L^p operator algebras with approximate identities I

We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L^p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L^2-operator algebras and Banach algebras, to algebras of bounded operators on Lp spaces.