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Cortiñas, G.; Tartaglia, G. "Operator ideals and assembly maps in K-theory" (2014) Proceedings of the American Mathematical Society. 142(4):1089-1099
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Abstract:

Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal Lp consisting of those operators whose sequence of singular values is p-summable; put S = ∪p Lp. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map HG * (ε(G, Vcyc),K(S)) → K*(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG * (ε(G,Fin),KH(Lp)) ⊗ ℚ → KH*(Lp[G]) ⊗ ℚ is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary. © 2014 American Mathematical Society.

Registro:

Documento: Artículo
Título:Operator ideals and assembly maps in K-theory
Autor:Cortiñas, G.; Tartaglia, G.
Filiación:Departamento de Matemática-IMAS, FCEyN-UBA, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Año:2014
Volumen:142
Número:4
Página de inicio:1089
Página de fin:1099
DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11837-X
Título revista:Proceedings of the American Mathematical Society
Título revista abreviado:Proc. Am. Math. Soc.
ISSN:00029939
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v142_n4_p1089_Cortinas

Referencias:

  • Bartels, A., Farrell, T., Jones, L., Reich, H., On the isomorphism conjecture in algebraic K-theory (2004) Topology, 43 (1), pp. 157-213. , DOI 10.1016/S0040-9383(03)00032-6. MR2030590 (2004m:19004)
  • Bartels, A., Lück, W., Isomorphism conjecture for homotopy K-theory and groups acting on trees (2006) J. Pure Appl. Algebra, 205 (3), pp. 660-696. , DOI 10.1016/j.jpaa.2005.07.020. MR2210223 (2007e:19005)
  • Cortiñas, G., The obstruction to excision in K-theory and in cyclic homology (2006) Invent. Math., 164 (1), pp. 143-173. , DOI 10.1007/s00222-005-0473-9. MR2207785 (2006k:19006)
  • Cortiñas, G., Ellis, E., Isomorphism conjectures with proper coefficients, , arXiv:1108.5196
  • Cortiñas, G., Thom, A., Comparison between algebraic and topological K-theory of locally convex algebras (2008) Adv. Math., 218 (1), pp. 266-307. , DOI 10.1016/j.aim.2007.12.007. MR2409415 (2009h:46136)
  • Cuntz, J., Cyclic theory and the bivariant Chern-Connes character (2004) Noncommutative geometry, Lecture Notes in Math, 1831, pp. 73-135. , Springer, Berlin, DOI 10.1007/978-3-540-39702-1 2. MR2058473 (2005e:58007)
  • Cuntz, J., Quillen, D., Excision in bivariant periodic cyclic cohomology (1997) Invent. Math., 127 (1), pp. 67-98. , DOI 10.1007/s002220050115. MR1423026 (98g:19003)
  • Davis, J.F., Lück, W., Spaces over a category and assembly maps in isomorphism conjectures in K-and L-theory (1998) K-Theory, 15 (3), pp. 201-252. , DOI 10.1023/A:1007784106877. MR1659969 (99m:55004)
  • Lück, W., Reich, H., Detecting K-theory by cyclic homology (2006) Proc. London Math. Soc. (3), 93 (3), pp. 593-634. , DOI 10.1017/S0024611506015954. MR2266961 (2007h:19008)
  • McCarthy, R., The cyclic homology of an exact category (1994) J. Pure Appl. Algebra, 93 (3), pp. 251-296. , DOI 10.1016/0022-4049(94)90091-4. MR1275967 (95b:19002)
  • Roe, J., Coarse cohomology and index theory on complete Riemannian manifolds (1993) Mem. Amer. Math. Soc., 104 (497), pp. x+90. , MR1147350 (94a:58193)
  • Suslin, A.A., Wodzicki, M., Excision in algebraic K-theory (1992) Ann. of Math., 136 (1-2), pp. 51-122. , DOI 10.2307/2946546. MR1173926 (93i:19006)
  • Thomason, R.W., Algebraic K-theory and étale cohomology (1985) Ann. Sci. École Norm. Sup., 18 (3-4), pp. 437-552. , MR826102 (87k:14016)
  • Weibel, C.A., An introduction to homological algebra (1994) Cambridge Studies in Advanced Mathematics, 38. , Cambridge University Press, Cambridge, MR1269324 (95f:18001)
  • Weibel, C.A., Homotopy algebraic K-theory, Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987) (1989) Contemp. Math., 83, pp. 461-488. , Amer. Math. Soc., Providence, RI, DOI 10.1090/conm/083/991991. MR991991 (90d:18006)
  • Wodzicki, M., Algebraic K-theory and functional analysis, First European Congress of Mathematics (1994) Progr. Math., 120, pp. 485-496. , Vol. II (Paris, 1992), Birkhäuser, Basel, MR1341858 (97f:46112)
  • Yu, G., The algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators, , arXiv:1106.3796

Citas:

---------- APA ----------
Cortiñas, G. & Tartaglia, G. (2014) . Operator ideals and assembly maps in K-theory. Proceedings of the American Mathematical Society, 142(4), 1089-1099.
http://dx.doi.org/10.1090/S0002-9939-2013-11837-X
---------- CHICAGO ----------
Cortiñas, G., Tartaglia, G. "Operator ideals and assembly maps in K-theory" . Proceedings of the American Mathematical Society 142, no. 4 (2014) : 1089-1099.
http://dx.doi.org/10.1090/S0002-9939-2013-11837-X
---------- MLA ----------
Cortiñas, G., Tartaglia, G. "Operator ideals and assembly maps in K-theory" . Proceedings of the American Mathematical Society, vol. 142, no. 4, 2014, pp. 1089-1099.
http://dx.doi.org/10.1090/S0002-9939-2013-11837-X
---------- VANCOUVER ----------
Cortiñas, G., Tartaglia, G. Operator ideals and assembly maps in K-theory. Proc. Am. Math. Soc. 2014;142(4):1089-1099.
http://dx.doi.org/10.1090/S0002-9939-2013-11837-X