A topology on \(E\)-theory (Q6561001)

The paper studies a topological enrichment of the asymptotic category introduced by \textit{A. Connes} and \textit{N. Higson} [C. R. Acad. Sci., Paris, Sér. I 311, No. 2, 101--106 (1990; Zbl 0717.46062)], whose objects are separable \(C^*\)-algebras with morphism sets \([[A,B]]\) given by asymptotic morphisms from \(A\) to \(B\), up to asymptotic homotopy. The topology on \([[A,B]]\) is based on the notion of semiprojectivity [\textit{B. Blackadar}, Math. Scand. 56, 249--275 (1985; Zbl 0615.46066)], using shape theoretic methods inspired by [\textit{M. Dadarlat}, Duke Math. J. 73, No. 3, 687--711 (1994; Zbl 0847.46028)]. A key point is a generalization of \textit{B. Blackadar}'s homotopy lifting property for semiprojective \(C^*\)-algebras [Math. Scand. 118, No. 2, 291--302 (2016; Zbl 1387.46043)].\N\NPulling back an asymptotic morphism along a semiprojective morphism gives an asymptotic morphism which can be represented by a \(*\)-homomorphism. This procedure yield a map \(\alpha^*\colon [[A,B]] \to H(A_0,B)\subseteq [[A_0,B]]\), where \(H(A_0,B)\) is the image of the natural map \(\mathrm{Hom}(A,B)\to [[A,B]]\). The set \([[A,B]]\) is equipped with the weakest topology which makes \(\alpha^*\) continuous, for any choice of semiprojective map \(\alpha\colon A_0\to A\). This topology is proved to be first countable, hence determined by its converging sequences, according to the following ``Pimsner's condition'': \(x_n\to x\in [[A,B]]\) if and only if there is \(y\in[[A,C(\mathbb{N}^+,B)]]\) with \(y(m)=x_m\) and \(y(\infty)=x\). The authors prove that this topology makes the categorical composition jointly continuous.\N\NThe Hausdorffized asymptotic category is introduced by replacing the morphism spaces \([[A,B]]\) with their Kolmogorov quotient (i.e., by declaring \(x,y\in[[A,B]]\) equivalent if the corresponding singletons have equal closures). This requires proving that composition descends to a well-defined and continuous map. The core results and applications of the paper revolve around these new spaces \([[A,B]]_{\text{Hd}}\). An important result describes these spaces as projective limits of discrete spaces, when using a shape system \((A_n,\alpha_n)\) for \(A\), in the following way:\N\[\N[[A,B]]_{\text{Hd}} \cong \varprojlim (H(A_n,B),\alpha_n^*).\N\]\NAs a result the Hausdorffized morphism sets are totally disconnected and completely metrizable. Elaborating on this results yields the continuity of the functor \(A\mapsto [[A,B]]_{\text{Hd}}\), i.e., it sends inductive limits to projective ones.\N\NThe authors go on to prove an equivalence of categories between the Hausdorffized asymptotic category and the shape category (the shape category is roughly speaking the full subcategory of the homotopy category of inductive systems of \(C^*\)-algebras given by shape systems). This results is analogous to a prior equivalence by Dadarlat [loc. cit.], between the asymptotic category and the \textit{strong} shape category.\N\NThe strategy is as follows: the strong shape category is a variation on the shape category which accounts for the data given by the sequences of homotopies involved in the definition. Dadarlat showed that this cateogry is the domain of a ''homotopy limit'' (or rather, colimit) construction with values in the asymptotic category (in other words, two inductive systems connected by a strong homotopy morphism get sent to their corresponding homotopy limits connected by an asymptotic morphism). This functor is an equivalence. The authors of this paper prove that this homotopy limit is independent of the choice of homotopies up to Hausdorffization, thus providing the aforementioned equivalence.\N\NThese results can be applied to the \(E\)-theory group \(E(A,B)\) because the latter are defined as \([[SA\otimes \mathbb{K},SA\otimes \mathbb{K}]]\) (suspension and stabilization). In particular it is possible to define groups \(EL(A,B)\) by passing to the Hausdorff quotient as above, in analogy with the \(KL\)-groups, a variation of the Kasparov groups which has proved itself useful in the classification program of nuclear \(C^*\)-algebras (e.g., in [\textit{N. C. Phillips}, Doc. Math. 5, 49--114 (2000; Zbl 0943.46037)]).\N\NSince isomorphisms in the shape cateogry lift to the strong shape category, the authors are able to prove that two \(C^*\)-algebras are \(E\)-equivalent if and only if they are \(EL\)-equivalent. The corresponding result for \(KK\)-theory is proved by \textit{M. Dadarlat} [J. Funct. Anal. 228, No. 2, 394--418 (2005; Zbl 1088.46042)] in the nuclear setting, by using the Kirchberg-Phillips theorem.