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THE GENERATOR PROBLEM FOR Z-STABLE C∗-ALGEBRAS 2 HANNESTHIELANDWILHELMWINTER 1 0 ABSTRACT. ThegeneratorproblemwasposedbyKadisonin1967,anditremains 2 openuntiltoday.WeprovideasolutionfortheclassofC∗-algebrasabsorbingthe n Jiang-SualgebraZtensorially.Moreprecisely,weshowthateveryunital,separa- a ble,Z-stableC∗-algebraAissinglygenerated,whichmeansthatthereexistsan J elementx∈Athatisnotcontainedinanypropersub-C∗-algebraofA. 8 Togiveapplicationsofourresult, weobservethatZ canbeembeddedinto 1 thereducedgroupC∗-algebraofadiscretegroupthatcontainsanon-cyclic,free subgroup.ItfollowsthatcertaintensorproductswithreducedgroupC∗-algebras ] aresinglygenerated.Inparticular,Cr∗(F∞)⊗Cr∗(F∞)issinglygenerated. A O . h t a 1. INTRODUCTION m Byanoperatoralgebrawemeana∗-subalgebraofB(H)thatiseitherclosedin [ thenorm topology (aconcreteC∗-algebra)or theweakoperatortopology (avon 1 Neumannalgebra).Onewayofrealizinganoperatoralgebraistotakeasubsetof v B(H)andconsiderthesmallestoperatoralgebracontainingit. 9 Inatrivialway,everyoperatoralgebracanbeobtainedthisway. Thesituation 7 becomesinterestingifoneimposesrestrictionsonthegeneratingset,andonenat- 8 ural possibility is to require that it consists of only one element, i.e., to consider 3 . operatoralgebrasthataregeneratedbyasingle operator. Itisanold problemto 1 determinewhichoperatoralgebrasarisethisway. 0 2 Moregenerally,onetriestocomputetheminimalnumberofelementsthatgen- 1 erate a given operator algebra, see 2.1. It is often convenient to consider self- : adjointgenerators. Notethattwoself-adjointelementsa,bgeneratethesameop- v i eratoralgebraastheelementa+ib.Thus,ifweaskwhetheranoperatoralgebrais X singlygenerated,itisequivalenttoaskwhetheritisgeneratedbytwoself-adjoint r elements. a In the case of von Neumann algebras, the generator problem wasincluded in Kadison’s famous ‘Problems on von Neumann algebras’, [Kad67]. This problem listhasturnedouttobeveryinfluential,yetitsoriginalformremainsunpublished. ItisindirectlyavailableinanarticlebyGe,[Ge03],whereabriefsummaryofthe developmentsaroundKadison’sfamousproblemsisgiven. Date:January19,2012. 2000MathematicsSubjectClassification. Primary46L05,46L85;Secondary46L35. Keywordsandphrases. C∗-algebras,generatorproblem,singlegeneration,Z-stability. ThisresearchwaspartiallysupportedbytheCentredeRecercaMatema`tica,Barcelona. Thefirst namedauthorwaspartiallysupportedbytheDanishNationalResearchFoundationthroughtheCen- treforSymmetryandDeformation,Copenhagen. Thesecondnamedauthorwaspartiallysupported byEPSRCGrantsEP/G014019/1andEP/I019227/1. 1 2 HANNESTHIELANDWILHELMWINTER Question1.1(Kadison,[Kad67,Problem14],seealso[Ge03]). Iseveryseparably- acting1vonNeumannalgebrasinglygenerated? As noted in [She09], there exist singly generated von Neumann algebras that arenot separably-acting. However, the separably-actingvon Neumann algebras arethenaturalclassforwhichonemightexpectsinglegeneration. Theanswerto Question1.1isstill openingeneral, butmanyauthorshavecontributed toshow thatlargeclassesofseparably-actingvonNeumannalgebrasaresinglygenerated. We just mention an incomplete list of results. It starts with von Neumann, [vN31],whoshowedthattheabelianoperatoralgebrasnamedafterhimaregener- atedbyasingleself-adjointelement,thusimplicitlyraisingthegeneratorproblem. Some thirty years later, this was extended by Pearcy, [Pea62], who showed that all von Neumann algebrasof type I are singly generated. Then Wogen, [Wog69, Theorem 2], proved that all properly infinite von Neumann algebras are singly generated,thusreducingthegeneratorproblemtothetypeII case. 1 Later,thiswasfurtherreducedtothecaseofaII -factorbyWillig,[Wil74],and 1 then to the case of a finitely-generated II -factor by Sherman, [She09, Theorem 1 3.8].ThismeansthatQuestion1.1hasapositiveanswerifeveryseparably-acting, finitelygeneratedII -factorissinglygenerated. 1 TherearemanypropertiesknowntoimplythataII -factorissinglygenerated. 1 WejustmentionthatGeandPopa,[GP98,Theorem6.2],showthateverytensori- allynon-prime2II -factorissinglygenerated.OurmainresultTheorem3.5canbe 1 consideredasapartialC∗-algebraicanalogofthisresult. Let us also mention that the free group factors W∗(F ) are the outstanding k examples of separably-acting von Neumann algebra for which it is not known whethertheyaresinglygenerated. In the case of C∗-algebras, the generator problem is more subtle. There is al- ready no obvious class of C∗-algebras for which one conjectures that they are singlygenerated. EverysinglygeneratedC∗-algebraisseparable3. However,the converseisfalse,andcounterexamplescanbefoundamongthecommutativeC∗- algebras. Infact,theC∗-algebraC (X)isgeneratedbynself-adjointelementsifandonly 0 ifX canbeembeddedintoRn. Thus,C (X)issinglygeneratedifandonlyifX is 0 planar,i.e.,canbeembeddedintotheplaneR2. ItiseasytoseethataC∗-algebraAisgeneratedbynself-adjointelementsifand onlyifitsminimalunitizationAisgeneratedbynself-adjointelements. Therefore, wewillmostlyconsiderthegeneratorproblemforseparable,unitalC∗-algebra.In e thatcase,takingthetensorproductwithamatrixalgebrahastheeffectofreduc- ing the necessary number of generators. If A is generated by n2 +1 self-adjoint elements,thenA⊗M issinglygenerated,seee.g. [Nag04,Theorem3]. n One derivesthe principle thata C∗-algebraneedslessgeneratorsif itis‘more non-commutative’. Consequently,onemightexpecta(separable)C∗-algebratobe singly generated if it is ‘maximally non-commutative’. As a non-unital instance 1A vonNeumannalgebraiscalled ‘separably-acting’, orjust‘separable’, ifit isasubalgebraof B(l2N),orequivalentlyifithasaseparablepredual. 2AII1-factorM iscalledtensoriallynon-primeifitisisomorphictoatensorproduct,M1⊗¯M2,of twoII1-factorsM1,M2. 3AC∗-algebraiscalled‘separable’ifitcontainsacountable,norm-densesubset THEGENERATORPROBLEMFORZ-STABLEC∗-ALGEBRAS 3 of this principle, we note that the stabilization, A⊗K, of a separable unital C∗- algebra A is singly generated, [OZ76, Theorem 8]. In the unital case, there are at least three natural cases when one considers a C∗-algebra A to be ‘maximally non-commutative’,whicharethefollowing: (1) Acontainsasimple,unital,nonelementarysub-C∗-algebra, (2) Acontainsasequenceofpairwiseorthogonal,fullelements, (3) Ahasnofinite-dimensionalirreduciblerepresentations. In general, the implications (1) ⇒ (2) ⇒ (3) hold; it is not known if the con- versesaretrue. Conditions(2)and(3)canalsobeconsideredforpossiblynon-unitalC∗-algebras,andwelet(2∗)betheweakerstatementthatAcontainstwoorthogonal,full elements. Theimplication‘(3) ⇒ (2)’holdsexactlyiftheimplication‘(3) ⇒ (2∗)’ holds. TheGlobalGlimmhalvingproblemasksthefollowing: Givena(possiblynon- unital) C∗-algebraA thatsatisfies condition (3), does there exista fullmap from theconeoverM toA? ItisnotknownwhethertheGlobalGlimmhalvingprob- 2 lemhasapositiveanswer,butifitdoesthenitshowsthatimplication‘(3) ⇒(2)’ holds,sincetheconeoverM containstwoorthogonal,fullelements. 2 Letusremarkthattheanalogsofconditions(1)−(3)forvonNeumannalgebras areallequivalent. Infact,ifavonNeumannalgebraM hasnofinite-dimensional representations,thenthehyperfiniteII -factorRunitallyembedsintoM. 1 Historically, the generator problem for C∗-algebras is mostly asked for C∗-al- gebrasthataresimple oremoregenerallyhavenofinite-dimensionalrepresenta- tions: Question1.2. Iseverysimple,separable,unitalC∗-algebrasinglygenerated? Question 1.3. Is a separable, unital C∗-algebra singly generated provided it has nofinite-dimensionalirreduciblerepresentations? The answers to both questions are open. A positive answer to Question1.3 impliesapositiveanswertoQuestion1.2,ofcourse. Theconverseisnotclear. Let us mention some results that solve the generator problem for particular classes of separable C∗-algebras. It was shown by Topping, [Top68], that every UHF-algebra is singly generated. This was generalized by Olsen and Zame, [OZ76,Theorem9],whoshowedthatthetensorproduct,A⊗B,ofanyseparable, unitalC∗-algebraAwithaUHF-algebraB issinglygenerated. Later,itwasshownbyLiandShen,[LS10,Theorem3.1],thateveryunital,ap- proximatelydivisible4C∗-algebraissingly generated. Thisgeneralizesthe result of Olsen and Zame, since the tensor product with a UHF-algebra is always ap- proximatelydivisible. In this article we prove that every separable, unital, Z-stable C∗-algebra is singly generated, see Theorem3.7. This generalizes the result of Li and Shen, since every approximately divisible C∗-algebra is Z-stable, see [TW08, Theorem 2.3]. The notion of Z-stability has proven to be very important in the classifica- tionprogramofnuclearC∗-algebras,seee.g.[Win07]or[ET08],anditishasbeen 4AunitalC∗-algebraAis‘approximately divisible’ifforeveryε > 0andfinitesubsetF ⊂ A thereexistsa finite-dimensional, unitalsub-C∗-algebra B ⊂ A such thatB hasnocharacters and kxb−bxk≤εkbkforallx∈F,b∈B. 4 HANNESTHIELANDWILHELMWINTER shown that many nuclear, simple C∗-algebras are Z-stable, see e.g. [Win10]. Z- stability is also relevant in the non-nuclear context; for example, unital Z-stable C∗-algebrassatisfyKadison’ssimilarityproperty,see[JW11]. Thispaperproceedsasfollows: InSection2wesetupournotationandgivesomebasicfactsaboutthegenera- torrank,see2.1,andC (X)-algebras,see2.4. 0 Section3 contains the proof of our main result, which states that the tensor product A⊗ B of two separable, unital C∗-algebras is singly generated, if A max satisfies condition (2) from above (e.g. A is simple and non-elementary) and B admitsaunitalembeddingoftheJiang-SualgebraZ,seeTheorem3.5. Wederivethateveryseparable,unital,Z-stableC∗-algebraissinglygenerated, see Theorem3.7. Our main result can be considered as a (partial) C∗-algebraic analog of a theorem of Ge and Popa, [GP98, Theorem 6.2], which shows that a tensorproduct,M⊗¯N,oftwoII -factorsM,N issinglygenerated. Infact,wecan 1 reprovetheirtheoremwithourmethods,seeCorollary3.11. In Section4 we give further applications of our main theorem to tensor prod- uctswithreducedgroupC∗-algebras.WefirstobservethatZembedsunitallyinto Cr∗(F∞), thereducedgroupC∗-algebraofthe freegrouponinfinitelymanygen- erators,seeLemma4.1. Consequently, ifadiscretegroupΓcontainsanon-cyclic freesubgroup,thenZ embedsunitallyintoC∗(Γ),seeProposition4.2. r WededucethattensorproductsoftheformA⊗ C∗(Γ)aresinglygenerated max r if A is a separable, unital C∗-algebra satisfying condition (2) from above, and Γ isagroupcontaininganon-cyclic freesubgroup, seeCorollary4.4. Forexample, Cr∗(F∞)⊗Cr∗(F∞) is singly generated, although this C∗-algebra is not Z-stable, seeExample4.5. 2. PRELIMINARIES By a morphism between C∗-algebras we mean a ∗-homomorphism, and by an ideal of a C∗-algebra we understand a closed, two-sided ideal. If A is a C∗-algebra, then we denote by A its minimal unitization. Often, we write M for the k C∗-algebraofk-by-kmatricesM (C). e k 2.1. Let A be a C∗-algebra, and A ⊂ A the subset of self-adjoint elements. We sa say that a set S ⊂ A generates A, denoted A = C∗(S), if the smallest sub-C∗- sa algebra of A containing S is A itself. We denote by gen(A) the smallest number n∈{1,2,3,...,∞}suchthatAcontainsageneratingsubsetS ⊂A ofcardinality sa n,andwecallgen(A)thegeneratingrankofA. We stress that for the definition of gen(A), the generators are assumed to be self-adjoint. Two self-adjoint elements a,b generate the same C∗-algebra as the (non-self-adjoint) element a+ib. Therefore, a C∗-algebra A is said to be singly generatedifgen(A)≤2. FormoredetailsonthegeneratingrankwereferthereadertoNagisa,[Nag04], wherealsothefollowingsimplefactsarenotedforC∗-algebrasAandB: (1) gen(A)=gen(A), e THEGENERATORPROBLEMFORZ-STABLEC∗-ALGEBRAS 5 (2) gen(C∗(A,B)) ≤ gen(A)+gen(B), if A,B aresub-C∗-algebrasof a common C∗-algebra, and where C∗(A,B) denotes the sub-C∗-algebra they generate together, (3) gen(A⊕B)=max{gen(A),gen(B)}ifatleastoneofthealgebrasisunital. Let I ✁ A be an ideal in a C∗-algebra A. It is easy to see that the generating rank of the quotient A/I is not bigger than the generating rank of A, i.e., gen(A/I) ≤ gen(A), and the generating rank of A can be estimated as gen(A) ≤ gen(I) + gen(A/I). The following result gives an estimate for gen(I), and it is probablywell-knowntoexperts; sincewecouldnotlocateitintheliterature,we includeashortproof. Proposition2.2. LetAbeaC∗-algebra,andI✁Aanideal. Thengen(I)≤gen(A)+1. Proof. We may assume gen(A) is finite. So let a ,...,a be a set of self-adjoint 1 k generatorsforA. ThenAandI areseparable,andsoI containsastrictlypositive element h. It follows that C∗(h) contains a quasi-central approximate unit, see [AP77,Corollary3.3]and[Arv77]. ItisstraightforwardtoshowthatIisgenerated bythek+1elementsh,ha h,...,ha h. (cid:3) 1 k ThefollowingresultisattributedtoKirchbergin[Nag04]. Theorem2.3(Kirchberg). Everyseparable,unital,properlyinfiniteC∗-algebraissingly generated. Proof. Wesketchaproofbasedontheproofof[OZ76,Theorem9]. LetAbeasep- arable,unital,properlyinfiniteC∗-algebra.Thenthereexistisometriess ,s ,...∈ 1 2 A with pairwise orthogonal ranges (i.e., A contains a unital copy of the Cuntz algebraO∞). Let a ,a ,... ∈ A be a sequence of (positive) generators for A such that their 1 2 spectrasatisfyσ(a )⊂[1/2·1/4k,1/4k]. AgeneratorforAisgivenby: k x:= (s a s∗+1/2ks ). k k k k Xk≥1 As in in the proof of [OZ76, Theorem 9], one can show that σ(x) ⊂ {0} ∪ [1/2·1/4k,1/4k]. LetB := C∗(x) ⊂ A. Proceeding inductively, one shows k≥1 tShatak,sk ∈ B. Weonlysketchthisfork = 1. Setp := s1s∗1. Letfn beasequence ofpolynomialsconverginguniformlyto1on[1/8,1/4]andto0on[0,1/16]. Then f (x) converges to an element y ∈ B of the form y = p + pb(1 − p) for some n b ∈ A. We compute yy∗ = p(1 +b(1−p)b∗)p. Then for a continuous function A f: R → R with f(0) = 0 and f(t) = 1 for t ≥ 1, we get f(yy∗) = p ∈ B. Then s a s∗ =pxp∈B ands =2·px(1−p)∈B,andthenalsoa ∈B. (cid:3) 1 1 1 1 1 2.4. LetX bealocallycompactσ-compactHausdorffspace. AC (X)-algebraisa 0 C∗-algebraAtogetherwithamorphismη: C (X)→Z(M(A)),fromthecommu- 0 tativeC∗-algebraC (X)tothecenterofthemultiplieralgebraofA,suchthatfor 0 anyapproximateunit(u ) ofC (X),η(u )a → aforanya ∈ A,orequivalently, λ Λ 0 λ the closure of η(C (X))A is all of A. Thus, if X is compact, then η is necessarily 0 unital. Wewillusuallysuppressreferencetothestructuremap,andsimplywrite faorf·ainsteadofη(f)afortheproductofafunctionf ∈C (X)andanelement 0 a∈A. 6 HANNESTHIELANDWILHELMWINTER LetY ⊂X beaclosedsubset,andU :=X\Y itscomplement(anopensubset). ThenC (U)·AisanidealofA,denotedbyA(U). ThequotientA/A(U)isdenoted 0 byA(Y). Given a point x ∈ X, we write A(x) for A({x}), and we call this C∗-algebra the fiber of A at x. For an element a ∈ A, we denote by a(x) the image of a in the fiber A(x). For each a ∈ A, we may consider the map aˇ: x 7→ ka(x)k. This isa real-valued,upper-semicontinuous function on X, vanishing atinfinity. The C (X)-algebraAiscalledcontinuousifaˇisacontinuousfunctionforeacha∈A. 0 For more information on C (X)-algebraswe refer the readerto [Kas88, §1] or 0 themorerecent[Dad09,§2]. 2.5. TheJiang-SualgebraZ wasconstructedin[JS99];itmayberegardedasaC∗- algebraic analog of the hyperfinite II -factor. It can be obtained as an inductive 1 limit of prime dimension drop algebras Z := {f: [0,1] → M ⊗M | f(0) ∈ p,q p q 1 ⊗M ,f(1)∈M ⊗1 }. p q p q Formoredetails,wereferthereaderto[Win11],whereZ ischaracterizedinan entirely abstract manner, and to [Rør04] and [RW10], where it is shown that the generalizeddimensiondropalgebraZ2∞,3∞ := {f: [0,1] → M2∞ ⊗M3∞ |f(0) ∈ 1⊗M3∞,f(1) ∈ M2∞ ⊗1}embedsunitallyintoZ; infact,Z canbewrittenasa stationaryinductivelimitofZ2∞,3∞. 3. RESULTS Lemma3.1. LetAbeaseparable,unitalC∗-algebra. Thengen(A⊗Z2∞,3∞)≤5. Proof. ConsidertheidealI := C0(0,1)⊗M6∞ inB := A⊗Z2∞,3∞. Thequotient B/Iisisomorphicto(A⊗M2∞)⊕(A⊗M3∞).Thus,wehaveashortexactsequence: // // A⊗C0(0,1)⊗M6∞ A⊗Z2∞,3∞ (A⊗M2∞)⊕(A⊗M3∞) It follows from [OZ76] that the tensor product of a unital, separable C∗-alge- brawithaUHF-algebraissinglygenerated.Inparticular,gen(A⊗M2∞),gen(A⊗ M3∞) ≤ 2. Thus, the quotientsatisfiesgen(B/I) = max{gen(A⊗M2∞),gen(A⊗ M3∞)}≤2,see2.1. Note that I is an ideal in the C∗-algebra C := A ⊗ C(S1) ⊗ M2∞. We have gen(C) ≤ 2, and then gen(I) ≤ gen(C) +1 ≤ 3, by Proposition2.2. Then, the extensionisgeneratedbyatmost2+3=5self-adjointelements. (cid:3) The following is a Stone-Weierstrass type result. We prove it using the factorial Stone-Weierstrassconjecture,whichstatesthatasub-C∗-algebraB ⊂ Aexhausts A if it separates the factorial states of A. The factorial Stone-Weierstrass conjec- turewasprovedforseparableC∗-algebrasindependentlybyLongo,[Lon84],and Popa,[Pop84]. See2.4forashortintroductiontoC (X)-algebras. 0 Lemma3.2. LetAbeaseparable, continuousC (X)-algebra,andB ⊂ Aasub-C∗-al- 0 gebrasuchthatthefollowingtwoconditionsaresatisfied: (i) Foreachx∈X,B exhauststhefiberA(x), (ii) B separates the points of X by full elements, i.e., for each distinct pair of points x ,x ∈ X thereexistssomeb ∈ B suchthatb(x )isfullinB(x ) = A(x )and 0 1 1 1 1 b(x )=0. 0 THEGENERATORPROBLEMFORZ-STABLEC∗-ALGEBRAS 7 ThenA=B. Condition (ii) is for instance satisfied if B contains the image of the structure map η: C (X)→Z(M(A)). 0 Proof. Set Y := Prim(Z(M(A))), and identify Z(M(A)) with C(Y). Let π: A → B(H) be a non-degenerate factor representation. Then π extends to a representation π˜: M(A) → B(H). It is straightforward to show π(A)′′ = π˜(M(A))′′, so that π˜ is a factor representation of M(A). For any c ∈ Z(M(A)), we have c ∈ π(A)′ ∩ π˜(M(A))′′ = C · 1 . Thus, there exists a point y ∈ Y such that H π˜(c)=c(y)·1 forallc∈Z(M(A)). Sinceη(C (X))containsanapproximateunit H 0 for A, we have that π˜ ◦η is non-zero. Thus, there existsa point x ∈ X suchthat π˜◦η(f)=f(x)·1 forallf ∈C (X). Thismeansthatπ˜◦ηvanishesontheideal H 0 A(X \{x}),sothatπfactorsthroughthefiberA(x). Letus show thatB ⊂ A separatesthe factorsstatesof A. Soletϕ ,ϕ be two 1 2 different, non-degenerate factors states of A. We have shown above that there are two points x ,x ∈ X such that ϕ factors through A(x ), and we denote by 1 2 i i ϕ¯ : A(x ) → CtheinducedfactorstateonA(x ),fori = 1,2. Wedistinguishtwo i i i cases: Case1: x = x . Inthiscase,sinceϕ 6= ϕ ,thereexistsanelementa ∈ Asuch 1 2 1 2 that ϕ (a) 6= ϕ (a). By condition (i), there exists some element b ∈ B such that 1 2 b(x ) =a(x ). Notethatϕ (b)= ϕ¯ (b(x )) =ϕ¯ (a(x ))= ϕ (a),fori =1,2. Thus, 1 1 i i 1 i 1 i bseparatesthetwostates. Case 2: x 6= x . In this case, by condition (ii), there exists an element b ∈ B 1 2 suchthatb(x )isfullinA(x )andb(x )=0. Sinceϕ 6=0,thereexistsanelement 2 2 1 2 a∈Asuchthat|ϕ (a)|=|ϕ¯ (a(x ))|≥1. 2 2 2 Since b(x ) is full, there exist finitely many elements g ,h ∈ A(x ) such that 2 i i 2 ka(x )− c b(x )d k < 1. Bycondition(i), thereexistelementsg˜,h˜ ∈ B such 2 i i 2 i i i thatg˜(x P) = g andh˜ (x ) = h . Setb′ := c˜bd˜. Then|ϕ (b′)| = |ϕ¯ (b′(x ))| > i 2 i i 2 i i i i 2 2 2 0,whileb′(x1)=0. Thisshowsthatb′sepaPratesthetwostates. It follows that B separates the factor states of A, and therefore B = A by the factorialStone-Weierstrass conjecture, proved independently by Longo, [Lon84], andPopa,[Pop84]. (cid:3) Lemma3.3. LetAbeaunitalC∗-algebrawithgen(A) ≤ 3. Thenthereexistapositive elementx ∈ A⊗Z andtwopositive,fullelementsy′,z′ ∈Z suchthatA⊗Z is 2,3 2,3 2,3 generatedbyxand1⊗y′,andfurthery′andz′areorthogonal. Proof. WeconsiderZ astheC∗-algebraofcontinuousfunctionsfrom[0,1]toM 2,3 6 withtheboundaryconditions Y Z f(0)= Y  f(1)=(cid:18) QZQ∗(cid:19), Y   whereY ∈ M and Z ∈ M arearbitrarymatrices, and Q ∈ M isthe following 2 3 3 fixedpermutationmatrix: 1 Q=1 . 1   8 HANNESTHIELANDWILHELMWINTER Thismeansthatf(0),f(1)∈M havethefollowingform: 6 µ µ λ λ λ 11 12 11 12 13  µ21 µ22   λ21 λ22 λ23  µ µ λ λ λ f(0)= 11 12 f(1)= 31 22 33 ,  µ µ   λ λ λ   21 22   33 31 32   µ11 µ12   λ13 λ11 λ12       µ µ   λ λ λ  21 22 23 21 22     fornumbersµ ,λ ∈C. i,j i,j NotethatZ isnaturallyacontinuousC([0,1])-algebra,withfibersZ (0) ∼= 2,3 2,3 M ,Z (1)∼=M ,andZ (t)∼=M forpointst∈(0,1)⊂[0,1]. 2 2,3 3 2,3 6 Leta,b,c∈Abeasetofinvertible,positivegeneratorsforA. Denotebye the i,j matrix units in M . To shorten notation, for indices i,j set f := e +e . For 6 i,j i,j j,i t∈[0,1]wedefinethefollowingelementofA⊗M : 6 x :=a⊗(e +(1−t)·e +e ) t 1,1 3,3 5,5 +b⊗(f +(1−t)·f +f ) 1,2 3,4 5,6 +c⊗(e +(1−t)·e +e ) 2,2 4.4 6,6 +1 ⊗(t·f +t·f +δ(t)·f ) A 2,3 4,5 1,3 where δ: [0,1] → [0,1] is a continuous function on [0,1] that takes the value 0 at theendpoints0and1,andisstrictlypositiveateachpointt ∈ (0,1),e.g.,δ could begivenbyδ(t) = 1/4−(t−1/2)2. Wealsodefinefor t ∈ [0,1]two elementsof M : 6 y′ :=e +(1−t)·e +e t 1,1 3,3 5,5 z′ :=e +(1−t)·e +e t 2,2 4,4 6,6 Itiseasytocheckthattheassignmentx: t7→x definesanelementx∈A⊗Z . t 2.3 Similarly,wegettwoelementsy′,z′ ∈Z definedviat7→y′andt7→z′.Inmatrix 2.3 t t form,theseelementslookasfollows: a b δ(t)  b c t  δ(t) t (1−t)a (1−t)b x :=  t  (1−t)b (1−t)c t     t a b     b c    1    1  (1−t) y′ :=  z′ :=  t   t  (1−t)       1           1      Sety :=1⊗y′,andletD :=C∗(x+1,y)bethesub-C∗-algebraofE :=A⊗Z 2,3 generatedby the twoself-adjointelements x+1and y. Sincex ≥ 0, we getthat both1andxlieinC∗(x+1). ItfollowsthatD =C∗(1,x,y),andwewillshowthat D = E. NotethatE hasanaturalcontinuousC([0,1])-algebrastructure(induced THEGENERATORPROBLEMFORZ-STABLEC∗-ALGEBRAS 9 bytheoneofZ ),withfibersE(0)∼=A⊗M ,E(1)∼=A⊗M ,andE(t)∼=A⊗M 2,3 2 3 6 forpointst∈(0,1)⊂[0,1]. LetJ :=E((0,1))✁Ebethenaturalidealcorrespondingtotheopenset(0,1)⊂ [0,1].NotethatJ ∼=A⊗C ((0,1))⊗M ,andJ isnaturallyacontinuousC ((0,1))- 0 6 0 algebra. We will show in two stepsthatD exhauststhe idealJ (i.e., D∩J = J) andthequotientE/J (i.e.,D/(D∩J)=E/J). Step 1: We want to apply Lemma3.2 to the C((0,1))-algebra J with sub-C∗- algebra D ∩J. To verify condition (ii), note that the C∗-algebra generated by y′ containsC ((0,1))⊗e . Therefore,D∩J contains1 ⊗C ((0,1))⊗e ,which 0 3,3 A 0 3,3 separatesthepointsof(0,1).Since1 ⊗e ∈E(t)∼=A⊗M isfull,condition(ii) A 3,3 6 ofLemma3.2holdsanditremainstoverifycondition(i). We need to show that D ∩J exhausts all fibers of J. Fix some t ∈ (0,1), and setD :=C∗(1,x ,y )⊂A⊗M . Tosimplifynotation,wewritee¯ forthematrix t t t 6 i,j units1 ⊗e ∈A⊗M . WeneedtoshowthatD isallofA⊗M .Thiswillfollow A i,j 6 t 6 ifD containsalle¯ ,andforthisitisenoughtoshowthattheoff-diagonalmatrix t i,j unitse¯ areinD ,fori=1,...,5. i,i+1 t Thespectrumofy is{0,1−t,1}. Applyingfunctionalcalculustoy weobtain t t thatthefollowingthreeelementslieinD : t u:=e¯ +e¯ 1,1 5,5 v :=e¯ 3,3 w :=1−v−u=e¯ +e¯ +e¯ 2,2 4,4 6,6 Then,weproceedasfollows: 1. e¯ =δ(t)−1ux v ∈D andsoe¯ ,e¯ ∈D . 1,3 t t 1,1 5,5 t 2. g := b⊗e = e¯ x w ∈ D . Itfollows b⊗e = (gg∗)1/2 ∈ D , cf. [OZ76]. 1,2 1,1 t t 1,1 t Then b−1 ⊗e ∈ C∗(b⊗e ) ⊂ D and so e¯ = (b−1 ⊗e )·g ∈ D and 1,1 1,1 t 1,2 1,1 t e¯ ∈D . 2,2 t 3. b⊗e = (1−t)−1e¯ x (w −e¯ ) ∈ D . Arguing as above, it follows that 3,4 3,3 t 2,2 t e¯ ∈D ,andthene¯ ,e¯ ∈D . 3,4 t 4,4 6,6 t 4. e¯ =t−1e¯ x e¯ ∈D . 2,3 2,2 t 3,3 t 5. e¯ =t−1e¯ x e¯ ∈D . 4,5 4,4 t 5,5 t 6. b⊗e =e¯ x e¯ ∈D andsoe¯ ∈D . 5,6 5,5 t 6,6 t 5,6 t ThisshowsthatD∩J exhauststhefibersofJ. WemayapplyLemma3.2and deduceD∩J =J,whichfinishesstep1. Step2:WewanttoshowthatD/J exhaustsE/J =E({0,1})∼=A⊗(M ⊕M ). 2 3 (0) Let us denote the matrix units in M by e , i = 1,2, and the matrix units in 2 i,j M by e(1), i = 1,2,3. To simplify notation, we write e¯(k) for the matrix units 3 i,j i,j 1 ⊗e(k) ∈A⊗(M ⊕M ). LetusdenotetheimageofxandyinD/J byvandw: A i,j 2 3 v =a⊗(e(0)+e(1))+b⊗(e(0)+e(0)+e(1)+e(1))+c⊗(e(0)+e(1))+e¯(1)+e¯(1) 1,1 1,1 1,2 2,1 1,2 2,1 2,2 2,2 2,3 3,2 a b a b = ⊕b c 1 (cid:18)b c(cid:19) 1   1 1 0 w =e¯(0)+e¯(1) = ⊕ 0 . 1,1 1,1 (cid:18)0 0(cid:19) 0   10 HANNESTHIELANDWILHELMWINTER Asinstep1,itisenoughtoshowthatD/J containstheoff-diagonalmatrixunits (0) (1) (1) e¯ ,e¯ ande¯ . Weargueasfollows: 1,2 1,2 2,3 1. g :=wv(1−w)=b⊗(e(0)+e(1))∈D/J.Asinstep1,itfollowsthatb⊗(e(0)+ 1,2 1,2 1,1 e(1)) = (gg∗)1/2 ∈ D/J. Thenb−1⊗(e(0) +e(1)) ∈ D/J, andso e¯(0) +e¯(1) = 1,1 1,1 1,1 1,2 1,2 (b−1⊗(e(0)+e(1)))·g ∈D/J. Itfollowsthate¯(0)+e¯(1) ∈D/J. 1,1 1,1 2,2 2,2 2. e¯(1) =1−w−(e¯(0)+e¯(1))∈D/J. 3,3 2,2 2,2 (1) (1) (1) 3. e¯ =ve¯ ∈D/J,andsoe¯ ∈D/J. 2,3 3,3 2,2 4. b⊗e(1) =wve¯(1) ∈D/J. Again,thisimpliese¯(1) ∈D/J andsoe¯(1) ∈D/J. 1,2 2,2 1,2 1,1 5. e¯(0) =w−e¯(1) ∈D/J. 1,1 1,1 6. e¯(0) =1−w−e¯(1)−e¯(1) ∈D/J. 2,2 2,2 3,3 7. b⊗e(0) =e¯(0)ve¯(0) ∈D/J. Again,thisimpliese¯(0) ∈D/J. 1,2 1,1 2,2 1,2 Thisfinishesstep2. We have seen that A⊗Z is generated by x+1 and y. Moreover, z′ is full, 2,3 positiveandorthogonaltoy′. (cid:3) Lemma3.4. LetAbeaseparable,unitalC∗-algebra. Thenthereexistapositiveelement x∈A⊗Z2∞,3∞ andtwopositive,fullelementsy′,z′ ∈ Z2∞,3∞ suchthatA⊗Z2∞,3∞ isgeneratedbyxandy :=1⊗y′,andfurthery′andz′areorthogonal. Proof. LetB :=A⊗Z2∞,3∞. NotethatZ2∞,3∞⊗Z2,3isnaturallyaC([0,1]×[0,1])- algebra. Then, the quotient corresponding to the diagonal {(t,t) | t ∈ [0,1]} ⊂ [0,1]×[0,1]isisomorphictoZ2∞,3∞,andwedenotetheresultingsurjectivemor- phismbyπ: Z2∞,3∞ ⊗Z2,3 →Z2∞,3∞. Weproceedintwosteps. Step1: Weshowthatgen(B) ≤ k+1impliesgen(B) ≤ kfork ≥ 2. Soassume Bisgeneratedbytheself-adjoint,invertibleelementsa ,...,a . Thesub-C∗-al- 1 k+1 gebraC :=C∗(ak−1,ak,ak+1)⊂Bisunitalandsatisfiesgen(C)≤3. Considerthe C∗-algebraB⊗Z . ByLemma3.3,thesub-C∗-algebraC ⊗Z isgeneratedby 2,3 2,3 twoself-adjointelements,sayb,c. One readily checks that B ⊗ Z is generated by the k self-adjoint elements 2,3 a1 ⊗ 1,...,ak−2 ⊗ 1,b,c. Since B = A⊗Z2∞,3∞ is isomorphic to a quotient of B⊗Z2,3 =A⊗Z2∞,3∞ ⊗Z2,3,weobtaingen(B)≤gen(B⊗Z2,3)≤k. Step2: ByLemma3.1,wehavegen(B) ≤ 5. ApplyingStep1severaltimes,we obtaingen(B)≤3. ItfollowsfromLemma3.3thatthereexistsapositiveelementx˜∈B⊗Z and 2,3 twopositive, fullelementsy˜′,z˜′ ∈ Z suchthatB ⊗Z isgeneratedbyx˜ and 2,3 2,3 1⊗y˜′,andfurthery˜′andz˜′areorthogonal. Consider the surjective morphism id⊗π: A⊗Z2∞,3∞ ⊗Z2,3 → A⊗Z2∞,3∞. Onechecksthattheelementsx:=(id⊗π)(x˜)∈A⊗Z2∞,3∞,andy′ :=π(y˜′),z′ := π(z˜′)∈Z2∞,3∞ havethedesiredproperties. (cid:3) Theorem3.5. LetA,B betwoseparable,unitalC∗-algebras. Assumethefollowing: (1) A contains a sequence a ,a ,... of full, positive elements that are pairwise or- 1 2 thogonal, (2) B admitsaunitalembeddingoftheJiang-SualgebraZ. ThenA⊗ B issinglygenerated. EveryothertensorproductA⊗ B isaquotientof max λ A⊗ B,andthereforeisalsosinglygenerated. max