Shape theory and asymptotic morphisms for \(C^*\)-algebras
We relate two topological invariants of separable \(C^*\)-algebras. The first is the shape invariant studied by \textit{E. G. Effros} and \textit{J. Kaminker} [Pitman Res. Notes Math. Ser. 123, 152-180 (1986; Zbl 0645.46051) and Publ. Math. Sci. Res. Inst. 6, 69-120 (1987; Zbl 0632.46064)] and then developed further by \textit{B. Blackadar} [Math. Scand. 56, 249-275 (1985; Zbl 0615.46066)]. The second invariant is the isomorphism class of a \(C^*\)-algebra in the asymptotic homotopy category \(\mathcal A\) 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)]. We prove that two separable \(C^*\)-algebras are shape equivalent if and only if they determine the same class in the category \(\mathcal A\) (see Theorem 3.9). The connection between these two invariants is established via a strong shape invariant that we introduce here. In particular, we show that any homotopy invariant functor on separable \(C^*\)-algebras that commutes with inductive limits automatically factors through the Connes-Higson category and is thus related to \(K\)-theory.
homotopy invariant functor on separable \(C^*\)-algebras
scientific article; zbMATH DE number 572048
scientific article