E-theory Spectra for graded C*-algebras

Abstract

This paper brings together C * -algebras and algebraic topology in terms of viewing a C * -algebraic invariant in terms of a topological spectrum. E-theory, E(A, B), is a bivariant functor in the sense that is a cohomology functor in the first variable and a homology functor in the second variable but underlying goes from the category of separable C * -algebras and * -homomorphisms to the category of abelian groups and group homomorphisms. Here we create a generalisation of a orthogonal spectrum to quasi-topological spaces for E-theory . This includes a rich product structure in the context of graded separable C * -algebras.

E-theory for $C^\ast$-Categories

“…K-theory for C * -algebras was generalised to C * -categories in [Joa03,Mit01] and we take a similar approach to generalising E-theory here. We build on the work of Guentner-Higson [HG04] and Browne [Bro17] on constructing E-theory for graded complex and real C *categories.…”

Compactly Generated Spaces and Quasi-spaces in Topology

“…The latter supercategory of Top, that is convenient for homotopy theory, was suitably used in the results of Booth [Boo73] and Day [Day81]. Although carrying a size illegitimacy demonstrated by Herrlich and Rajagopalan [HR83], the category of quasitopological spaces provided a useful tool in current works as the ones by Dadarlat and Meyer [DM12] and Browne [Bro17] on E-theory of C * -algebras, which share a common topic with a past work of Dubuc and Porta [DP80]; we also refer to Petrakis' PhD thesis [Pet15] that highlights the relation between quasi-topological spaces and Bishop spaces. Moreover, the category QsTop also serves as a paradigm for the results of Escardó and Xu [XE13], and Dubuc and Español [Dub79,DE06] who presented a much more general context of quasi-topologies using the notion of Grothendieck topologies.…”