One-sided ideals and approximate identities in operator algebras

A left ideal of any C-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed, left ideals in C-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C-algebra. We study left ideals from the perspective of ‘Hamana theory ’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approximate identity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’. 2000 Mathematics subject classification: primary 46L05, 46L07, 47L30; secondary 46H10, 47L75. 1. Introduction an