Real positivity and approximate identities in Banach algebras
  Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-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. 
David P. Blecher and Narutaka Ozawa
math.FA math.OA