A separable Brown-Douglas-Fillmore theorem and weak stability

Author:
Huaxin Lin

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 2889-2925

MSC (2000):
Primary 46L05, 46L80

Published electronically:
March 2, 2004

MathSciNet review:
2052601

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Abstract: We give a separable Brown-Douglas-Fillmore theorem. Let be a separable amenable -algebra which satisfies the approximate UCT, be a unital separable amenable purely infinite simple -algebra and be two monomorphisms. We show that and are approximately unitarily equivalent if and only if We prove that, for any and any finite subset , there exist and a finite subset satisfying the following: for any amenable purely infinite simple -algebra and for any contractive positive linear map such that

for all there exists a homomorphism such that

provided, in addition, that are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number and any there is such that in any unital amenable purely infinite simple -algebra if

for a pair of unitaries, then there exists a pair of unitaries and in such that

**[BG]**Robert G. Bartle and Lawrence M. Graves,*Mappings between function spaces*, Trans. Amer. Math. Soc.**72**(1952), 400–413. MR**0047910**, 10.1090/S0002-9947-1952-0047910-X**[B]**Bruce Blackadar,*𝐾-theory for operator algebras*, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR**1656031****[Br1]**Lawrence G. Brown,*Stable isomorphism of hereditary subalgebras of 𝐶*-algebras*, Pacific J. Math.**71**(1977), no. 2, 335–348. MR**0454645****[BDF]**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Extensions of 𝐶*-algebras and 𝐾-homology*, Ann. of Math. (2)**105**(1977), no. 2, 265–324. MR**0458196****[CE]**Man Duen Choi and Edward G. Effros,*The completely positive lifting problem for 𝐶*-algebras*, Ann. of Math. (2)**104**(1976), no. 3, 585–609. MR**0417795****[DE1]**Marius Dadarlat and Søren Eilers,*On the classification of nuclear 𝐶*-algebras*, Proc. London Math. Soc. (3)**85**(2002), no. 1, 168–210. MR**1901373**, 10.1112/S0024611502013679**[DE2]**Marius Dadarlat and Søren Eilers,*Asymptotic unitary equivalence in 𝐾𝐾-theory*, 𝐾-Theory**23**(2001), no. 4, 305–322. MR**1860859**, 10.1023/A:1011930304577**[DL1]**Marius Dădărlat and Terry A. Loring,*𝐾-homology, asymptotic representations, and unsuspended 𝐸-theory*, J. Funct. Anal.**126**(1994), no. 2, 367–383. MR**1305073**, 10.1006/jfan.1994.1151**[DL2]**Marius Dadarlat and Terry A. Loring,*A universal multicoefficient theorem for the Kasparov groups*, Duke Math. J.**84**(1996), no. 2, 355–377. MR**1404333**, 10.1215/S0012-7094-96-08412-4**[DR]**Kenneth J. Dykema and Mikael Rørdam,*Purely infinite, simple 𝐶*-algebras arising from free product constructions*, Canad. J. Math.**50**(1998), no. 2, 323–341. MR**1618318**, 10.4153/CJM-1998-017-x**[Ell1]**George A. Elliott,*On the classification of 𝐶*-algebras of real rank zero*, J. Reine Angew. Math.**443**(1993), 179–219. MR**1241132**, 10.1515/crll.1993.443.179**[EG]**George A. Elliott and Guihua Gong,*On the classification of 𝐶*-algebras of real rank zero. II*, Ann. of Math. (2)**144**(1996), no. 3, 497–610. MR**1426886**, 10.2307/2118565**[F]**László Fuchs,*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673****[GL1]**Guihua Gong and Huaxin Lin,*Almost multiplicative morphisms and almost commuting matrices*, J. Operator Theory**40**(1998), no. 2, 217–275. MR**1660385****[GL2]**Guihua Gong and Huaxin Lin,*Almost multiplicative morphisms and 𝐾-theory*, Internat. J. Math.**11**(2000), no. 8, 983–1000. MR**1797674**, 10.1142/S0129167X0000043X**[H]**Nigel Higson,*A characterization of 𝐾𝐾-theory*, Pacific J. Math.**126**(1987), no. 2, 253–276. MR**869779****[HR]**Uffe Haagerup and Mikael Rørdam,*Perturbations of the rotation 𝐶*-algebras and of the Heisenberg commutation relation*, Duke Math. J.**77**(1995), no. 3, 627–656. MR**1324637**, 10.1215/S0012-7094-95-07720-5**[Ka]**G. G. Kasparov,*Hilbert 𝐶*-modules: theorems of Stinespring and Voiculescu*, J. Operator Theory**4**(1980), no. 1, 133–150. MR**587371****[K2]**E. Kirchberg,*The classification of purely infinite simple**-algebras using Kasparov's theory*, to appear in the Fields Institute Communication series.**[KP]**Eberhard Kirchberg and N. Christopher Phillips,*Embedding of exact 𝐶*-algebras in the Cuntz algebra 𝒪₂*, J. Reine Angew. Math.**525**(2000), 17–53. MR**1780426**, 10.1515/crll.2000.065**[Ln1]**Hua Xin Lin,*Almost commuting unitary elements in purely infinite simple 𝐶*-algebras*, Math. Ann.**303**(1995), no. 4, 599–616. MR**1359951**, 10.1007/BF01461007**[Ln2]**Huaxin Lin,*Extensions of 𝐶(𝑋) by simple 𝐶*-algebras of real rank zero*, Amer. J. Math.**119**(1997), no. 6, 1263–1289. MR**1481815****[Ln3]**Huaxin Lin,*Stable approximate unitary equivalence of homomorphisms*, J. Operator Theory**47**(2002), no. 2, 343–378. MR**1911851****[Ln4]**Huaxin Lin,*Tracially AF 𝐶*-algebras*, Trans. Amer. Math. Soc.**353**(2001), no. 2, 693–722 (electronic). MR**1804513**, 10.1090/S0002-9947-00-02680-5**[Ln5]**Huaxin Lin,*Classification of simple tracially AF 𝐶*-algebras*, Canad. J. Math.**53**(2001), no. 1, 161–194. MR**1814969**, 10.4153/CJM-2001-007-8**[Ln6]**Huaxin Lin,*The tracial topological rank of 𝐶*-algebras*, Proc. London Math. Soc. (3)**83**(2001), no. 1, 199–234. MR**1829565**, 10.1112/plms/83.1.199**[Ln7]**Huaxin Lin,*An introduction to the classification of amenable 𝐶*-algebras*, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR**1884366****[Ln10]**H. Lin,*An approximate Universal Coefficient Theorem*, preprint.**[Lo1]**Terry A. Loring,*𝐶*-algebras generated by stable relations*, J. Funct. Anal.**112**(1993), no. 1, 159–203. MR**1207940**, 10.1006/jfan.1993.1029**[Lo2]**Terry A. Loring,*Stable relations. II. Corona semiprojectivity and dimension-drop 𝐶*-algebras*, Pacific J. Math.**172**(1996), no. 2, 461–475. MR**1386627****[Lo3]**Terry A. Loring,*Lifting solutions to perturbing problems in 𝐶*-algebras*, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. MR**1420863****[Ped]**Gert K. Pedersen,*𝐶*-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006****[P1]**N. Christopher Phillips,*Approximation by unitaries with finite spectrum in purely infinite 𝐶*-algebras*, J. Funct. Anal.**120**(1994), no. 1, 98–106. MR**1262248**, 10.1006/jfan.1994.1025**[P2]**N. Christopher Phillips,*Approximate unitary equivalence of homomorphisms from odd Cuntz algebras*, Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 243–255. MR**1424965****[P3]**N. Christopher Phillips,*A classification theorem for nuclear purely infinite simple 𝐶*-algebras*, Doc. Math.**5**(2000), 49–114 (electronic). MR**1745197****[Ro1]**Mikael Rørdam,*Classification of inductive limits of Cuntz algebras*, J. Reine Angew. Math.**440**(1993), 175–200. MR**1225963**, 10.1515/crll.1993.440.175**[Ro2]**Mikael Rørdam,*Classification of certain infinite simple 𝐶*-algebras*, J. Funct. Anal.**131**(1995), no. 2, 415–458. MR**1345038**, 10.1006/jfan.1995.1095**[Ro3]**Mikael Rørdam,*A short proof of Elliott’s theorem: 𝒪₂⊗𝒪₂≅𝒪₂*, C. R. Math. Rep. Acad. Sci. Canada**16**(1994), no. 1, 31–36. MR**1276341****[Ro4]**M. Rørdam,*Classification of nuclear, simple 𝐶*-algebras*, Classification of nuclear 𝐶*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR**1878882**, 10.1007/978-3-662-04825-2_1**[RS]**Jonathan Rosenberg and Claude Schochet,*The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized 𝐾-functor*, Duke Math. J.**55**(1987), no. 2, 431–474. MR**894590**, 10.1215/S0012-7094-87-05524-4**[S1]**Claude Schochet,*Topological methods for 𝐶*-algebras. III. Axiomatic homology*, Pacific J. Math.**114**(1984), no. 2, 399–445. MR**757510****[S2]**Claude Schochet,*Topological methods for 𝐶*-algebras. IV. Mod 𝑝 homology*, Pacific J. Math.**114**(1984), no. 2, 447–468. MR**757511****[S3]**C. L. Schochet,*The fine structure of the Kasparov groups I: Continuity of the**-paring*, preprint, Sept. 1996.**[S4]**C. L. Schochet,*The fine structure of the Kasparov groups II: relative quasidiagonality*, preprint, Sept. 1996.**[Zh1]**Shuang Zhang,*A property of purely infinite simple 𝐶*-algebras*, Proc. Amer. Math. Soc.**109**(1990), no. 3, 717–720. MR**1010004**, 10.1090/S0002-9939-1990-1010004-X**[Zh2]**Shuang Zhang,*On the structure of projections and ideals of corona algebras*, Canad. J. Math.**41**(1989), no. 4, 721–742. MR**1012625**, 10.4153/CJM-1989-033-4**[Zh3]**Shuang Zhang,*On the exponential rank and exponential length of 𝐶*-algebras*, J. Operator Theory**28**(1992), no. 2, 337–355. MR**1273050**

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Additional Information

**Huaxin Lin**

Affiliation:
Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China

Address at time of publication:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

DOI:
https://doi.org/10.1090/S0002-9947-04-03558-5

Keywords:
Weakly semiprojective $C^*$-algebras,
purely infinite simple $C^*$-algebras

Received by editor(s):
September 18, 2002

Received by editor(s) in revised form:
April 29, 2003

Published electronically:
March 2, 2004

Additional Notes:
This research was partially supported by NSF grant DMS 0097903

Article copyright:
© Copyright 2004
American Mathematical Society