Real positivity and approximate identities in Banach algebras

David P. Blecher and Narutaka Ozawa

Vol. 277 (2015), No. 1, 1–59

DOI: 10.2140/pjm.2015.277.1

Abstract

Blecher and Read recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^{\ast }$-algebraic results and theories to more general algebras. In the present paper, we generalize some part of this, and some other facts, to larger classes of Banach algebras.

Keywords

Banach algebra, approximate identity, unitization, real-positive, states, quasistate, ideals, hereditary subalgebra, ordered linear spaces, $M$-ideal, accretive operator, sectorial operator, operator roots, noncommutative Tietze theorem

Mathematical Subject Classification 2010

Primary: 46H10, 46H99, 46J99, 47L10, 47L30

Secondary: 47L75

Milestones

Received: 21 May 2014

Revised: 6 January 2015

Accepted: 13 January 2015

Published: 6 August 2015