  We investigate symmetry properties of vector-valued diffusion and
Schr\"odinger equations. For a separable Hilbert space $H$ we characterize the
subspaces of $L^2(\Omega, H)$ that are local (i.e., defined pointwise) and
discuss the issue of their invariance under the time evolution of the
differential equation. In this context, the possibility of a connection between
our results and the theory of gauge symmetries in mathematical physics is
explored.
