# A completely bounded non-commutative Choquet boundary for operator spaces

## TL;DR

This paper introduces a new framework for the non-commutative Choquet boundary in operator spaces, focusing on a specialized class of completely bounded maps with dilation properties, leading to the construction of $C^*$-envelopes.

## Contribution

It develops a bounded non-commutative Choquet boundary theory by identifying a subset of maps with dilation properties, and constructs associated $C^*$-envelopes with potential isomorphism conjectures.

## Key findings

- Existence of extremal maps with unique extension properties
- Construction of $C^*$-envelopes from operator space representations
- Verification of isomorphism conjecture in specific cases

## 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.