  A well-known result going back to the 1930s states that all bounded linear
operators mapping scalar-valued $L^1$-spaces into $L^\infty$-spaces are kernel
operators and that in fact this relation induces an isometric isomorphism
between the space of such operators and the space of all bounded kernels. We
extend this result to the case of spaces of vector-valued functions.
  A recent result due to Arendt and Thomaschewski states that the local
operators acting on $L^p$-spaces of functions with values in separable Banach
spaces are precisely the multiplication operators. We extend this result to
non-separable dual spaces.
  Moreover, we relate positivity and other order properties of the operators to
corresponding properties of the representations.
