Relatively spectral homomorphisms and K-injectivity

TL;DR

This paper demonstrates that a unital continuous homomorphism between Banach algebras, which is relatively spectral and has dense range, induces injective maps on algebraic K-theory groups, revealing new structural insights.

Contribution

It establishes that relatively spectral homomorphisms with dense range induce injective K-theory maps, a novel connection between spectral properties and K-theory.

Findings

Relatively spectral homomorphisms induce monomorphisms on K-theory groups.

Dense range condition is crucial for K-injectivity.

Provides new links between spectral theory and algebraic K-theory.

Abstract

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