Lp-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 L-p-spaces. In the process we answer some open questions on real positivity in Banach algebras from work of Blecher and Ozawa.