Onesided ideals and approximate identities in operator algebras
Abstract
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 introduced by the author. More generally, we develop some general theory for operator algebras which have a 1sided identity or approximate identity, including a BanachStone theorem for these algebras, and an analysis of the `multiplier operator algebra'.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:math/0111189
 Bibcode:
 2001math.....11189B
 Keywords:

 Operator Algebras;
 Functional Analysis;
 47L30 47L45 47L25