A completely bounded non-commutative Choquet boundary for operator spaces

Abstract

We develop a completely bounded counterpart to the non-commutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps on an operator space admitting a dilation of the same norm which is multiplicative on the generated $C^*$-algebra. We view such maps as analogues of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps which are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to $*$-homomorphisms which we use to associate to any representation of an operator space an entire scale of $C^*$-envelopes. We conjecture that these $C^*$-envelopes are all $*$-isomorphic, and verify this in some important cases.