Extreme states and boundary representations of operator spaces in ternary rings of operators

Abstract

This paper explores different notions of extreme states on operator systems and on operator spaces. We prove that boundary points of the generalized state space of an operator system in finite dimensions are precisely the restrictions of boundary representations of the generated [Formula: see text]-algebra. We study weak TRO-extreme states on operator spaces in ternary rings of operators as a non self-adjoint version of [Formula: see text]-extreme states on operator systems in [Formula: see text]-algebras. Rectangular boundary point for a rectangular matrix convex set is introduced and its connection with boundary representations of the associated operator space is established in finite dimensional setting. Also, we show that the set of all rectangular boundary points is the minimal spanning set for a compact rectangular matrix convex set in finite dimensions.