MEMOIRS of the American Mathematical Society Volume 235 • Number 1107 (second of 5 numbers) • May 2015 Locally AH-Algebras Huaxin Lin ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society MEMOIRS of the American Mathematical Society Volume 235 • Number 1107 (second of 5 numbers) • May 2015 Locally AH-Algebras Huaxin Lin ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society Providence, Rhode Island Library of Congress Cataloging-in-Publication Data Lin,Huaxin,1956- LocallyAH-algebras/HuaxinLin. pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 235, number1107) Includesbibliographicalreferences. ISBN978-1-4704-1466-5(alk. paper) 1.C*-algebras. 2.Unitarygroups. 3.Algebra. I.Title. QA326.L575 2015 512(cid:2).55–dc23 2014049970 DOI:http://dx.doi.org/10.1090/memo/1107 Memoirs of the American Mathematical Society Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics. Subscription information. 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(cid:2) ThispublicationisindexedinMathematicalReviewsR,Zentralblatt MATH,ScienceCitation Index(cid:2)R,ScienceCitation IndexTM-Expanded,ISI Alerting ServicesSM,SciSearch(cid:2)R,Research (cid:2) (cid:2) (cid:2) AlertR,CompuMathCitation IndexR,Current ContentsR/Physical, Chemical& Earth Sciences. ThispublicationisarchivedinPortico andCLOCKSS. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 201918171615 Contents Chapter 1. Introduction 1 Chapter 2. Preliminaries 5 Chapter 3. Definition of C 13 g Chapter 4. C∗-algebras in C 15 g Chapter 5. Regularity of C∗-algebras in C 23 1 Chapter 6. Traces 35 Chapter 7. The unitary group 41 Chapter 8. Z-stability 49 Chapter 9. General Existence Theorems 53 Chapter 10. The uniqueness statement and the existence theorem for Bott map 67 Chapter 11. The Basic Homotopy Lemma 75 Chapter 12. The proof of the uniqueness theorem 10.4 89 Chapter 13. The reduction 97 Chapter 14. Appendix 103 Bibliography 107 iii Abstract A unital separable C∗-algebra A is said to be locally AH with no dimension growth if there is an integer d > 0 satisfying the following: for any (cid:2) > 0 and any compact subset F ⊂A, there is a unital C∗-subalgebra B of A with the form PC(X,M )P,whereX isacompactmetricspacewithcoveringdimensionnomore n than d and P ∈C(X,M ) is a projection, such that n dist(a,B)<(cid:2) for all a∈F. We prove that the class of unital separable simple C∗-algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple C∗-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with nodimensiongrowth. Infact,weshowthateveryunitalamenableseparablesimple C∗-algebra withfinitetracialrankwhichsatisfiestheUCThastracialrankatmost one. Therefore, by the author’s previous result, the class of those unital separable simple amenable C∗-algebras A satisfying the UCT which have rationally finite tracial rank can be classified by their Elliott invariant. We also show that unital separable simple C∗-algebras which are “tracially”locallyAHwithslow dimension growth are Z-stable. ReceivedbytheeditorJune3,2012and,inrevisedform,April25,2013. ArticleelectronicallypublishedonOctober1,2014. DOI:http://dx.doi.org/10.1090/memo/1107 2010 MathematicsSubjectClassification. Primary: 46L35,46L05. The author is affiliated with The Research Center for Operator Algebras, Department of Mathematics, East China Normal University, Shanghai, 20062, China — and — Department of MathematicsUniversityofOregon,Eugene,Oregon97405. (cid:3)c2014 American Mathematical Society v CHAPTER 1 Introduction The program of classification of amenable C∗-algebras, or the Elliott program, istoclassifyamenableC∗-algebrasuptoisomorphismsbytheirK-theoreticaldata. One of the high lights of the success of the Elliott program is the classification of unital simple AH-algebras (inductive limits of homogeneous C∗-algebras) with no dimension growth by their K-theoretical data (known as the Elliott invariant) ([16]). Theproofofthisfirstappearedneartheendofthelastcentury. Immediately aftertheproofappeared, amongmanyquestionsraisedisthequestionwhetherthe sameresultholdsforunitalsimplelocallyAH-algebras(seethedefinition3.5below) with no dimension growth. It should be noted that AF-algebras are locally finite dimensional. But(separable)AF-algebrasareinductivelimitsoffinitedimensional C∗-algebras. Theso-calledAT-algebrasareinductivelimitsofcirclealgebras. More than often, these AT-algebras arise as local circle algebras (approximated by circle algebras). Fortunately, due to the weak-semi-projectivity of circle algebras, locally AT-algebras are AT-algebras. However, the situation is completely different for locally AH algebras. In fact it was proved in [11] that there are unital C∗-algebras which are inductive limits of AH-algebras but themselves are not AH-algebras. So in general, a locally AH algebra is not an AH algebra. Ontheotherhand,however,itwasprovedin[27]thataunitalseparablesimple C∗-algebra which is locally AH is a unital simple AH-algebra, if, in addition, it has real rank zero, stable rank one and weakly unperforated K -group and which 0 has countably many extremal traces. In fact these C∗-algebras have tracial rank zero. The tracial condition was later removed in [59]. In particular, if A is a unitalseparablesimpleC∗-algebra whichislocallyAHwithno(orslow)dimension growthandwhichhasrealrankzeromustbeaunitalAH-algebra. InfactsuchC∗- algebras have stable rank one and have weakly unperforated K (A). The condition 0 of real rank zero forces these C∗-algebras to have tracial rank zero. More recently, classification theory extends to those C∗-algebras that have rationally tracial rank atmost one ([60], [33], [38] and[36]). These areunital separable simple amenable C∗-algebras A such that A⊗U have tracial rank at most one for some infinite dimensional UHF algebra U. An important subclass of this (which includes, for example, the Jiang-Su algebra Z) is the class of those unital separable simple C∗- algebrasAsuchthatA⊗U havetracialrankzero. Bynowwehavesomemachinery to verify that certain C∗-algebras to have tracial rank zero, (see [27], [5], [59] and [39]) and based on these results, we have some tools to verify when a unital simple C∗-algebra is rationally tracial rank zero ([56] and [55]). However, these results could not be applied to the case that C∗-algebras are of tracial rank one, or rationally tracial rank one. Until now, there has been no effective way, besides Gong’s decomposition result ([19]), to verify when a unital separable simple C∗- algebra has tracial rank one (but not tracial rank zero). In fact, as mentioned 1 2 HUAXINLIN above, we did not even know when a unital simple separable locally AH algebra with no dimension growth has tracial rank one. This makes it much harder to decide when a unital simple separable C∗-algebra is rationally tracial rank one. AcloselyrelatedproblemiswhetheraunitalseparablesimpleC∗-algebra with finitetracialrankisinfactoftracialrankatmostone. Thisisanopenproblemfor a decade. If the problem has an affirmative answer, it will make it easier, in many cases, to decide whether certain unital simple C∗-algebras to have tracial rank at most one. The purpose of this research is to solve these problems. Our main results include the following: Theorem 1.1. Let A be a unital separable simple C∗-algebra which is locally AHwith no dimensiongrowth. Then Ais isomorphicto a unital simpleAH-algebra with no dimension growth. We actually prove the following. Theorem 1.2. Let A be a unital separable simple amenable C∗-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem. Then A is isomorphic to a unital simple AH-algebra with no dimension growth. In particular, A has tracial rank at most one. To establish the above, we also prove the following Theorem 1.3. Let A be a unital amenable separable simple C∗-algebra in C 1 then A is Z-stable, i.e., A∼=A⊗Z. (See 3.6 below for the definition of C .) 1 The article is organized as follows. Section 2 serves as a preliminary which includes a number of conventions that will be used throughout this article. Some facts about a subgroup SU(M (C(X))/CU(M (C(X))) will be discussed. The n n detection of those unitaries with trivial determinant at each point which are not in the closure of commutator subgroup plays a new role in the Basic Homotopy Lemma, which will be presneted in section 11. In section 3, we introduce the class C of simple C∗-algebras which may be described as tracially locally AH algebras 1 of slow dimension growth. Several related definitions are given. In section 4, we discuss some basic properties of C∗-algebras in class C . In section 5, we prove, 1 among other things, that C∗-algebras in C have stable rank one and the strict 1 comparison for positive elements. In section 6, we study the tracial state space of a unital simple C∗-algebra in C . In particular, we show that every quasi-trace of 1 a unital separable simple C∗-algebra in C extends to a trace. Moreover, we show 1 that, for a unital simple C∗-algebra A in C , the affine map from the tracial state 1 space to state space of K (A) maps the extremal points onto the extremal points. 0 In section 7, we discuss the unitary groups of simple C∗-algebras in a subclass of C . Insection8, using what have beenestablished in previoussections, we combine 1 an argument of Winter ([61]) and an argument of Matui and Sato ([40]) to prove Theorem 1.3 above. In section 9 we present some versions of so-called existence theorem. In section 10, we present a uniqueness statement that will be proved in section 12 and an existence type result regarding the Bott map. The uniqueness theoremholdsforthecaseofY beingafiniteCWcomplexofdimensionzeroaswell as the case of Y =[0,1]. An induction on the dimension d will be presented in the nexttwosections. Insection11,wepresentaversionofTheBasicHomtopyLemma, 1. INTRODUCTION 3 which was first studied intensively in [4] and later in [32]. A new obstruction for theBasicHomotopyLemmainthisversionwillbedealtwith,whichwasmentioned earlier in section 2. In section 12, we prove the uniqueness statement in section 10. In section 13 we present the proofs for Theorem 1.1 and 1.3. Section 14 serves as an appendix to this article. Acknowledgement: The mostpartofthisworkwasinitiallydoneduring the summer 2010 when the author was in East China Normal University. A significant revised work was done in the spring of 2012 when the author was at the Research Center for Operator Algebras in ECNU. This research was partially supported by grants from East China Normal University, including the Changjiang Lectureship in the summer 2010. It was also partially supported by a grant from NSF.