Homomorphisms of C*-algebras and their K-theory

TL;DR

This paper investigates the properties of homomorphisms between C*-algebras and their effects on K-theory, especially focusing on conditions for injectivity, torsion-freeness, and embeddability.

Contribution

It provides new results on the relationship between algebra homomorphisms and their induced K-theory maps, including conditions for torsion-free co-images and embeddability.

Findings

Co-image is torsion free when A and B are commutative, unital, with B having real rank zero, and the homomorphism is unital and injective.

A is embeddable in B if the K-theory map is injective, A has stable rank one, and real rank zero.

The paper characterizes properties of kernels and images of K-theory maps induced by *-homomorphisms.

Abstract

Let $A$ and $B$ be C*-algebras and $\varphi\colon A\to B$ be a $*$-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map $\mathrm{K}_{0}(\varphi)\colon \mathrm{K}_{0}(A) \to \mathrm{K}_{0}(B)$ on the level of K-theory. In particular, we are interested in the case that the co-image is torsion free, and show that it holds when $A$ and $ B $ are commutative and unital, $B$ has real rank zero, and $\varphi$ is unital and injective. We also show that $ A$ is embeddable in $B$ if $ \mathrm{K}_{0}(\varphi)$ is injective and $A$ has stable rank one and real rank zero.