Lifting isomorphisms in K-theory through gradings of C^*-algebras

We show that every strongly Z-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous 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 $ϕ$ of C*-algebras A graded by torsion-free abelian groups H, if the restriction ϕ₀ of $ϕ$ to the zero-graded component A₀ of A induces isomorphisms in K-theory, so does $ϕ$ itself. When H is free abelian, we show how to pick out smaller subalgebras of A₀ on which it suffices to check that $ϕ$ induces isomorphisms in K-theory.