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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cellular filtration of K-theory
and determinants of $C^*$-algebras


Author: Liangqing Li
Journal: Proc. Amer. Math. Soc. 125 (1997), 2637-2642
MSC (1991): Primary 46L80, 46M20, 19K56
MathSciNet review: 1389529
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Abstract: In this note, we will disprove the following conjecture raised by Exel-Loring: Let $A$ be a $C^{*}$-algebra with trace $\tau $ and let $\det : U_{\infty } \to \mathbb {R}/\tau _{*}(K_{0}(A))$ be a determinant associated to $\tau $. If $\phi _{t}: C(S^{3}) \to A ~ (0 \leq t \leq 1)$ is a continuous family of homomorphisms and $b\in C(S^{3})\otimes M_{2}$ is the canonical matrix valued function on $S^{3}$ which represents the Bott element in $K_{1}(C(S^{3}))$, then $\det (\phi _{0}(b)) = \det (\phi _{1}(b))$. It should be noticed that the conjecture has been proved by Exel-Loring for the case that $\phi _{t}$ is a smooth family of homomorphisms.


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Additional Information

Liangqing Li
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Address at time of publication: Department of Mathematics, University of Puerto Rico, Rio Piedras, P. O. Box 23355, San Juan, Puerto Rico 00931
Email: li@fields.utoronto.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03857-4
PII: S 0002-9939(97)03857-4
Received by editor(s): January 22, 1996
Received by editor(s) in revised form: March 18, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society