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Let $\A$ and $\B$ be unital Banach algebras and $\phi\colon\A\to\B$ be aunital 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 $K_i(\B)$, $i=0,1$.

Relatively spectral homomorphisms and K-injectivity