Relative $K$-theory for $C^*$-algebras

TL;DR

This paper develops a new perspective on relative K-theory for C*-algebras using Banach categories, establishing exact sequences and showing its isomorphism to the K-theory of the mapping cone.

Contribution

It introduces a Banach category framework for relative K-theory and proves its equivalence to the K-theory of the mapping cone, with simplified element representation.

Findings

Established two six-term exact sequences.

Proved the natural isomorphism to the K-theory of the mapping cone.

Provided a simplified representation of elements in the relative groups.

Abstract

Given C$^*$-algebras $A$ and $B$ and a $^*$-homomorphism $\phi:A\rightarrow B$, we adopt the portrait of the relative $K$-theory $K_*(\phi)$ due to Karoubi using Banach categories and Banach functors. We show that the elements of the relative groups may be represented in a simple form. We prove the existence of two six-term exact sequences, and we use these sequences to deduce the fact that the relative theory is isomorphic, in a natural way, to the $K$-theory of the mapping cone.