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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
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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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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