2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230553This 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.27 pages, http://www.math.uci.edu/~mkaneda/Operator AlgebrasFunctional Analysis47L07 (Primary); 47L30, 47L25, 46L07, 47L50, 46M10, 46L05 (Secondary)text