Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ C\sp \infty$ loop algebras and noncommutative Bott periodicity


Author: N. Christopher Phillips
Journal: Trans. Amer. Math. Soc. 325 (1991), 631-659
MSC: Primary 58G12; Secondary 19K99, 46L80, 55R50
DOI: https://doi.org/10.1090/S0002-9947-1991-1016810-5
MathSciNet review: 1016810
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct the noncommutative analogs $ {\Omega _\infty }A$ and $ {\Omega _{{\text{lip}}}}A$ of the $ {C^\infty }$ and Lipschitz loop spaces for a pro-$ {C^\ast}$-algebra $ A$ equipped with a suitable dense subalgebra. With $ {U_{{\text{nc}}}}$ and $ P$ being the classifying algebras for $ K$-theory earlier introduced by the author, we then prove that there are homotopy equivalences $ {\Omega _\infty }{U_{{\text{nc}}}} \simeq P$ and $ {\Omega _\infty }P \simeq {U_{{\text{nc}}}}$. This result is a noncommutative analog of Bott periodicity in the form $ \Omega U \simeq {\mathbf{Z}} \times BU$ and $ \Omega ({\mathbf{Z}} \times BU) \simeq U$.


References [Enhancements On Off] (What's this?)

  • [1] R. G. Bartle and L. M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413. MR 0047910 (13:951i)
  • [2] B. Blackadar, Shape theory for $ {C^\ast}$-algebras, Math. Scand. 56 (1985), 249-275. MR 813640 (87b:46074)
  • [3] -, $ K$-theory for operator algebras, MSRI publications, no. 5, Springer-Verlag, New York and Berlin, 1986. MR 859867 (88g:46082)
  • [4] A. Connes, $ {C^\ast}$-algèbres et géométrie différentielle, C.R. Acad. Sci. Paris (Sér. A) 290 (1980), 599-604. MR 572645 (81c:46053)
  • [5] J. Cuntz, A new look at $ KK$-theory, $ K$-Theory 1 (1987), 31-51. MR 899916 (89a:46142)
  • [6] M. Karoubi, $ K$-theory, Grundlehren der math. Wissenschaften, no. 226, Springer-Verlag, Berlin, Heidelberg and New York, 1978. MR 0488029 (58:7605)
  • [7] G. G. Kasparov, The operator $ K$-functor and extensions of $ {C^\ast}$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 571-636; English transl., Math. USSR-Izv. 16 (1981), 513-572. MR 582160 (81m:58075)
  • [8] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math., no. 5, Springer-Verlag, Berlin, Heidelberg and New York, 1971. MR 0354798 (50:7275)
  • [9] N. C. Phillips, Inverse limits of $ {C^\ast}$-algebras, J. Operator Theory 19 (1988), 159-195. MR 950831 (90c:46090)
  • [10] -, Representable $ K$-theory for $ \sigma {\text{-}}{C^\ast}$-algebras, $ K$-Theory 3 (1989), 441-478. MR 1050490 (91k:46082)
  • [11] -, Inverse limits of $ {C^\ast}$-algebras and applications, Operator Algebras and Applications. Vol. 1: Structure Theory; $ K$-Theory, Geometry and Topology, L.M.S. Lecture Notes Ser., vol. 135, Cambridge Univ. Press, 1988, pp. 127-185.
  • [12] -, Classifying algebras for the $ K$-theory of $ \sigma {\text{-}}{C^\ast}$-algebras, Canad. J. Math. 41 (1989), 1021-1089. MR 1018451 (91h:46119)
  • [13] J. Rosenberg, The role of $ K$-theory in noncommutative algebraic topology, Operator Algebras and $ K$-Theory, Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155-182. MR 658514 (84h:46097)
  • [14] C. Schochet, Topological methods for $ {C^\ast}$-algebras III: axiomatic homology, Pacific J. Math. 114 (1984), 399-445. MR 757510 (86g:46102)
  • [15] -, Topological methods for $ {C^\ast}$-algebras IV: $ \bmod\, p$ homology, Pacific J. Math. 114 (1984), 447-468. MR 757511 (86g:46103)
  • [16] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [17] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 0210075 (35:970)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G12, 19K99, 46L80, 55R50

Retrieve articles in all journals with MSC: 58G12, 19K99, 46L80, 55R50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1016810-5
Keywords: Noncommutative loop space, $ {C^\infty }$ loop algebra, Bott periodicity, pro-$ {C^\ast}$-algebra, representable $ K$-theory
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society