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

Mitch Haslehurst

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

Operator Algebras 2023-04-18 v3

Authors: Mitch Haslehurst

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.

Keywords

graph algebras algebraic k-theory c*-algebras

Cite

@article{arxiv.2106.02620,
  title  = {Relative $K$-theory for $C^*$-algebras},
  author = {Mitch Haslehurst},
  journal= {arXiv preprint arXiv:2106.02620},
  year   = {2023}
}

Comments

Final version to appear in the Rocky Mountain Journal of Mathematics; restructured main results; new results added

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