In foundational work of Choi, Effros, and Ruan, it is shown that linear subspaces of $B(H)$ (i.e. operator spaces) are abstractly characterized as matrix normed spaces, while self-adjoint unital linear subspaces of $B(H)$ (i.e. operator systems) are abstractly characterized as matricially-ordered spaces with an Archimedean matrix order unit. Given an abstract operator space or operator system, one can produce a complete isometry into $B(H)$ realizing the unit as the identity operator, the adjoint as the operator adjoint, and positive elements as positive operators. However, some multiplicative structures may not be preserved by such representations. For example, if $p$ is a non-trivial projection in an operator system, one can always construct a unital complete order embedding into $B(H)$ which fails to map $p$ to a projection operator.

In this talk, we show that a variety of multiplicative structures in operator systems and unital operator spaces can be characterized by the matrix order and matrix norm structures. In particular, we present abstract characterizations for projections and unitaries in operator systems and a new abstract characterization for unital operator spaces. We also characterize the situation when a partially defined product may be defined on an operator system or a unital operator space. This talk is based on joint work with Roy Araiza and Adam Dor-On.