Abstract

 It has recently been discovered that there are algebras on L^{pspaces 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}pspace).
 
We have analogs of some of the most common examples of Hilbert space operator algebras, such as the  AF Algebras, the irrational rotation algebras, group C-algebras and von Neumann algebras, more general crossed products, the Cuntz algebras, and a few others. We have been able to prove analogs of some of the standard theorems about these algebras. We also have some ideas towards when an operator algebra on an L^p space  deserves to be considered the analog of a C-algebra or a von Neumann algebra. However, there is little general theory and there are many open questions, particularly for the analogs of von Neumann algebras.
 
In this talk, we will try to give an overview of some of what is known and some of the interesting open questions.