L^p operator algebras with approximate identities I

TL;DR

This paper explores how the theory of operator algebras, especially real positivity, extends from Hilbert spaces to L^p spaces, focusing on algebras of bounded operators on Lp spaces.

Contribution

It initiates the study of L^p operator algebras with approximate identities and examines the applicability of real positivity theory in this context.

Findings

Extended real positivity concepts to L^p operator algebras.

Analyzed the structure of algebras acting on L^p spaces.

Identified limitations and potential for further generalization.

Abstract

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.