Scalability and asymptotic adjunction
Georgii S. Makeev
TL;DR
The paper addresses the lack of a genuine adjunction for the endofunctor $C_{0}(oldsymbol{X})$ on $C^{*}$-algebras by developing a robust asymptotic-adjoint framework built from good endofunctors. It introduces Roe and relative Roe functors within this framework and proves that tensoring with $C_{oldsymbol{oldsymbol X}}$ has a right asymptotic adjoint given by $\, rak{N}_{oldsymbol{f X}}^{u}$ for scalable pairs, with explicit unit and counit. This asymptotic adjunction yields unsuspended descriptions of Connes–Higson $E$-theory, bridges between $E_{1}$-theory and extensions, and a $K$-homology interpretation in terms of metric cones, while providing technical simplifications in the separable case. The results connect coarse geometric methods with operator algebraic $K$-theory and $E$-theory, offering new computational tools and perspectives for noncommutative topology. Overall, the work extends categorical techniques to asymptotic settings, enabling practical and conceptual advances in $E$-theory and related invariants.
Abstract
In this paper, we introduce relative Roe functors and show that for every pair of scalable proper metric spaces, the functor of continuous functions and the relative Roe functor, both associated with this pair, are asymptotically adjoint. While this asymptotic adjunction is weaker than the genuine one, it retains sufficient categorical properties to be intuitive and useful in applications. These results can be used to provide an unsuspended description of the Connes-Higson $E$-theory, establish connections between $E_{1}$-theory and extension theory, and express $K$-homology of compact metric spaces in terms the corresponding metric cones.