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

Abstract

We define an $A{W}^{\ast }$-TRO as an off-diagonal corner of an $A{W}^{\ast }$-algebra, and show that the unit ball of an $A{W}^{\ast }$-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 $A{W}^{\ast }$-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 $A{W}^{\ast }$-algebra (respectively, an injective $C^{\ast }$-algebra). In particular, we observe that $A{W}^{\ast }$-TRO, hence an injective operator space, has an algebrization which admits a quasi-identity.

4 pages

References in corpus (1)

• Diagonalizing matrices over AW*-algebras