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

## TL;DR

This paper proves that the unit ball of an injective operator space has an extreme point by introducing $AW^*$-TROs and demonstrating their ideal decompositions, thus answering a previously open question.

## Contribution

It establishes that the unit ball of an injective operator space has an extreme point and introduces the concept of $AW^*$-TROs with ideal decompositions.

## Key findings

- The unit ball of an $AW^*$-TRO has an extreme point.
- Injective operator spaces can be decomposed into ideals within an $AW^*$-algebra.
- $AW^*$-TROs admit an algebrization with a quasi-identity.

## Abstract

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 the author's previous work [Journal of Operator Theory, 76(2) (2016), 219-248] 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 $AW^*$-TRO, hence an injective operator space, has an algebrization which admits a quasi-identity.