Exact sequences in the Enchilada category

TL;DR

This paper introduces the concept of exact sequences in the enchilada category of C*-algebras, demonstrating the non-exactness of the reduced-crossed-product functor and exploring implications for the Baum-Connes conjecture.

Contribution

It defines exact sequences within the enchilada category and investigates their properties, revealing the category's unusual behavior and its impact on understanding the Baum-Connes conjecture.

Findings

The reduced-crossed-product functor is not exact in the enchilada category.

The enchilada category exhibits unusual and 'strange' properties.

Results suggest limitations in using enchilada categories to analyze the Baum-Connes conjecture.

Abstract

We define exact sequences in the enchilada category of $C^*$-algebras and correspondences, and prove that the reduced-crossed-product functor is not exact for the enchilada categories. Our motivation was to determine whether we can have a better understanding of the Baum-Connes conjecture by using enchilada categories. Along the way we prove numerous results showing that the enchilada category is rather strange.