J. Aust. Math. Soc.
76 (2004), 425447

Onesided ideals and approximate identities in operator algebras

David P. Blecher
Department of Mathematics
University of Houston
4800 Calhoun
Houston, TX 772043008
USA
dblecher@math.uh.edu



Abstract

A left ideal of any
algebra is an example of an operator algebra
with a right contractive approximate identity
(r.c.a.i.). Indeed, left ideals in
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 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 1sided identity or approximate
identity, including a BanachStone theorem for
these algebras, and an analysis of the
`multiplier operator algebra'.

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