$L^p$-operator algebras with approximate identities, I

David P. Blecher and N. Christopher Phillips

Vol. 303 (2019), No. 2, 401–457

DOI: 10.2140/pjm.2019.303.401

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

Keywords

$L^p$-operator algebra, accretive, approximate identity, Banach algebra, Kaplansky density, M-ideal, real positive, smooth Banach space, strictly convex Banach space, state of a Banach algebra, unitization

Mathematical Subject Classification 2010

Primary: 46E30, 46H10, 46H99, 47L10, 47L30

Secondary: 46H35, 47B38, 47B44, 47L75

Milestones

Received: 3 April 2018

Revised: 18 May 2019

Accepted: 19 May 2019

Published: 4 January 2020