Cellular filtration of K-theory and determinants of $C^*$-algebras
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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.References
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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
- Received by editor(s): January 22, 1996
- Received by editor(s) in revised form: March 18, 1996
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2637-2642
- MSC (1991): Primary 46L80, 46M20, 19K56
- DOI: https://doi.org/10.1090/S0002-9939-97-03857-4
- MathSciNet review: 1389529