𝐶^{∞} loop algebras and noncommutative Bott periodicity

Phillips, N. Christopher

doi:10.1090/S0002-9947-1991-1016810-5

$C^ \infty$ loop algebras and noncommutative Bott periodicity
HTML articles powered by AMS MathViewer

by N. Christopher Phillips

Trans. Amer. Math. Soc. 325 (1991), 631-659

DOI: https://doi.org/10.1090/S0002-9947-1991-1016810-5

PDF | Request permission

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

Robert G. Bartle and Lawrence M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400–413. MR 47910, DOI 10.1090/S0002-9947-1952-0047910-X
Bruce Blackadar, Shape theory for $C^\ast$-algebras, Math. Scand. 56 (1985), no. 2, 249–275. MR 813640, DOI 10.7146/math.scand.a-12100
Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
Alain Connes, $C^{\ast }$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604 (French, with English summary). MR 572645
Joachim Cuntz, A new look at $KK$-theory, $K$-Theory 1 (1987), no. 1, 31–51. MR 899916, DOI 10.1007/BF00533986
Max Karoubi, $K$-theory, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR 0488029
G. G. Kasparov, The operator $K$-functor and extensions of $C^{\ast }$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636, 719 (Russian). MR 582160
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
N. Christopher Phillips, Inverse limits of $C^*$-algebras, J. Operator Theory 19 (1988), no. 1, 159–195. MR 950831
N. Christopher Phillips, Representable $K$-theory for $\sigma$-$C^*$-algebras, $K$-Theory 3 (1989), no. 5, 441–478. MR 1050490, DOI 10.1007/BF00534137

Inverse limits of

algebras and applications

N. Christopher Phillips, Classifying algebras for the $K$-theory of $\sigma$-$C^*$-algebras, Canad. J. Math. 41 (1989), no. 6, 1021–1089. MR 1018451, DOI 10.4153/CJM-1989-046-2
Jonathan Rosenberg, The role of $K$-theory in noncommutative algebraic topology, Operator algebras and $K$-theory (San Francisco, Calif., 1981) Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182. MR 658514
Claude Schochet, Topological methods for $C^{\ast }$-algebras. III. Axiomatic homology, Pacific J. Math. 114 (1984), no. 2, 399–445. MR 757510
Claude Schochet, Topological methods for $C^{\ast }$-algebras. IV. Mod $p$ homology, Pacific J. Math. 114 (1984), no. 2, 447–468. MR 757511
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075