Elements\alpha\in A\otimes E of the tensor product of a Banach algebra A and a Banach space E induce systems \{\psi(\alpha):\psi\in E^*\} of elements of A indexed by the dual space E^*, whose joint spectrum belongs to the second dual E^{**}. In this note we investigate when the spectrum actually lies in E\subseteq E^{**}, and extend the spectral mapping theorem P\sigma_A(\alpha)=\sigma_AP(\alpha) to polynomial mappings P:E\to F between Banach spaces. When the algebra A is commutative and the Banach space E=B is another algebra we also reach a sort of vector-valued Gelfand theory.