Unsuspended Connective $E$-Theory

We prove connective versions of results by Shulman [Shu10] and Dadarlat-Loring [DL94]. As a corollary, we see that two separable $C^*$-algebras of the form $C_0(X) \otimes A$, where $X$ is a based, connected, finite CW-complex and $A$ is a unital properly infinite algebra, are $\bu$-equivalent if and only if they are asymptotic matrix homotopy equivalent.

💡 Research Summary

The paper revisits two classical results—Shulman’s theorem that qA⊗K and Σ²A⊗K are asymptotically equivalent, and Dadarlat‑Loring’s identification of unsuspended E‑theory with asymptotic matrix homotopy—by placing them in a connective framework. The author works throughout with separable C*-algebras and uses the Connes‑Higson asymptotic homotopy category S together with the compact operators K to define the usual (non‑connective) bivariant E‑theory as E(A,B)=S(ΣA, ΣB⊗K).

The first technical step is to introduce the “asymptotic matrix homotopy category” m, defined as the colimit over matrix stabilizations: \