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Transactions of the American Mathematical Society

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

 
 

 

$C^ \infty$ 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
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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 $\Omega ({\mathbf {Z}} \times BU) \simeq U$.


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Keywords: Noncommutative loop space, <!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${C^\infty }$"> loop algebra, Bott periodicity, pro-<IMG WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img37.gif" ALT="${C^\ast }$">-algebra, representable <IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$K$">-theory
Article copyright: © Copyright 1991 American Mathematical Society