗ 1 n ) = v ∗ v , whence 1 A − α (1 n +1 ) ≤ 1 A − v ∗ v = ρ (1 n ). This prov es (y). Finally , (z) follows fro m (y) and the fact that ρ (1 n ) ¯ ρ ( x ) = ρ ( x ) ∈ A . It follows from (x) and (z) above, together with P rop osition 2.3 (iii), that β ( x ) = (1 A − α (1 n +1 )) ¯ ρ ( x ) , x ∈ M n , defines an or der zero c.p.c. ma p from M n int o A . Use (y) to see that (1 A − α (1 n +1 )) ¯ ρ (1 n ) = 1 A − α (1 n +1 ), whence α (1 n +1 ) + β (1 n ) = 1 A . T o complete the pro of we must show that the images of α and β commut e. F o r brevity , put h = λ ( g ⊗ 1 n ), r ecall that α (1 n +1 ) = h + ϕ ( h ) and 13 that h ⊥ ϕ ( h ) (the latter b y (m)). F or k, l , i = 1 , 2 , . . . , n w e have: β ( e ( n ) kl ) α ( e ( n +1) i,n +1 ) (q) = (1 − h − ϕ ( h )) ¯ ρ ( e ( n ) kl ) wϕ ( h ) ρ ( e ( n ) i 1 ) ¯ v (i),(l),(o) = (1 − ϕ ( h )) ϕ ( h ) ¯ ρ ( e ( n ) kl ) w ¯ ρ ( e ( n ) i 1 ) ρ (1 n ) ¯ v = γ   (1 − g ) g ⊗ 1 n ⊗ 1 n  (1 ⊗ 1 n ⊗ e ( n ) kl ) u (1 ⊗ 1 n ⊗ e ( n ) i 1  ρ (1 n ) ¯ v (p) = γ   (1 − g ) g ⊗ 1 n ⊗ 1 n  u (1 ⊗ 1 n ⊗ e ( n ) i 1 )(1 ⊗ e ( n ) kl ⊗ 1 n  ρ (1 n ) ¯ v = (1 − ϕ ( h )) ϕ ( h ) w ¯ ρ ( e ( n ) i 1 )( ϕ ◦ λ )(1 ⊗ e ( n ) kl ) ρ (1 n ) ¯ v (k) = ϕ ( h )(1 − ϕ ( h )) w ¯ ρ ( e ( n ) i 1 ) ρ (1 n )( ϕ ◦ λ )(1 ⊗ e ( n ) kl ) ¯ v (i),(j) = ϕ ( h )(1 − ϕ ( h )) wρ ( e ( n ) i 1 ) ¯ v λ (1 ⊗ e ( n ) kl ) (o), (k), (j) = ϕ ( h ) wρ ( e ( n ) i 1 ) ¯ v (1 − h ) ¯ ρ ( e ( n ) kl ) (m) = ϕ ( h ) wρ ( e ( n ) i 1 ) ¯ v (1 − h − ϕ ( h )) ¯ ρ ( e ( n ) kl ) = α ( e ( n +1) i,n +1 ) β ( e ( n ) kl ) The image of α is contained in the C ∗ -algebra , E , g enerated by { α ( e i,n +1 ) | i = 1 , 2 , . . . n } , whic h again, b y the argument above, is contained in the commutan t of the image of β . (T o see this use that α (1 n +1 ) ∈ E , that E is con tained in the her editary sub- C ∗ -algebra of A genera ted by α (1 n +1 ), a nd tha t α ( x ) α ( y ) = α (1 n +1 ) α ( xy ) for all x, y ∈ M n +1 , cf. P r opo sition 2 .3 (ii).) It has now b een v erified that the images o f α and β commute. (iv) ⇒ (i). This follows from [1 5, Lemma 4.2].  Remark 5.2 An y stably finite unital Z -sta ble C ∗ -algebra satisfies co nditions (i) throug h (iv) of Prop osition 5.1. Quite surprising ly there is a very re cen t example of a unital, s imple infinite dimensional C ∗ -al- gebra that do es not admit a unital embedding of the Jia ng–Su a lgebra or for that matter of any dimension drop C ∗ -algebra Z n,m with n, m ≥ 2 (see [4]). This example is ba sed on Example 4.8 of [7] of a unital C ( X )-a lgebra who s e fibres abs o rb the J iang–Su algebra, but which does not itself absorb the Jia ng–Su algebra. This C ∗ -algebra has no finite dimensio nal quotient, and one ca n quite easily see that o ne canno t unita lly em b ed the Jiang–Su algebr a or Z n,m (with n, m ≥ 2) int o this C ∗ -algebra . In other words, simple infinite dimensiona l C ∗ -algebra s ca n fa il to have the (very weak) divis i- bilit y prop erty 5.1 (i). Nonetheless, prompted by the equiv a lence of (i) and (iv) of the prop osition ab o ve, one might ask the following: Question 5.3 Does the Jiang– Su algebra Z embed unitally into any unital C ∗ -algebra A fo r whic h its Cuntz semigroup W ( A ) has the following divisibility prop erty: F or every natural num be r n there exis ts x ∈ W ( A ) such that nx ≤ h 1 A i ≤ ( n + 1) x ? The Jia ng–Su algebra has the divisibity prop erty of the question ab ov e (cf. [1 5, Lemma 4.2]), and hence so do es any unital C ∗ -algebra that admits a unital embedding of Z . The question abov e has a n affirmative answer when A is strongly self-a bsorbing and of stable rank one: Prop osition 5.4 L et D b e a str ongly self-absorbing C ∗ -algebr a of stable r ank one su ch that for e ach natur al numb er n ther e is x in the Cuntz semigr oup W ( D ) with nx ≤ h 1 D i ≤ ( n + 1) x . Then the Jiang–Su algebr a Z emb e ds unital ly into D . Pro of: One can wr ite Z as an inductive limit of prime dimension- dr op C ∗ -algebra s o f the form Z n,n +1 . By a ssumption a nd Pro positio n 5.1, e a c h Z n,n +1 maps unitally into D . As D is strongly self-absor bing, D embeds unitally into D ∞ ∩ D ′ , where D ∞ = ℓ ∞ ( D ) /c 0 ( D ), whence Z n,n +1 maps 14 unitally in to D ∞ ∩ D ′ for a ll n . It no w follows from [16, Prop osition 2.2 ] that D ∼ = D ⊗ Z , and hence that Z embeds unitally into D .  W e are no w r e a dy to prove our ma in result of this section: Theorem 5.5 L et D b e a unital C ∗ -algebr a. Then D ∼ = Z if and o nly if (i) D is str ongly self-absorbing, (ii) the stable r ank of D is one, (iii) for al l n ther e is an element x ∈ W ( D ) s u ch that nx ≤ h 1 D i ≤ ( n + 1) x , (iv) D ⊗ B ∼ = B for all UHF-algebr as B . Pro of: It is well-kno wn that Z s atisfies (i)–(iv). T o prov e the “ if ” pa rt it suffices to s how that D embeds unitally into Z a nd that Z embeds unitally into D , cf. P ropo sition 2.1. It follows from Prop osition 3.1 that D ⊗ Z 2 ∞ , 3 ∞ ∼ = Z 2 ∞ , 3 ∞ . Hence D em b eds unita lly into Z 2 ∞ , 3 ∞ which ag ain embeds unitally into Z by P rop osition 3.3. That Z embeds in to D follows fro m Prop osition 5.4.  6 Strongly self-absorbing C ∗ -algebras wit h almost unp erfo- rated Cunt z semigroup In this section, we r ephrase Theo rem 5.5 in terms of an algebr aic condition on the Cuntz se mig roup of a str ongly self-absor bing C ∗ -algebra . Along the wa y , we show that a strongly s elf-absorbing C ∗ -algebra has almost unp erfora ted Cun tz se mig roup if and only if it abso rbs the Jia ng–Su algebra. Remark 6.1 (Dimension functions) A dimension function on a C ∗ -algebra A is a function d : M ∞ ( A ) + → R + which satisfies d ( a ⊕ b ) = d ( a ) + d ( b ), a nd d ( a ) ≤ d ( b ) if a - b for all a, b ∈ M ∞ ( A ) + . It is lower semic ontinuous if, for every monotone incre a sing sequence ( a n ) in M ∞ ( A ) + with a n → a for s ome a ∈ A , o ne has d ( a n ) → d ( a ). If τ is a (p ositiv e) tr ace on A , then d τ ( a ) = lim n →∞ τ ( a 1 /n ) = lim ε → 0+ τ ( f ε ( a )) , a ∈ M ∞ ( A ) + , defines a dimension function on A (wher e f ε is as defined in Notation 4 .4, and when τ is ex tended in the ca nonical wa y to M ∞ ( A )). Every low er semicontin uous dimension function on an exact C ∗ -algebra arises in this w ay . Every dimension function d on A factors throug h the Cunt z semigroup, i.e., it g iv es rise to an additive order preserving mapping ˜ d : W ( A ) → R + given by ˜ d ( h a i ) = d ( a ) for a ∈ M ∞ ( A ) + . The functional ˜ d is called a state (or a dimension function) on W ( A ). If there is no r is k of confusion, then we use the sa me symbol to deno te the dimension function on A a nd the corresp onding state on W ( A ). It is well-known that a stably finite stro ngly s e lf-absorbing C ∗ -algebra D has precisely one trace (whic h we shall usually denote by τ ); this determines a unique lo wer semico n tin uous dimension function (a lso denoted by d τ in the sequel). When identifying D with D ⊗ D o ne has d τ ( a ⊗ b ) = d τ ( a ) · d τ ( b ) (6.1) for a ll a, b ∈ D + . 15 Remark 6.2 (Almost unp erforation and strict c ompariso n) The Cuntz semigroup W ( A ) of a C ∗ -algebra A is s aid to b e almost unp erfor ate d , cf. [15], if for all x, y ∈ W ( A ) and for a ll natural n um b ers n one has ( n + 1) x ≤ ny ⇒ x ≤ y . If A is simple and unital, then W ( A ) is almost unperforated if and only if A has strict c om- p arison , i.e., whenever x, y ∈ W ( A ) are s uc h that d ( x ) < d ( y ) for all dimension functions d on A (that ca n b e taken to b e nor malized: d ( h 1 A i ) = 1), then x ≤ y (se e [15, Prop osition 3.2 ]). If A is simple, ex a ct and unital, then W ( A ) is almost unperfor ated if and only if A has strict c omp arison given by tr ac es : F or a ll x, y ∈ W ( A ) one has that x ≤ y if d τ ( x ) < d τ ( y ) for a ll tracia l states τ on A , (see [15, Cor ollary 4.6]). Lemma 6.3 L et A 6 = C b e a unital C ∗ -algebr a with a faithful tr acial s t ate τ . Then ther e ar e 0 < λ < 1 and p ositive elements e and f in A such that e ⊥ f and d τ ( e ) = λ, d τ ( f ) = 1 − λ. Pro of: Cho ose a po sitiv e nor malized element d ∈ A such that { 0 , 1 } ⊆ σ ( d ); suc h an e le men t exists in any C ∗ -algebra of vector space dimensio n strictly la rger than 1. If σ ( d ) 6 = [0 , 1], then A contains a nontrivial pro jection p , and we can take λ = τ ( p ), e = p and f = 1 − p . Suppose now that σ ( d ) = [0 , 1]. The trace τ induces a probability measur e µ on σ ( d ) = [0 , 1] whic h is non-zero on any non-empty o pen subset of [0 , 1] (because τ is assumed to b e faithful). T ake t in the op en int erv al (0 , 1) such that µ ( { t } ) = 0. Then λ = µ ([0 , t ]), e = ( d − t ) − , and f = ( d − t ) + are as desired.  In the lemmas b elow it is established that the Cunt z s emigroup of a strongly self-a bsorbing C ∗ - algebra has a rather str ong divisibility prop erty . Lemma 6.4 L et D b e a str ongly self-absorb ing C ∗ -algebr a. Ther e ar e p ositive elements b, c ∈ D such that h b i = h c i , b ⊥ c , and d τ ( b ) = d τ ( c ) = 1 / 2 . Pro of: W e can identify D with ( D 0 ) ⊗∞ , where D 0 is (isomorphic to) D . By Lemma 6.3, there are 0 < λ < 1 and p ositive elements e, f in D (that we can assume to ha ve nor m e q ual to 1) such that e ⊥ f , d τ ( e ) = λ , and d τ ( f ) = 1 − λ . Set ¯ λ = λ (1 − λ ) > 0, and s et b 0 = e ⊗ f ⊗ 1 D 0 ⊗ · · · ∈ D , c 0 = f ⊗ e ⊗ 1 D 0 ⊗ · · · ∈ D . Then b 0 ⊥ c 0 , and h b 0 i = h c 0 i because D is strongly self-absor bing (which implies that ther e is a sequence ( u n ) o f unitaries in D suc h that u ∗ n b 0 u n → c 0 ). Moreov er, by (6 .1) , we hav e d τ ( b 0 ) = d τ ( c 0 ) = ¯ λ . Set d = e ⊗ e + f ⊗ f ∈ D 0 ⊗ D 0 , and for eac h na tural num ber n set b n = d ⊗ · · · ⊗ d ⊗ e ⊗ f ⊗ 1 D 0 ⊗ · · · ∈ D , c n = d ⊗ · · · ⊗ d ⊗ f ⊗ e ⊗ 1 D 0 ⊗ · · · ∈ D , where d a ppears n times. Then, as a bov e, we hav e tha t h b n i = h c n i ; and the elemen ts b 0 , b 1 , b 2 , . . . , c 0 , c 1 , c 2 , . . . are pa irwise or thogonal. Mo reov er, by (6.1), we hav e d τ ( d ) = 1 − 2 ¯ λ a nd hence d τ ( b n ) = d τ ( c n ) = (1 − 2 ¯ λ ) n ¯ λ. It fo llows tha t ∞ X n =0 d τ ( b n ) = ∞ X n =0 d τ ( c n ) = 1 / 2 , whence the nor m- con vergent sums b := ∞ X n =0 1 n + 1 b n , c := ∞ X n =0 1 n + 1 c n , define elemen ts b and c in D with the desir ed prop erties.  16 Lemma 6.5 L et A b e a C ∗ -algebr a which c ontains an incr e asing se quen c e ( A n ) of sub- C ∗ -algebr as whose union is dense in A . L et x ∈ W ( A ) and ε > 0 b e gi ven. L et τ b e a t r ac e on A . Then ther e exist natu r al nu mb ers k and r and a p ositive element a ∈ M r ( A k ) ⊆ M r ( A ) such that h a i ≤ x and d τ ( h a i ) ≥ d τ ( x ) − ε . Pro of: The element x is represented by a po sitiv e elemen t b in a ma tr ix a lg ebra M r ( A ) ov er A . Since d τ is lo wer semicontin uous ther e is δ > 0 such that d τ (( b − δ ) + ) ≥ d τ ( b ) − ε . Find k and a po sitiv e element a 0 in M r ( A k ) such that k a 0 − b k < δ / 2. Put a = ( a 0 − δ / 2) + ∈ M r ( A k ). Then h a i ≤ h b i = x (by [1 3, Section 2]). Mor eov e r, k a − b k < δ , so ag ain by [1 3, Section 2], w e ha ve h a i ≥ h ( b − δ ) + i , which implies that d τ ( h a i ) ≥ d τ ( h ( b − δ ) + i ) ≥ d τ ( x ) − ε .  Lemma 6.6 L et D b e a str ongly self-absorbing C ∗ -algebr a. L et x ∈ W ( D ) and 0 6 = k ∈ N b e given. Then, for e ach ε > 0 , ther e is y ∈ W ( D ) su ch that k y ≤ x and kd τ ( y ) ≥ d τ ( x ) − ε . Pro of: Let us first prov e the lemma for k = 2 (for k = 1, there is no thing to sho w). F or each natural num b er r , identify M r ( D ) with M r ( D 0 ) ⊗ ( D 0 ) ⊗∞ , where D 0 is (isomorphic to) D . By Lemma 6.5 it suffices to consider the case where x = h d i for some po sitiv e element d ∈ M r ( D 0 ) ⊗ ( D 0 ) ⊗ k ⊗ 1 D 0 ⊗ · · · , for suitable natural num b ers k and r , that is d = d 0 ⊗ 1 D 0 ⊗ · · · for some d 0 ∈ M r ( D 0 ) ⊗ ( D 0 ) ⊗ k . Let b and c b e a s in Lemma 6.4, and set b ′ = d 0 ⊗ b ∈ M r ( D ) , c ′ = d 0 ⊗ c ∈ M r ( D ) , where we ha ve identified M r ( D ) with M r ( D 0 ) ⊗ ( D 0 ) ⊗ k ⊗ ( D 0 ) ⊗∞ . Then b ′ and c ′ are or thog- onal, b elong to the her editary sub- C ∗ -algebra of M r ( D ) generated b y d , and satisfy h b ′ i = h c ′ i . Moreov er, by (6.1), d τ ( b ′ ) = d τ ( c ′ ) = d τ ( d ) / 2 . Set y = h b ′ i . Then 2 y = h b ′ + c ′ i ≤ h d i = x , and 2 d τ ( y ) = 2 d τ ( b ′ ) = d τ ( x ). (Note that in this case, i.e., for k = 2 and for x = h d i of the sp ecial for m considered ab ov e , we prove the lemma with ε = 0.) Next, a repe ated a pplica tion o f the c a se k = 2 yields that the lemma ho lds for k = 2 j , for any j ∈ N . T o derive the lemma fo r an arbitrary na tural num ber k , cho ose m, j ∈ N such that 1 k − ε 2 k d τ ( x ) ≤ m 2 j ≤ 1 k . Then 2 j  1 − ε 2 d τ ( x )  ≤ mk ≤ 2 j . Cho ose ε 0 > 0 such that ( d τ ( x ) − ε 0 )  1 − ε 2 d τ ( x )  ≥ d τ ( x ) − ε. Now apply the lemma with 2 j and ε 0 in the place of k and ε to o btain y 0 ∈ W ( D ) with 2 j y 0 ≤ x and 2 j d τ ( y 0 ) ≥ d τ ( x ) − ε 0 . P ut y = my 0 . Then k y = k my 0 ≤ 2 j y 0 ≤ x and k d τ ( y ) = mk d τ ( y 0 ) ≥ 2 j  1 − ε 2 d τ ( x )  d τ ( y 0 ) ≥  1 − ε 2 d τ ( x )  ( d τ ( x ) − ε 0 ) ≥ d τ ( x ) − ε.  Prop osition 6.7 L et D b e a str ongly self-absorb ing C ∗ -algebr a. Then W ( D ) is almost un p erfo- r ate d if and only if D absorbs the Jiang–Su algebr a tensorial ly. 17 Pro of: By [15], Z -stability implies tha t the Cuntz semig roup is almos t unp erfora ted. T o show the c o n verse, it will be enough to co nsider finite D , for if D is infinite, it is well kno wn to abso rb O ∞ , hence Z . W e s how that Pro p osition 5.1(ii) holds for each na tur al num ber n , whic h then, b y Prop osition 5.1, will imply that Z n,n +1 embeds unitally into D . As in the pr oof of Prop osition 5 .4, this entails that Z embeds unitally into D . W e can fina lly use Pr opos ition 2.1 to conclude tha t D is Z - s table. Our pro of of 5.1 (ii) follo ws to a large extent that of (i) ⇒ (ii) of P r opo sition 5.1; how ever, w e will ha ve to av oid use o f Lemma 4.5, since we do no t assume D to be of stable rank one. Let n ∈ N b e given. By Lemma 6 .6 there is x ∈ W ( D ) such that nx ≤ h 1 D i and d τ ( x ) > 1 / ( n + 1 ). Now follow the proo f of (i) ⇒ (ii) of Prop osition 5.1 to the p oin t wher e δ > 0, d ∈ M k ( D ), and pairwise or thogonal po s itiv e elemen ts e 1 , e 2 , . . . , e n in D hav e b e en c o nstructed such that x = h d i and e j ∼ ( d − δ ) + . (Note that the assumption of stable r ank one was not used up to that p oin t.) Up on c ho osing δ > 0 s mall enough, and using lo wer s e micon tinuit y of d τ , one ca n further o bta in that d τ ( e 1 ) = d τ (( d − δ ) + ) > 1 / ( n + 1) (reca lling that d τ ( d ) = d τ ( x ) > 1 / ( n + 1)). F or η > 0, let f η be as in Notation 4.4. As 1 D − f η ( e 1 + e 2 + · · · + e n ) ⊥ ( e 1 + e 2 + · · · + e n − η ) + we ge t lim η → 0+ d τ (1 D − f η ( e 1 + e 2 + · · · + e n )) ≤ 1 − lim η → 0+ d τ (( e 1 + e 2 + · · · + e n − η ) + ) = 1 − nd τ ( e 1 ) < d τ ( e 1 ) = lim η → 0+ d τ (( e 1 − η ) + ) (the first inequalit y is actually equality). Th us we infer, by Remark 6.2 and the ass umption that W ( D ) is a lmost unp erforated, tha t 1 D − f η ( e 1 + e 2 + · · · + e n ) - ( e 1 − η ) + for s ome η > 0. W e can now follow the last thr ee lines of the pro of o f (i) ⇒ (ii) of Pro positio n 5 .1 to ar rive at the co nclusion that 5.1(ii) holds .  Corollary 6.8 In The or em 5.5, c onditions (ii) and (ii i) may as wel l b e r eplac e d by (ii’) D is finite (iii’) W ( D ) is almost unp erfor ate d. Pro of: By [8] and [1 5], Z satisfies (ii’) and (iii’), so we have to chec k that (ii’) a nd (iii’) (together with the other hypo theses) imply conditions (ii) and (iii) of 5.5. But (iii’) entails that D is Z - stable b y P rop o sition 6.7, and (ii’) together with Z -stability yields stable ra nk one, cf. [1 5]. Now by Remar k 5.2, D satisfies 5.1(i), hence 5.5(iii).  7 Strongly self-absorbing C ∗ -algebras with finite decomp o- sition rank In this final s ection we single out the Jiang–Su a lgebra among str ongly self-abso rbing C ∗ -algebra s with finite de c o mpositio n rank . Recall that the latter is a notion of topolog ical dimens io n for nu clear C ∗ -algebra s that w as in tro duced by E. Kirch ber g and the second named author in [11]. The order on the Cun tz semigroup is not the algebraic order (i.e., if x ≤ y , then we do not necessarily ha v e z in the Cunt z semigroup such that y = x + z ). The follo wing lemma, whic h is needed for the pro of of Prop osition 7.5 below, seeks to remedy this situation. 18 Lemma 7.1 L et A b e a C ∗ -algebr a. (i) L et a, b b e p ositive elements in A such that a - b , and let ε > 0 b e given. Then ther e ar e p ositive ele ments a 0 and c in bAb such that a 0 ⊥ c, a 0 ∼ ( a − 2 ε ) + , b - ( a − ε ) + ⊕ c. Mor e over, if d is a lower semic ontinuous dimension function on A and if δ > 0 is given, then ther e exists ε 0 > 0 such that if 0 < ε ≤ ε 0 , then d ( b ) − d ( a ) ≤ d ( c ) ≤ d ( b ) − d ( a ) + δ. (ii) L et d b e a lower s emic ontinuous d imension function on W ( A ) , and le t x, y ∈ W ( A ) b e such that x ≤ y . Then, for e ach δ > 0 , ther e is z ∈ W ( A ) such t hat x + z ≥ y and d ( z ) ≤ d ( y ) − d ( x ) + δ . Pro of: (i). By [13, Prop osition 2.4] there is v ∈ A such that v ∗ v = ( a − ε ) + and v v ∗ belo ngs to bAb . With h ε as de fined in (4.1) we hav e h ε ( v v ∗ ) ⊥ ( v v ∗ − ε ) + . (W e remark that h ε ( v v ∗ ) b elongs to A if A is unital, a nd that it otherwis e b elongs to the unitiza tio n of A .) P ut a 0 = ( v v ∗ − ε ) + ∼ ( v ∗ v − ε ) + = ( a − 2 ε ) + , c = h ε ( v v ∗ ) bh ε ( v v ∗ ) , and note that a 0 and c b oth b elong to bAb . Moreov er, a 0 ⊥ c , a nd vv ∗ + c is s trictly p ositive in bAb . The latter implies that b - v v ∗ + c - v v ∗ ⊕ c ∼ ( a − ε ) + ⊕ c. If d is a lower semicontin uo us dimension function on A , then for each δ > 0 there is ε 0 > 0 such that d (( a − 2 ε 0 ) + ) ≥ d ( a ) − δ . As a 0 ⊥ c we have d ( a 0 ) + d ( c ) = d ( a 0 + c ) ≤ d ( b ), whence d ( c ) ≤ d ( b ) − d ( a 0 ) = d ( b ) − d (( a − 2 ε ) + ) ≤ d ( b ) − d (( a − 2 ε 0 ) + ) ≤ d ( b ) − d ( a ) + δ, whenever 0 < ε ≤ ε 0 . O n the other hand, since b - ( a − ε ) + ⊕ c w e have d ( b ) ≤ d (( a − ε ) + ) + d ( c ) ≤ d ( a ) + d ( c ), which entails that d ( c ) ≥ d ( b ) − d ( a ). (ii). Up on replacing A w ith a matrix algebra ov er A we can assume that x = h a i and y = h b i for some p ositive ele ments a, b ∈ A . No w use (i) to find ε > 0 and c such that b - ( a − ε ) + ⊕ c - a ⊕ c and such tha t d ( c ) ≤ d ( b ) − d ( a ) + δ . W e can then take z to be h c i .  W e quote b elow a res ult by Andrew T oms and the seco nd named autho r sta ting that C ∗ -algebra s with finite decomp o sition rank satisfy a w eak v ersion of strict comparison. The origina l lemma was stated in a s ligh tly differen t manner; the v ersion below employs the fact that decompositio n rank is inv a riant under taking matr ix algebra s. Lemma 7.2 (T oms –Win ter, [17, Lemma 6.1]) L et A b e a s imple, sep ar able and unit al C ∗ - algebr a with de c omp osition r ank n < ∞ . Supp ose that x, y 0 , y 1 , . . . , y n ∈ W ( A ) satisfy d ( x ) < d ( y j ) for al l j = 0 , 1 , . . . , n and fo r any lower semic ontinuous dimensio n function d on A . Then x ≤ y 0 + y 1 + · · · + y n . The lemma ab ov e ha s the following tw o sharpe r versions fo r strongly self-absorbing C ∗ -algebra s: Lemma 7.3 L et D b e st r ongly self-absorbi ng with de c omp osition r ank n < ∞ , and let x, y ∈ W ( D ) with ( n + 1 ) d τ ( x ) < d τ ( y ) b e given. Then x ≤ y . Pro of: Apply Lemma 6.6 with k = n + 1 to o btain z ∈ W ( D ) such that ( n + 1) z ≤ y and ( n + 1) d τ ( z ) > d τ ( y ) − ( d τ ( y ) − ( n + 1) d τ ( x )) = ( n + 1) d τ ( x ); we then hav e d τ ( x ) < d τ ( z ). No w from Lemma 7.2 we o btain x ≤ ( n + 1) z ≤ y .  19 Lemma 7.4 L et D b e str ongly self-absorbing with de c omp osition r ank n < ∞ , and let x, y , z ∈ W ( D ) b e such that x ≤ y and ( n + 1) d τ ( z ) < d τ ( y ) − d τ ( x ) . Then x + z ≤ y . Pro of: W e may a ssume that x = h a i , y = h b i a nd z = h e i , where a , b and e ar e p ositive elements in some matrix algebra M r ( D ) over D . T o show that x + z ≤ y it suffices to show that ( a − 2 ε ) + ⊕ e - b for a ll ε > 0. By Lemma 7 .1 (i) there a re mutually orthogonal pos itiv e elemen ts a 0 and c in the he r editary sub- C ∗ -algebra of M r ( D ) genera ted b y b suc h that a 0 ∼ ( a − 2 ε ) + and d τ ( c ) ≥ d τ ( b ) − d τ ( a ) > ( n + 1) d τ ( e ). But then it follows from Lemma 7.3 that e - c , whence ( a − 2 ε ) + ⊕ e - ( a − 2 ε ) + ⊕ c ∼ a 0 ⊕ c ∼ a 0 + c - b , as des ired.  Prop osition 7.5 A ny str ongly self-absorbing C ∗ -algebr a D with fin ite de c omp osition r ank absorbs the Jiang–Su algebr a Z , i.e., D ⊗ Z ∼ = D . Pro of: By Remar k 6.2 a nd P rop osition 6.7 it suffices to show that for a ll x, y ∈ W ( D ) with d τ ( x ) < d τ ( y ) o ne has x ≤ y , where τ is the unique trace on D . Put δ = ( d τ ( y ) − d τ ( x )) / ( n + 1), where n is the dec o mpositio n rank of D . Choose an in teger k ≥ n such that ( n + 1) d τ ( x ) /k < δ . By Lemma 6.6 th ere is x 0 ∈ W ( D ) suc h that k x 0 ≤ x and k d τ ( x 0 ) ≥ d τ ( x ) − δ / 2, a nd b y Lemma 7.1 (ii) there is z ∈ W ( D ) such that kx 0 + z ≥ x and d τ ( z ) < d τ ( x ) − k d τ ( x 0 ) + δ / 2 ≤ δ . F or each j = 0 , 1 , . . . , n − 1 w e ha ve ( n + 1) d τ ( x 0 ) ≤ ( n + 1) d τ ( x ) /k < δ = d τ ( y ) − d τ ( x ) ≤ d τ ( y ) − d τ ( j x 0 ) . Lemma 7 .4 therefore yields j x 0 ≤ y ⇒ ( j + 1) x 0 ≤ y for j = 0 , 1 , . . . , n − 1. Hence nx 0 ≤ y . Next, ( n + 1) d τ ( z ) < ( n + 1) δ ≤ d τ ( y ) − d τ ( x ) ≤ d τ ( y ) − d τ ( nx 0 ) , so, a gain by Lemma 7.4, we get x ≤ nx 0 + z ≤ y a s desired.  Theorem 7.6 L et D b e a unital C ∗ -algebr a. Then D ∼ = Z if and o nly if (i) D is str ongly self-absorbing, (ii) the de c omp osition r ank of D is fi nite, (iii) D is K K -e quivalent to C . Pro of: It is well-know that Z sa tisfies prop erties (i)–(iii) a b ov e. Assume now that (i)–(iii) holds. T o show that D ⊗ Z ∼ = Z , note that D ⊗ Z a nd Z b oth ha ve (lo cally) finite decomp osition rank, and ar e Z -stable. Since D ⊗ Z is K K -e quiv alent to C , K 0 ( D ⊗ Z ) = Z a nd K 1 ( D ⊗ Z ) = 0. Since D ⊗ Z is stably finite and Z -stable, the order structure of its K -theor y is determined by the unique tracial state (see [6]), whence D ⊗ Z ∼ = Z by [22, Corollary 8.1 ]. That D ⊗ Z ∼ = D simply follows from P rop osition 7.5.  Remarks 7.7 F ormally , Theorems 5.5 and 7 .6 a re very similar , a nd it is in teresting to compare them. Conditions (ii) of b oth theorems re fer to notions o f noncommutativ e covering dimension; how ev er, one should keep in mind that decomp osition rank has a muc h mo re topolog ical fla vor than stable rank one. F urthermo re, using [7 ] and the fact that (genera lized) prime dimension drop C ∗ -algebra s a re K K -equiv ale n t to C (cf. [8]), 7 .6(iii) follows fro m 5.5(iv). B ecause of these conditions, neither 5.5 nor 7.6 are completely intrinsic characteriza tions Condition 5.5(iii) ma y be interpreted as a K -theo ry type c o ndition in the bro a dest sense ; it remains an in teresting poss ibilit y that it is redundant in 5.5. Similar ly , it migh t be the case that 7.6 still holds when only asking for lo cally finite (a s opp osed to finite) decomp osition rank in 20 7.6(ii). Conditions 5.5(iii) and 7.6(ii) are (implicitly) b oth used to ensure notions o f co mparison of po sitiv e elements. So , the que s tion is whether (stably finite) strongly self-abso rbing C ∗ -algebra s automatically have s o me sort of compariso n pro perty . (In the infinite case, this ha s an a ffir mativ e answer, sinc e an infinite stro ngly self-absor bing C ∗ -algebra is alwa ys pure ly infinite by a result of Kirch be r g.) References [1] N. B rown, F. Perera, and A. S. 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[22] , L o c alizing t he El liott c onje ctur e at st ro ngly self-absorbing C ∗ -algebr as , P reprint, Math. Archiv e math.O A/0708 .0283v3 , 2007 . Dep ar tment of Ma thema tical Sciences, U niversity of Copenhag en, U niversitets- p ark en 5, DK-2100 Copenha gen, Denmark E-mail address: rord am@math. ku.dk Int ernet home pa ge: www.m ath.ku.d k/ e r ordam School of Ma thema tical Sciences, University of Nottingham, University P ark, Nottingham NG7 2RD, United Kingdom E-mail address: wilh elm.wint er@nottingham.ac.uk 22