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

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.