A Milnor exact sequence for $E$-theory

Abstract

For separable $C^{\ast }$-algebras $A$ and $B$, the $E$-theory group $E(A,B)$ carries a natural, generally non-Hausdorff, second-countable group topology, Hausdorff exactly when the closure of zero $\overline{\{0\}}\subseteq E(A,B)$ is trivial. The nonzero elements of $\overline{\{0\}}$ are the phantom classes, invisible to every continuous Hausdorff-valued invariant. We identify $\overline{\{0\}}$ as a derived inverse limit over any shape system $(C_n)$ of $SA\otimes \mathcal{K}$, giving a natural Milnor sequence $0\to {\underleftarrow{\lim }}^1[C_n,S^{2}B\otimes \mathcal{K}]\to E(A,B)\to \underleftarrow{\lim }[C_n,SB\otimes \mathcal{K}]\to 0$ for all separable $A$ and $B$, with no UCT or nuclearity hypothesis. For nuclear $A$ it agrees with the Willett-Yu controlled-$KK$ Milnor sequence, and when $A$ satisfies the UCT, nuclear or not, its ${\underleftarrow{\lim }}^1$ term is the pure-extension group ${\mathrm{Pext}}_{\mathbb{Z}}^1(K_{\ast +1}(A),K_{\ast }(B))$, as in Schochet's fine-structure description of the Kasparov groups.

14 pages