Lifting isomorphisms in $K$-theory through gradings of $C^{\ast }$-algebras

We show that every strongly $\mathbb{Z}$-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous $\mathbb{T}$-action with full spectral subspaces) is a Cuntz–Pimsner algebra, and describe subalgebras and subspaces that can be used as the coefficient algebra and module in the construction. We deduce that for surjective graded homomorphisms $\phi$ of C*-algebras $A$ graded by torsion-free abelian groups $H$, if the restriction ${\phi }_0$ of $\phi$ to the zero-graded component $A_0$ of $A$ induces isomorphisms in K-theory, so does $\phi$ itself. When $H$ is free abelian, we show how to pick out smaller subalgebras of $A_0$ on which it suffices to check that $\phi$ induces isomorphisms in $K$-theory.