  This paper is a revision and an enlargement of the previous version titled
"Extreme points of the unit ball of a quasi-multiplier space" which had been
circulated since 2004. We study extreme points of the unit ball of an operator
space by introducing the new notion (approximate) "quasi-identities". Then we
characterize an operator algebra with a contractive approximate quasi-
(respectively, left, right, two-sided) identity in terms of quasi-multipliers
and extreme points. Furthermore, we give a very neat necessary and sufficient
condition for a given operator space to become a $C^*$-algebra or a one-sided
ideal in a $C^*$-algebra in terms of quasi-multipliers. An extreme point is
also used to show that any TRO with predual can be decomposed to the direct sum
of a two-sided ideal, a left ideal, and a right ideal in some von Neumann
algebra.
