The unit ball of an injective operator space has an extreme point

We define an AW⁎-TRO as an off-diagonal corner of an AW⁎-algebra, and show that the unit ball of an AW⁎-TRO has an extreme point. In particular, the unit ball of an injective operator space has an extreme point, which answers a question raised in [8] affirmatively. We also show that an AW⁎-TRO (respectively, an injective operator space) has an ideal decomposition, that is, it can be decomposed into the direct sum of a left ideal, a right ideal, and a two-sided ideal in an AW⁎-algebra (respectively, an injective C⁎-algebra). In particular, we observe that an AW⁎-TRO, hence an injective operator space, has an algebrization which admits a quasi-identity.