Abstract
The homotopy symmetric \(C^*\)-algebras are those separable \(C^*\)-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear \(C^*\)-algebras and use it to show that the property of being homotopy symmetric passes to nuclear \(C^*\)-subalgebras and it has a number of other significant permanence properties. As an application, we show that if I(G) is the kernel of the trivial representation \(\iota :C^*(G)\rightarrow \mathbb {C}\) for a countable discrete torsion free nilpotent group G, then I(G) is homotopy symmetric and hence the Kasparov group KK(I(G), B) can be realized as the homotopy classes of asymptotic morphisms \([[I(G),B \otimes \mathcal {K}]]\) for any separable \(C^*\)-algebra B.
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References
Adem, A., Cheng, M.C.: Representation spaces for central extensions and almost commuting unitary matrices. Preprint 2015, mathematics. arXiv:1502.05092
Blanchard, É.: Subtriviality of continuous fields of nuclear \(C^*\)-algebras. J. Reine Angew. Math. 489, 133–149 (1997)
Brown, N.P.: Herrero’s approximation problem for quasidiagonal operators. J. Funct. Anal. 186(2), 360–365 (2001)
Connes, A.: Noncommutative geometry. Academic Press, San Diego (1994)
Connes, A., Gromov, M., Moscovici, H.: Conjecture de Novikov et fibrés presque plats. C. R. Acad. Sci. Paris Sér. I Math. 310(5), 273–277 (1990)
Connes, A., Higson, N.: Déformations, morphismes asymptotiques et \({K}\)-théorie bivariante. C. R. Acad. Sci. Paris Sér. I Math. 311(2), 101–106 (1990)
Dadarlat, M.: On the asymptotic homotopy type of inductive limit \(C^*\)-algebras. Math. Ann. 297(4), 671–676 (1993)
Dadarlat, M.: A note on asymptotic homomorphisms. K-Theory 8(5), 465–482 (1994)
Dadarlat, M.: Shape theory and asymptotic morphisms for \({C}^*\)-algebras. Duke Math. J. 73(3), 687–711 (1994)
Dadarlat, M.: Morphisms of simple tracially AF algebras. Intern. J. Math. 15(9), 919–957 (2004)
Dadarlat, M.: Group quasi-representations and almost flat bundles. J. Noncommut. Geom. 8(1), 163–178 (2014)
Carrión, J.R., Dadarlat, M.: Almost flat K-theory of classifying spaces Preprint 2015, mathematics. arXiv:1503.06170
Dadarlat, M., Loring, T.A.: \({K}\)-homology, asymptotic representations, and unsuspended \({E}\)-theory. J. Funct. Anal. 126(2), 367–383 (1994)
Echterhoff, S., Williams, D.P.: Crossed products by \(C_0(X)\)-actions. J. Funct. Anal. 158(1), 113–151 (1998)
Green, P.: The structure of imprimitivity algebras. J. Funct. Anal. 36(1), 88–104 (1980)
Higson, N., Kasparov, G.: \({E}\)-theory and \({K}{K}\)-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144(1), 23–74 (2001)
Houghton-Larsen, T.G., Thomsen, K.: Universal (co)homology theories. K-Theory 16(1), 1–27 (1999)
Jennings, S.A.: The group ring of a class of infinite nilpotent groups. Can. J. Math. 7, 169–187 (1955)
Kasparov, G.G.: Hilbert \(C^*\)-modules: theorems of Stinespring and Voiculescu. J. Oper. Theory 4, 133–150 (1980)
Kasparov, G.G.: Equivariant \(KK\)-theory and the Novikov conjecture. Invent. Math. 91(1), 147–201 (1988)
Kitaev, S.: Periodic table for topological insulators and superconductors. Preprint 2010, physics. arXiv:0901.2686
Lee, S.T., Packer, J.A.: \(K\)-theory for \(C^*\)-algebras associated to lattices in Heisenberg Lie groups. J. Oper. Theory 41(2), 291–319 (1999)
Loring, T.A.: \(K\)-theory and pseudospectra for topological insulators. Ann. Phys. 356, 383–416 (2015)
Packer, J.A., Raeburn, I.: On the structure of twisted group \(C^*\)-algebras. Trans. Am. Math. Soc. 334(2), 685–718 (1992)
Rieffel, M.A.: Deformation quantization for actions of \({ R}^d\). Memoirs of the American Mathematical Society, vol. 106, no. 506. American Mathematical Society, Providence (1993)
Robinson, D.J.S.: A course in the theory of groups, vol. 80 of Graduate Texts in Mathematics, Second ed. edn. Springer, New York (1996)
Rørdam, M.: Classification of nuclear, simple \(C^*\)-algebras. Encyclopaedia Mathematical Sciences, vol. 126. Springer, Berlin (2002)
Rosenberg, J.: The role of \(K\)-theory (San Francisco, California, 1981), vol. 10 of Contemporary Math., pp. 155–182. American Mathematical Society, Providence (1982)
Tikuisis, A., White, S., Winter, W.: Quasidiagonality of nuclear C*-algebras, Preprint 2015, mathematics. arXiv:1509.08318
Thomsen, K.: Discrete asymptotic homomorphisms in \(E\)-theory and \(KK\)-theory. Math. Scand. 92(1), 103–128 (2003)
Trout, J.: Asymptotic morphisms and elliptic operators over \(C^*\)-algebras. K-theory 18(3), 277–315 (1999)
Voiculescu, D.: A note on quasidiagonal \(C^*\)-algebras and homotopy. Duke Math. J. 62(2), 267–271 (1991)
Whitehead, G.W.: Elements of homotopy theory, Volume 61 of graduate texts in mathematics, Springer, New York (1978)
Tu, J.-L.: The gamma element for groups which admit a uniform embedding into Hilbert space. In: Recent advances in operator theory, operator algebras, and their applications. Oper. Theory Adv. Appl., vol. 153, Birkhäuser, Basel, pp. 271–286 (2005)
Yu, G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139(1), 201–240 (2000)
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M.D. was partially supported by NSF Grant #DMS–1362824. U.P. was partially supported by the SFB 878—“Groups, Geometry & Actions”.
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Dadarlat, M., Pennig, U. Deformations of nilpotent groups and homotopy symmetric \(C^*\)-algebras. Math. Ann. 367, 121–134 (2017). https://doi.org/10.1007/s00208-016-1379-0
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DOI: https://doi.org/10.1007/s00208-016-1379-0