Corner embeddings into algebras of compact operators in $K$-theory

TL;DR

This paper demonstrates that in certain $K$-theory categories, many corner embeddings into discrete compact operator algebras are invertible, impacting the faithfulness of functors in algebraic $KK$-theory.

Contribution

It establishes the invertibility of corner embeddings in $K$-theory-like categories and characterizes faithfulness of functors on split-exact algebraic $KK$-theory.

Findings

Corner embeddings into discrete compact operator algebras are invertible in $K$-theory-like categories.

Functor faithfulness on split-exact algebraic $KK$-theory is equivalent to faithfulness on the subcategory generated by homomorphisms.

Abstract

We show that in $K$-theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic $KK$-theory are faithful if and only if they are faithful on the subcategory generated by the homomorphisms.