Relatively spectral homomorphisms and K-injectivity

Abstract

Let $\A$ and $\B$ be unital Banach algebras and $ϕ\colon\A\to\B$ be a unital continuous homomorphism. We prove that if $\varphi$ is relatively spectral (i.e., there is a dense subalgebra $X$ of $\A$ such that $\sp_\B(ϕ(a))=\sp_\A(a)$ for every $a\in X$) and has dense range, then $\varphi$ induces monomorphisms from $K_i(\A)$ to $K_i(\B)$, $i=0,1$.

6 pages