It has recently been discovered that there are algebras on $L^p$ spaces which deserve to be thought of as analogs of selfadjoint operator algebras on Hilbert spaces (even though there is no adjoint on the algebra of bounded operators on an $L^p$ space). So far, most work has concentrated on norm closed (“C*-like”) algebras. There are scattered results on the literature which can be interpreted as being on “von Nuemann algebra like” algebras of operators on $L^p$ spaces, but no systematic consideration.

In this talk, we present the beginnings of a systematic consideration of weak operator closed algebras on $L^p$ spaces which more or less deserve to be thought of as analogs of von Neumann algebras. We focus mainly on the analogs of type I factors and especially of type ${\mathrm{II}}_1$ factors, presenting both old and new results. There are both considerable similarities with, and considerable differences from, the Hilbert space theory. There are also many open problems.

This is joint work with Eusebio Gardella.