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A Dixmier-Douady theory for strongly self-absorbing \(C^\ast\)-algebras. (English) Zbl 1431.46051

Summary: We show that the Dixmier-Douady theory of continuous fields of \(C^\ast\)-algebras with compact operators \(\mathbb{K}\) as fibers extends significantly to a more general theory of fields with fibers \(A\otimes \mathbb{K}\) where \(A\) is a strongly self-absorbing \(C^\ast\)-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient \(K\)-theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension.

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
55N15 Topological \(K\)-theory
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[1] Adams J. F., Infinite loop spaces, Ann. of Math. Stud. 90, Princeton University Press, Princeton 1978.; Adams, J. F., Infinite loop spaces (1978) · Zbl 0398.55008
[2] Arlettaz D., The order of the differentials in the Atiyah-Hirzebruch spectral sequence, K-Theory 6 (1992), no. 4, 347-361.; Arlettaz, D., The order of the differentials in the Atiyah-Hirzebruch spectral sequence, K-Theory, 6, 4, 347-361 (1992) · Zbl 0768.55012
[3] Atiyah M. and Segal G., Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287-330.; Atiyah, M.; Segal, G., Twisted K-theory, Ukr. Mat. Visn., 1, 3, 287-330 (2004) · Zbl 1151.55301
[4] Atiyah M. and Segal G., Twisted K-theory and cohomology, Inspired by S. S. Chern, Nankai Tracts Math. 11, World Scientific, Hackensack (2006), 5-43.; Atiyah, M.; Segal, G., Twisted K-theory and cohomology, Inspired by S. S. Chern, 5-43 (2006) · Zbl 1138.19003
[5] Bellissard J., K-theory of \({C^{\ast}}\)-algebras in solid state physics, Statistical mechanics and field theory: Mathematical aspects (Groningen 1985), Lecture Notes in Phys. 257, Springer-Verlag, Berlin (1986), 99-156.; Bellissard, J., K-theory of \({C^{\ast}}\)-algebras in solid state physics, Statistical mechanics and field theory: Mathematical aspects, 99-156 (1986)
[6] Benson D. J., Kumjian A. and Phillips N. C., Symmetries of Kirchberg algebras, Canad. Math. Bull. 46 (2003), no. 4, 509-528.; Benson, D. J.; Kumjian, A.; Phillips, N. C., Symmetries of Kirchberg algebras, Canad. Math. Bull., 46, 4, 509-528 (2003) · Zbl 1079.46047
[7] Blackadar B., K-theory for operator algebras, 2nd ed., Math. Sci. Res. Inst. Publ. 5, Cambridge University Press, Cambridge 1998.; Blackadar, B., K-theory for operator algebras (1998) · Zbl 0913.46054
[8] Blackadar B., Operator algebras, Encyclopaedia Math. Sci. 122, Springer-Verlag, Berlin 2006.; Blackadar, B., Operator algebras (2006) · Zbl 1092.46003
[9] Bouwknegt P. and Mathai V., D-branes, B-fields and twisted K-theory, J. High Energy Phys. 11 (2000), no. 3, Paper No. 7.; Bouwknegt, P.; Mathai, V., D-branes, B-fields and twisted K-theory, J. High Energy Phys., 11, 3 (2000) · Zbl 0959.81037
[10] Brown L. G., Stable isomorphism of hereditary subalgebras of \({C^*}\)-algebras, Pacific J. Math. 71 (1977), no. 2, 335-348.; Brown, L. G., Stable isomorphism of hereditary subalgebras of \({C^*}\)-algebras, Pacific J. Math., 71, 2, 335-348 (1977) · Zbl 0362.46042
[11] Brown L. G., Green P. and Rieffel M. A., Stable isomorphism and strong Morita equivalence of \({C^*}\)-algebras, Pacific J. Math. 71 (1977), no. 2, 349-363.; Brown, L. G.; Green, P.; Rieffel, M. A., Stable isomorphism and strong Morita equivalence of \({C^*}\)-algebras, Pacific J. Math., 71, 2, 349-363 (1977) · Zbl 0362.46043
[12] Connes A., Noncommutative geometry, Academic Press, San Diego 1994.; Connes, A., Noncommutative geometry (1994) · Zbl 0681.55004
[13] Cuntz J. and Higson N., Kuiper’s theorem for Hilbert modules, Operator algebras and mathematical physics (Iowa City 1985), Contemp. Math. 62, American Mathematical Society, Providence (1987), 429-435.; Cuntz, J.; Higson, N., Kuiper’s theorem for Hilbert modules, Operator algebras and mathematical physics, 429-435 (1987) · Zbl 0616.46066
[14] Dadarlat M., The homotopy groups of the automorphism group of Kirchberg algebras, J. Noncommut. Geom. 1 (2007), no. 1, 113-139.; Dadarlat, M., The homotopy groups of the automorphism group of Kirchberg algebras, J. Noncommut. Geom., 1, 1, 113-139 (2007) · Zbl 1144.46047
[15] Dadarlat M., Fiberwise KK-equivalence of continuous fields of \({C^*}\)-algebras, J. K-Theory 3 (2009), no. 2, 205-219.; Dadarlat, M., Fiberwise KK-equivalence of continuous fields of \({C^*}\)-algebras, J. K-Theory, 3, 2, 205-219 (2009) · Zbl 1173.46050
[16] Dadarlat M. and Pennig U., Unit spectra of K-theory from strongly self-absorbing \({C^*}\)-algebras, preprint 2013, .; Dadarlat, M.; Pennig, U., Unit spectra of K-theory from strongly self-absorbing \(C^*\)-algebras (2013) · Zbl 1332.46068
[17] Dadarlat M. and Winter W., Trivialization of \({C(X)}\)-algebras with strongly self-absorbing fibres, Bull. Soc. Math. France 136 (2008), no. 4, 575-606.; Dadarlat, M.; Winter, W., Trivialization of \({C(X)}\)-algebras with strongly self-absorbing fibres, Bull. Soc. Math. France, 136, 4, 575-606 (2008) · Zbl 1170.46051
[18] Dadarlat M. and Winter W., On the KK-theory of strongly self-absorbing \({C^*}\)-algebras, Math. Scand. 104 (2009), no. 1, 95-107.; Dadarlat, M.; Winter, W., On the KK-theory of strongly self-absorbing \({C^*}\)-algebras, Math. Scand., 104, 1, 95-107 (2009) · Zbl 1170.46065
[19] Dixmier J., \({C^*}\)-algebras, North Holland, Amsterdam 1982.; Dixmier, J., \({C^*}\)-algebras (1982)
[20] Dixmier J. and Douady A., Champs continus d’espaces hilbertiens et de \({C^{\ast}}\)-algèbres, Bull. Soc. Math. France 91 (1963), 227-284.; Dixmier, J.; Douady, A., Champs continus d’espaces hilbertiens et de \({C^{\ast}}\)-algèbres, Bull. Soc. Math. France, 91, 227-284 (1963) · Zbl 0127.33102
[21] Donovan P. and Karoubi M., Graded Brauer groups and K-theory with local coefficients, Publ. Math. Inst. Hautes Études Sci. 38 (1970), 5-25.; Donovan, P.; Karoubi, M., Graded Brauer groups and K-theory with local coefficients, Publ. Math. Inst. Hautes Études Sci., 38, 5-25 (1970) · Zbl 0207.22003
[22] Eckmann B. and Hilton P. J., Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145 (1961/1962), 227-255.; Eckmann, B.; Hilton, P. J., Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann., 145, 227-255 (19611962) · Zbl 0099.02101
[23] Eilenberg S. and Steenrod N., Foundations of algebraic topology, Princeton University Press, Princeton 1952.; Eilenberg, S.; Steenrod, N., Foundations of algebraic topology (1952) · Zbl 0047.41402
[24] Herman R. H. and Rosenberg J., Norm-close group actions on \({C^{\ast}}\)-algebras, J. Operator Theory 6 (1981), no. 1, 25-37.; Herman, R. H.; Rosenberg, J., Norm-close group actions on \({C^{\ast}}\)-algebras, J. Operator Theory, 6, 1, 25-37 (1981) · Zbl 0502.46043
[25] Hilton P., General cohomology theory and K-theory, London Math. Soc. Lecture Note Ser. 1, Cambridge University Press, London 1971.; Hilton, P., General cohomology theory and K-theory (1971) · Zbl 1386.55001
[26] Hirshberg I., Rørdam M. and Winter W., \({\mathcal{C}_0(X)} \)-algebras, stability and strongly self-absorbing \({C^*} \)-algebras, Math. Ann. 339 (2007), no. 3, 695-732.; Hirshberg, I.; Rørdam, M.; Winter, W., \({\mathcal{C}_0(X)} \)-algebras, stability and strongly self-absorbing \({C^*} \)-algebras, Math. Ann., 339, 3, 695-732 (2007) · Zbl 1128.46020
[27] Huber P. J., Homotopical cohomology and Čech cohomology, Math. Ann. 144 (1961), 73-76.; Huber, P. J., Homotopical cohomology and Čech cohomology, Math. Ann., 144, 73-76 (1961) · Zbl 0096.37504
[28] Jiang X., Nonstable K-theory for \({\mathcal{Z}} \)-stable \(C^*\)-algebras, preprint 1997, .; Jiang, X., Nonstable K-theory for \(<mml:mi mathvariant=''script``>\mathcal{Z} \)-stable \(C^*\)-algebras (1997)
[29] Kodaka K., Full projections, equivalence bimodules and automorphisms of stable algebras of unital \({C^*}\)-algebras, J. Operator Theory 37 (1997), no. 2, 357-369.; Kodaka, K., Full projections, equivalence bimodules and automorphisms of stable algebras of unital \({C^*}\)-algebras, J. Operator Theory, 37, 2, 357-369 (1997) · Zbl 0889.46050
[30] Landsman N. P., Mathematical topics between classical and quantum mechanics, Springer Monogr. Math., Springer-Verlag, New York 1998.; Landsman, N. P., Mathematical topics between classical and quantum mechanics (1998) · Zbl 0923.00008
[31] Lundell A. T. and Weingram S., The topology of CW complexes, Univ. Ser. Higher Math., Van Nostrand Reinhold, New York 1969.; Lundell, A. T.; Weingram, S., The topology of CW complexes (1969) · Zbl 0207.21704
[32] May J. P., The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer-Verlag, Berlin 1972.; May, J. P., The geometry of iterated loop spaces (1972) · Zbl 0244.55009
[33] May J. P., \({E_{\infty}}\) spaces, group completions, and permutative categories, New developments in topology (Oxford 1972), London Math. Soc. Lecture Note Ser. 11, Cambridge University Press, London (1974), 61-93.; May, J. P., \({E_{\infty}}\) spaces, group completions, and permutative categories, New developments in topology, 61-93 (1974)
[34] May J. P., Classifying spaces and fibrations, Mem. Amer. Math. Soc. 155 (1975).; May, J. P., Classifying spaces and fibrations, Mem. Amer. Math. Soc., 155 (1975) · Zbl 0321.55033
[35] May J. P., The spectra associated to permutative categories, Topology 17 (1978), no. 3, 225-228.; May, J. P., The spectra associated to permutative categories, Topology, 17, 3, 225-228 (1978) · Zbl 0417.55011
[36] Mingo J. A., K-theory and multipliers of stable \({C^{\ast}}\)-algebras, Trans. Amer. Math. Soc. 299 (1987), no. 1, 397-411.; Mingo, J. A., K-theory and multipliers of stable \({C^{\ast}}\)-algebras, Trans. Amer. Math. Soc., 299, 1, 397-411 (1987) · Zbl 0616.46067
[37] Nawata N. and Watatani Y., Fundamental group of simple \({C^*}\)-algebras with unique trace, Adv. Math. 225 (2010), no. 1, 307-318.; Nawata, N.; Watatani, Y., Fundamental group of simple \({C^*}\)-algebras with unique trace, Adv. Math., 225, 1, 307-318 (2010) · Zbl 1202.46068
[38] Nistor V., Fields of \({{\rm AF}}\)-algebras, J. Operator Theory 28 (1992), no. 1, 3-25.; Nistor, V., Fields of \({{\rm AF}}\)-algebras, J. Operator Theory, 28, 1, 3-25 (1992) · Zbl 0822.46082
[39] Rieffel M. A., Quantization and \({C^{\ast}}\)-algebras, \({C^{\ast}}\)-algebras: 1943-1993 (San Antonio 1993), Contemp. Math. 167, American Mathematical Society, Providence (1994), 66-97.; Rieffel, M. A., Quantization and \({C^{\ast}}\)-algebras, \({C^{\ast}}\)-algebras: 1943-1993, 66-97 (1994)
[40] Rørdam M., Classification of nuclear, simple \({C^*}\)-algebras, Encyclopaedia Math. Sci. 126, Springer-Verlag, Berlin 2002.; Rørdam, M., Classification of nuclear, simple \({C^*}\)-algebras (2002) · Zbl 1016.46037
[41] Rørdam M., The stable and the real rank of \({\mathcal{Z}} \)-absorbing \({C^*} \)-algebras, Internat. J. Math. 15 (2004), no. 10, 1065-1084.; Rørdam, M., The stable and the real rank of \({\mathcal{Z}} \)-absorbing \({C^*} \)-algebras, Internat. J. Math., 15, 10, 1065-1084 (2004) · Zbl 1077.46054
[42] Rosenberg J., Continuous-trace algebras from the bundle theoretic point of view, J. Aust. Math. Soc. Ser. A 47 (1989), no. 3, 368-381.; Rosenberg, J., Continuous-trace algebras from the bundle theoretic point of view, J. Aust. Math. Soc. Ser. A, 47, 3, 368-381 (1989) · Zbl 0695.46031
[43] Schochet C., The Dixmier-Douady invariant for dummies, Notices Amer. Math. Soc. 56 (2009), no. 7, 809-816.; Schochet, C., The Dixmier-Douady invariant for dummies, Notices Amer. Math. Soc., 56, 7, 809-816 (2009) · Zbl 1189.46048
[44] Schön R., Fibrations over a CWh-base, Proc. Amer. Math. Soc. 62 (1977), no. 1, 165-166.; Schön, R., Fibrations over a CWh-base, Proc. Amer. Math. Soc., 62, 1, 165-166 (1977) · Zbl 0346.55020
[45] Segal G., Classifying spaces and spectral sequences, Publ. Math. Inst. Hautes Études Sci. 34 (1968), 105-112.; Segal, G., Classifying spaces and spectral sequences, Publ. Math. Inst. Hautes Études Sci., 34, 105-112 (1968) · Zbl 0199.26404
[46] Segal G., Categories and cohomology theories, Topology 13 (1974), 293-312.; Segal, G., Categories and cohomology theories, Topology, 13, 293-312 (1974) · Zbl 0284.55016
[47] Segal G., Topological structures in string theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1784, 1389-1398.; Segal, G., Topological structures in string theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 359, 1784, 1389-1398 (2001) · Zbl 1041.81094
[48] Strøm A., Note on cofibrations, Math. Scand. 19 (1966), 11-14.; Strøm, A., Note on cofibrations, Math. Scand., 19, 11-14 (1966) · Zbl 0145.43604
[49] Thomsen K., Homotopy classes of \({*}\)-homomorphisms between stable \({C^*}\)-algebras and their multiplier algebras, Duke Math. J. 61 (1990), no. 1, 67-104.; Thomsen, K., Homotopy classes of \({*}\)-homomorphisms between stable \({C^*}\)-algebras and their multiplier algebras, Duke Math. J., 61, 1, 67-104 (1990) · Zbl 0718.46054
[50] Toms A. S. and Winter W., Strongly self-absorbing \({C^*}\)-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999-4029.; Toms, A. S.; Winter, W., Strongly self-absorbing \({C^*}\)-algebras, Trans. Amer. Math. Soc., 359, 8, 3999-4029 (2007) · Zbl 1120.46046
[51] Winter W., Strongly self-absorbing \({C^*}\)-algebras are \({\mathcal{Z}} \)-stable, J. Noncommut. Geom. 5 (2011), no. 2, 253-264.; Winter, W., Strongly self-absorbing \({C^*} \)-algebras are \({\mathcal{Z}} \)-stable, J. Noncommut. Geom., 5, 2, 253-264 (2011) · Zbl 1227.46041
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