The unit ball of an injective operator space has an extreme point (Q1746765)

An operator space \(X\) is called a triple system or a ternary ring of operators (TRO) if there exists a complete isometry \(\iota\) from \(X\) into a \(C^*\)-algebra such that \(\iota(x) \iota(y)^* \iota(z) \subset \iota(X)\) for all \(x, y, z\in X\). \textit{Z.-J. Ruan} [Trans. Am. Math. Soc. 315, No. 1, 89--104 (1989; Zbl 0669.46029)] and \textit{M. Hamana} [Math. J. Toyama Univ. 22, 77--93 (1999; Zbl 0948.46045)] proved in two independent papers that an operator space \(X\) is injective if and only if it is an off-diagonal corner of an injective \(C^*\)-algebra, that is, there exist an injective \(C^*\)-algebra \(A\) and two projections (self-adjoint idempotents) \(p, q\in A\) such that \(X\) is completely isometric to \(pAq\). In particular, an injective operator space is a TRO. Motivated by the fact that every injective \(C^*\)-algebra is monotone complete and hence an \(AW^*\)-algebra, the author considers the notion of an \(AW^*\)-TRO. An operator space \(X\) is an \(AW^*\)-TRO if there exist an \(AW^*\)-algebra \(A\) and projections \(p, q\in A\) such that \(X\) is completely isometric to \(pAq\). The main result of the paper under review shows that the unit ball of an \(AW^*\)-TRO contains an extreme point. In particular, the unit ball of an injective operator space has an extreme point, which gives a positive answer to a question posed by the same author in [J. Oper. Theory 76, No. 2, 219--248 (2016; Zbl 1389.47182)]. This result closes a natural and interesting question in the theory of operator spaces.