Equivalence of the $C^*$-Algebras $q\mathbb C$ and $C_0(\mathbb R^2)$ in the Asymptotic Category

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors $[[q\mathbb C\otimes K,B\otimes K]]=[[C_0(\mathbb R^2)\otimes K,B\otimes K]]$ for any $C^*$-algebra $B$; here $[[A,B]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A$ to $B$. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras $q\mathbb C\otimes K$ and $C_0(\mathbb R^2)\otimes K$ are equivalent in the category whose objects are $C^*$-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras $q\mathbb C\otimes K$ and $C_0(\mathbb R^2)\otimes K$ are represented as fields of $C^*$-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.