Quasi-multipliers of operator spaces were defined by V. I. Paulsen in late
2002, and their correspondence to operator algebra products was discovered by
me in early 2003. That is, possible operator algebra products a given operator
space can be equipped with are precisely the bilinear mappings that are
implemented by the contractive quasi-multipliers [Kaneda-Paulsen 2003, to
appear in Journal of Functional Analysis]. In GPOTS 2003, after my talk,
G. K. Pedersen asked what the extreme points of the contractive
quasi-multipliers are. This gave me a further direction to study
quasi-multipliers. In this talk, we give a characterization of extreme points
of contractive quasi-multipliers by introducing the new notion: an
(approximate) quasi-identity. This is a natural generalization of an
(approximate) identity or an (approximate) one-sided identity. Furthermore, we
give a necessary and sufficient condition for an operator space to become an
operator algebra with a left, right, or two-sided contractive approximate
identity, or a $C^*$-algebra, or its left or right ideal, respectively, in
terms of (extreme points) of contractive quasi-multipliers.