Perturbations of nuclear C*-algebras

Erik Christensen; Allan M. Sinclair; Roger R. Smith; Stuart A. White; Wilhelm Winter

doi:10.1007/s11511-012-0075-5

2012 Perturbations of nuclear C*-algebras

Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart A. White, Wilhelm Winter

Author Affiliations +

Acta Math. 208(1): 93-150 (2012). DOI: 10.1007/s11511-012-0075-5

Abstract

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.

This paper provides the details of the results announced in Christensen et al. Proc. Natl. Acad. Sci. USA 107 (2010), 587–591.

References

1.

A rveson, W. B., Subalgebras of C^∗-algebras. Acta Math., 123 (1969), 141–224. MR253059 0194.15701 10.1007/BF02392388A rveson, W. B., Subalgebras of C^∗-algebras. Acta Math., 123 (1969), 141–224. MR253059 0194.15701 10.1007/BF02392388

2.

— Notes on extensions of C^∗ -algebras. Duke Math.J., 44 (1977), 329–355.— Notes on extensions of C^∗ -algebras. Duke Math.J., 44 (1977), 329–355.

3.

B ratteli, O., Inductive limits of finite dimensional C^∗-algebras. Trans. Amer.Math. Soc., 171 (1972), 195–234. MR312282 0264.46057B ratteli, O., Inductive limits of finite dimensional C^∗-algebras. Trans. Amer.Math. Soc., 171 (1972), 195–234. MR312282 0264.46057

4.

B rown, N. P. & O zawa, N., C^∗-Algebras andFinite-Dimensional Approximations. Graduate Studies in Mathematics, 88. Amer. Math. Soc., Providence, RI, 2008.B rown, N. P. & O zawa, N., C^∗-Algebras andFinite-Dimensional Approximations. Graduate Studies in Mathematics, 88. Amer. Math. Soc., Providence, RI, 2008.

5.

B unce, J. W. & P aschke, W. L., Derivations on a C^∗-algebra and its double dual. J. Funct.Anal., 37 (1980), 235–247. MR578934 0433.46052 10.1016/0022-1236(80)90043-9B unce, J. W. & P aschke, W. L., Derivations on a C^∗-algebra and its double dual. J. Funct.Anal., 37 (1980), 235–247. MR578934 0433.46052 10.1016/0022-1236(80)90043-9

6.

C hoi, M. D. & C hristensen, E., Completely order isomorphic and close C^∗-algebras need not be ∗-isomorphic. Bull. LondonMath. Soc., 15 (1983), 604–610. MR720750 0541.46046 10.1112/blms/15.6.604C hoi, M. D. & C hristensen, E., Completely order isomorphic and close C^∗-algebras need not be ∗-isomorphic. Bull. LondonMath. Soc., 15 (1983), 604–610. MR720750 0541.46046 10.1112/blms/15.6.604

7.

C hoi, M. D. & E ffros, E. G., Nuclear C^∗-algebras and the approximation property. Amer. J.Math., 100 (1978), 61–79. MR482238 0397.46054 10.2307/2373876C hoi, M. D. & E ffros, E. G., Nuclear C^∗-algebras and the approximation property. Amer. J.Math., 100 (1978), 61–79. MR482238 0397.46054 10.2307/2373876

8.

C hristensen, E., Perturbations of type I von Neumann algebras. J. LondonMath. Soc., 9 (1974/75), 395–405.C hristensen, E., Perturbations of type I von Neumann algebras. J. LondonMath. Soc., 9 (1974/75), 395–405.

9.

— Perturbations of operator algebras. Invent. Math., 43 (1977), 1–13.— Perturbations of operator algebras. Invent. Math., 43 (1977), 1–13.

10.

— Perturbations of operator algebras. II. Indiana Univ.Math. J., 26 (1977), 891–904.— Perturbations of operator algebras. II. Indiana Univ.Math. J., 26 (1977), 891–904.

11.

— Near inclusions of C^∗-algebras. Acta Math., 144 (1980), 249–265.— Near inclusions of C^∗-algebras. Acta Math., 144 (1980), 249–265.

12.

C hristensen, E., S inclair, A. M., S mith, R. R. & W hite, S. A., Perturbations of C^∗-algebraic invariants. Geom. Funct.Anal., 20 (2010), 368–397. MR2671282 1205.46036 10.1007/s00039-010-0070-yC hristensen, E., S inclair, A. M., S mith, R. R. & W hite, S. A., Perturbations of C^∗-algebraic invariants. Geom. Funct.Anal., 20 (2010), 368–397. MR2671282 1205.46036 10.1007/s00039-010-0070-y

13.

C hristensen, E., S inclair, A. M., S mith, R. R., W hite, S. A. & W inter, W., The spatial isomorphism problem for close separable nuclear C^∗-algebras. Proc. Natl.Acad. Sci.USA, 107 (2010), 587–591. MR2608650 1206.46045 10.1073/pnas.0913281107C hristensen, E., S inclair, A. M., S mith, R. R., W hite, S. A. & W inter, W., The spatial isomorphism problem for close separable nuclear C^∗-algebras. Proc. Natl.Acad. Sci.USA, 107 (2010), 587–591. MR2608650 1206.46045 10.1073/pnas.0913281107

14.

C onnes, A., Classification of injective factors. Cases II₁, II_∞, III_λ, λ≠1. Ann. ofMath., 104 (1976), 73–115.C onnes, A., Classification of injective factors. Cases II₁, II_∞, III_λ, λ≠1. Ann. ofMath., 104 (1976), 73–115.

15.

— On the cohomology of operator algebras. J. Funct.Anal., 28 (1978), 248–253.— On the cohomology of operator algebras. J. Funct.Anal., 28 (1978), 248–253.

16.

D adarlat, M., H irshberg, I., T oms, A. S. & W inter, W., The Jiang-Su algebra does not always embed. Math. Res.Lett., 16 (2009), 23–26. MR2480557 1180.46048D adarlat, M., H irshberg, I., T oms, A. S. & W inter, W., The Jiang-Su algebra does not always embed. Math. Res.Lett., 16 (2009), 23–26. MR2480557 1180.46048

17.

D avidson, K. R., Perturbations of reflexive operator algebras. J. OperatorTheory, 15 (1986), 289–305. MR833213 0601.47038D avidson, K. R., Perturbations of reflexive operator algebras. J. OperatorTheory, 15 (1986), 289–305. MR833213 0601.47038

18.

E ffros, E. G. & H aagerup, U., Lifting problems and local reflexivity for C^∗-algebras. Duke Math.J., 52 (1985), 103–128. MR791294 0613.46047 10.1215/S0012-7094-85-05207-XE ffros, E. G. & H aagerup, U., Lifting problems and local reflexivity for C^∗-algebras. Duke Math.J., 52 (1985), 103–128. MR791294 0613.46047 10.1215/S0012-7094-85-05207-X

19.

E ffros, E. G. & K aminker, J., Some homotopy and shape calculations for C^∗-algebras, in Group Representations,Ergodic Theory,Operator Algebras,and MathematicalPhysics (Berkeley, CA, 1984), Math. Sci. Res. Inst. Publ., 6, pp. 69–120. Springer, New York, 1987.E ffros, E. G. & K aminker, J., Some homotopy and shape calculations for C^∗-algebras, in Group Representations,Ergodic Theory,Operator Algebras,and MathematicalPhysics (Berkeley, CA, 1984), Math. Sci. Res. Inst. Publ., 6, pp. 69–120. Springer, New York, 1987.

20.

E ffros, E. G. & R osenberg, J., C^∗-algebras with approximately inner flip. Pacific J.Math., 77 (1978), 417–443. MR510932 0412.46052E ffros, E. G. & R osenberg, J., C^∗-algebras with approximately inner flip. Pacific J.Math., 77 (1978), 417–443. MR510932 0412.46052

21.

E ilers, S., L oring, T. A. & P edersen, G. K., Stability of anticommutation relations: an application of noncommutative CW complexes. J. ReineAngew. Math., 499 (1998), 101–143. MR1631120 0897.46056E ilers, S., L oring, T. A. & P edersen, G. K., Stability of anticommutation relations: an application of noncommutative CW complexes. J. ReineAngew. Math., 499 (1998), 101–143. MR1631120 0897.46056

22.

E lliott, G. A., On the classification of C^∗-algebras of real rank zero. J. ReineAngew. Math., 443 (1993), 179–219. MR1241132 0809.46067E lliott, G. A., On the classification of C^∗-algebras of real rank zero. J. ReineAngew. Math., 443 (1993), 179–219. MR1241132 0809.46067

23.

H aagerup, U., All nuclear C^∗-algebras are amenable. Invent. Math., 74 (1983), 305–319. MR723220 0529.46041 10.1007/BF01394319H aagerup, U., All nuclear C^∗-algebras are amenable. Invent. Math., 74 (1983), 305–319. MR723220 0529.46041 10.1007/BF01394319

24.

— The Grothendieck inequality for bilinear forms on C^∗-algebras. Adv. Math., 56 (1985), 93–116.— The Grothendieck inequality for bilinear forms on C^∗-algebras. Adv. Math., 56 (1985), 93–116.

25.

J iang, X. & S u, H., On a simple unital projectionless C^∗-algebra. Amer. J.Math., 121 (1999), 359–413. MR1680321 0923.46069 10.1353/ajm.1999.0012J iang, X. & S u, H., On a simple unital projectionless C^∗-algebra. Amer. J.Math., 121 (1999), 359–413. MR1680321 0923.46069 10.1353/ajm.1999.0012

26.

J ohanesova, M. & W inter, W., The similarity problem for $ {\mathcal z} $ -stable C^∗-algebras. Preprint, 2011. arXiv:1104.2067 [math.OA].J ohanesova, M. & W inter, W., The similarity problem for $ {\mathcal z} $ -stable C^∗-algebras. Preprint, 2011. arXiv:1104.2067 [math.OA].

27.

J ohnson, B. E., Approximate diagonals and cohomology of certain annihilator Banach algebras. Amer. J.Math., 94 (1972), 685–698. MR317050 0246.46040 10.2307/2373751J ohnson, B. E., Approximate diagonals and cohomology of certain annihilator Banach algebras. Amer. J.Math., 94 (1972), 685–698. MR317050 0246.46040 10.2307/2373751

28.

— Cohomology inBanach Algebras. Memoirs of the American Mathematical Society, 127. Amer. Math. Soc., Providence, RI, 1972.— Cohomology inBanach Algebras. Memoirs of the American Mathematical Society, 127. Amer. Math. Soc., Providence, RI, 1972.

29.

— A counterexample in the perturbation theory of C^∗-algebras. Canad. Math.Bull., 25 (1982), 311–316.— A counterexample in the perturbation theory of C^∗-algebras. Canad. Math.Bull., 25 (1982), 311–316.

30.

— Approximately multiplicative maps between Banach algebras. J. LondonMath. Soc., 37 (1988), 294–316.— Approximately multiplicative maps between Banach algebras. J. LondonMath. Soc., 37 (1988), 294–316.

31.

— Near inclusions for subhomogeneous C^∗-algebras. Proc. LondonMath. Soc., 68 (1994), 399–422.— Near inclusions for subhomogeneous C^∗-algebras. Proc. LondonMath. Soc., 68 (1994), 399–422.

32.

K adison, R. V. & K astler, D., Perturbations of von Neumann algebras. I. Stability of type. Amer. J.Math., 94 (1972), 38–54. MR296713 0239.46070K adison, R. V. & K astler, D., Perturbations of von Neumann algebras. I. Stability of type. Amer. J.Math., 94 (1972), 38–54. MR296713 0239.46070

33.

K hoshkam, M., On the unitary equivalence of close C^∗-algebras. Michigan Math.J., 31 (1984), 331–338. MR767612 0601.46056 10.1307/mmj/1029003077K hoshkam, M., On the unitary equivalence of close C^∗-algebras. Michigan Math.J., 31 (1984), 331–338. MR767612 0601.46056 10.1307/mmj/1029003077

34.

— Perturbations of C^∗-algebras and K-theory. J. OperatorTheory, 12 (1984), 89–99.— Perturbations of C^∗-algebras and K-theory. J. OperatorTheory, 12 (1984), 89–99.

35.

— Close C^∗-algebras are KK-equivalent, in Operator Theory:Operator Algebrasand Applications (Durham, NH, 1988), Part 2, Proc. Sympos. Pure Math., 51, pp. 159-162. Amer. Math. Soc., Providence, RI, 1990.— Close C^∗-algebras are KK-equivalent, in Operator Theory:Operator Algebrasand Applications (Durham, NH, 1988), Part 2, Proc. Sympos. Pure Math., 51, pp. 159-162. Amer. Math. Soc., Providence, RI, 1990.

36.

K irchberg, E. & R ørdam, M., Infinite non-simple C^∗-algebras: absorbing the Cuntz algebras $ {\mathcal O} $_∞. Adv. Math., 167 (2002), 195–264. MR1906257 1030.46075 10.1006/aima.2001.2041K irchberg, E. & R ørdam, M., Infinite non-simple C^∗-algebras: absorbing the Cuntz algebras $ {\mathcal O} $_∞. Adv. Math., 167 (2002), 195–264. MR1906257 1030.46075 10.1006/aima.2001.2041

37.

K irchberg, E. & W inter, W., Covering dimension and quasidiagonality. Internat. J.Math., 15 (2004), 63–85. MR2039212 1065.46053 10.1142/S0129167X04002119K irchberg, E. & W inter, W., Covering dimension and quasidiagonality. Internat. J.Math., 15 (2004), 63–85. MR2039212 1065.46053 10.1142/S0129167X04002119

38.

K ishimoto, A., O zawa, N. & S akai, S., Homogeneity of the pure state space of a separable C^∗-algebra. Canad. Math.Bull., 46 (2003), 365–372. MR1994863 1066.46052 10.4153/CMB-2003-038-3K ishimoto, A., O zawa, N. & S akai, S., Homogeneity of the pure state space of a separable C^∗-algebra. Canad. Math.Bull., 46 (2003), 365–372. MR1994863 1066.46052 10.4153/CMB-2003-038-3

39.

L ance, E. C., Cohomology and perturbations of nest algebras. Proc. LondonMath. Soc., 43 (1981), 334–356. MR628281 0477.47030 10.1112/plms/s3-43.2.334L ance, E. C., Cohomology and perturbations of nest algebras. Proc. LondonMath. Soc., 43 (1981), 334–356. MR628281 0477.47030 10.1112/plms/s3-43.2.334

40.

L oring, T. A., Lifting Solutionsto PerturbingProblems inC^∗-Algebras. Fields Institute Monographs, 8. Amer. Math. Soc., Providence, RI, 1997.L oring, T. A., Lifting Solutionsto PerturbingProblems inC^∗-Algebras. Fields Institute Monographs, 8. Amer. Math. Soc., Providence, RI, 1997.

41.

Murphy, G.J., C^∗-Algebras andOperator Theory. Academic Press, Boston, MA, 1990.Murphy, G.J., C^∗-Algebras andOperator Theory. Academic Press, Boston, MA, 1990.

42.

P hillips, J., Perturbations of C^∗-algebras. Indiana Univ.Math. J., 23 (1973/74), 1167–1176.P hillips, J., Perturbations of C^∗-algebras. Indiana Univ.Math. J., 23 (1973/74), 1167–1176.

43.

— Perturbations of type I von Neumann algebras. Pacific J.Math., 52 (1974), 505–511.— Perturbations of type I von Neumann algebras. Pacific J.Math., 52 (1974), 505–511.

44.

P hillips, J. & R aeburn, I., Perturbations of AF-algebras. Canad. J.Math., 31 (1979), 1012–1016. MR546954 0373.46074 10.4153/CJM-1979-093-8P hillips, J. & R aeburn, I., Perturbations of AF-algebras. Canad. J.Math., 31 (1979), 1012–1016. MR546954 0373.46074 10.4153/CJM-1979-093-8

45.

— Perturbations of C^∗-algebras. II. Proc. LondonMath. Soc., 43 (1981), 46–72.— Perturbations of C^∗-algebras. II. Proc. LondonMath. Soc., 43 (1981), 46–72.

46.

P isier, G., Grothendieck’s theorem for noncommutative C^∗-algebras, with an appendix on Grothendieck’s constants. J. Funct.Anal., 29 (1978), 397–415. MR512252 0388.46043 10.1016/0022-1236(78)90038-1P isier, G., Grothendieck’s theorem for noncommutative C^∗-algebras, with an appendix on Grothendieck’s constants. J. Funct.Anal., 29 (1978), 397–415. MR512252 0388.46043 10.1016/0022-1236(78)90038-1

47.

P itts, D. R., Close CSL algebras are similar. Math. Ann., 300 (1994), 149–156. MR1289835 0805.46056 10.1007/BF01450480P itts, D. R., Close CSL algebras are similar. Math. Ann., 300 (1994), 149–156. MR1289835 0805.46056 10.1007/BF01450480

48.

P owers, R. T., Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. ofMath., 86 (1967), 138–171. MR218905 0157.20605 10.2307/1970364P owers, R. T., Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. ofMath., 86 (1967), 138–171. MR218905 0157.20605 10.2307/1970364

49.

R aeburn, I. & T aylor, J. L., Hochschild cohomology and perturbations of Banach algebras. J. Funct.Anal., 25 (1977), 258–266. MR634038 0349.46043 10.1016/0022-1236(77)90072-6R aeburn, I. & T aylor, J. L., Hochschild cohomology and perturbations of Banach algebras. J. Funct.Anal., 25 (1977), 258–266. MR634038 0349.46043 10.1016/0022-1236(77)90072-6

50.

R osenberg, J., Amenability of crossed products of C^∗-algebras. Comm. Math.Phys., 57 (1977), 187–191. MR467331 0399.46046 10.1007/BF01625777R osenberg, J., Amenability of crossed products of C^∗-algebras. Comm. Math.Phys., 57 (1977), 187–191. MR467331 0399.46046 10.1007/BF01625777

51.

S inclair, A. M. & S mith, R. R., Hochschild cohomologyof vonNeumann algebras. London Mathematical Society Lecture Note Series, 203. Cambridge University Press, Cambridge, 1995.S inclair, A. M. & S mith, R. R., Hochschild cohomologyof vonNeumann algebras. London Mathematical Society Lecture Note Series, 203. Cambridge University Press, Cambridge, 1995.

52.

T oms, A. S. & W inter, W., Strongly self-absorbing C^∗-algebras. Trans. Amer.Math. Soc., 359 (2007), 3999–4029. MR2302521 1120.46046 10.1090/S0002-9947-07-04173-6T oms, A. S. & W inter, W., Strongly self-absorbing C^∗-algebras. Trans. Amer.Math. Soc., 359 (2007), 3999–4029. MR2302521 1120.46046 10.1090/S0002-9947-07-04173-6

53.

— Minimal dynamics and the classification of C^∗-algebras. Proc. Natl.Acad. Sci.USA, 106:40 (2009), 16942–16943.— Minimal dynamics and the classification of C^∗-algebras. Proc. Natl.Acad. Sci.USA, 106:40 (2009), 16942–16943.

54.

W inter, W., Covering dimension for nuclear C^∗-algebras. J. Funct.Anal., 199 (2003), 535–556. MR1971906 1026.46049 10.1016/S0022-1236(02)00109-XW inter, W., Covering dimension for nuclear C^∗-algebras. J. Funct.Anal., 199 (2003), 535–556. MR1971906 1026.46049 10.1016/S0022-1236(02)00109-X

55.

— Covering dimension for nuclear C^∗-algebras. II. Trans. Amer.Math. Soc., 361 (2009), 4143–4167.— Covering dimension for nuclear C^∗-algebras. II. Trans. Amer.Math. Soc., 361 (2009), 4143–4167.

56.

W inter, W. & Z acharias, J., Completely positive maps of order zero. Münster J.Math., 2 (2009), 311–324. MR2545617 1190.46042W inter, W. & Z acharias, J., Completely positive maps of order zero. Münster J.Math., 2 (2009), 311–324. MR2545617 1190.46042

57.

— The nuclear dimension of C^∗-algebras. Adv. Math., 224 (2010), 461–498.— The nuclear dimension of C^∗-algebras. Adv. Math., 224 (2010), 461–498.

Citation Download Citation

Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart A. White, and Wilhelm Winter "Perturbations of nuclear C*-algebras," Acta Mathematica 208(1), 93-150, (2012). https://doi.org/10.1007/s11511-012-0075-5

Received: 27 October 2009; Published: 2012

Access the abstract

JOURNAL ARTICLE
58 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY