J. Aust. Math. Soc.  76 (2004), 425-447
One-sided ideals and approximate identities in operator algebras

David P. Blecher
  Department of Mathematics
  University of Houston
  4800 Calhoun
  Houston, TX 77204-3008

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'.
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