  We construct classes of von Neumann algebra modules by considering ``column
sums" of noncommutative L^p spaces. Our abstract characterization is based on
an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and
representations on Hilbert space. While the (single) representation theory is
similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be
nearly trivial.
