[Paper Review] Boundary Representations of Operator Spaces, and Compact Rectangular Matrix Convex Sets

Adam H. Fuller, Michael Hartz+1 more|arXiv (Cornell University)|Jan 1, 2016

Advanced Operator Algebra ResearchMathematics11 references22 citations

TL;DR

This paper introduces compact rectangular matrix convex sets as a framework for studying matrix convexity in operator spaces, establishes analogs of the Krein-Milman and bipolar theorems, and proves Arveson's conjecture: every operator space is completely normed by its boundary representations, leading to a canonical construction of the triple envelope.

ABSTRACT

We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.

Motivation & Objective

• To develop a theory of matrix convexity tailored to operator spaces, extending classical convexity to non-self-adjoint settings.
• To define and characterize compact rectangular matrix convex sets as a natural generalization of compact convex sets in matrix spaces.
• To establish foundational theorems—Krein-Milman and bipolar—within the matrix convex setting for operator spaces.
• To introduce the concept of boundary representations for operator spaces and prove their role in completely norming the space.
• To provide a canonical construction of the triple envelope of an operator space using its boundary representations.

Proposed method

• Define compact rectangular matrix convex sets as closed, bounded, matrix-convex sets closed under direct sums and *-homomorphic images.
• Prove the matrix convex analog of the Krein-Milman theorem: every compact rectangular matrix convex set is the matrix convex hull of its extreme points.
• Establish the matrix bipolar theorem, showing that a set is a compact rectangular matrix convex set if and only if it is the bipolar of its bounded, matrix-convex subset.
• Introduce boundary representations as irreducible representations that completely norm the operator space.
• Use the duality between matrix convex sets and operator spaces to prove that every operator space is completely normed by its boundary representations.
• Construct the triple envelope as the operator space generated by its boundary representations, using the canonical duality established.

Experimental results

Research questions

• RQ1 How can classical convexity theorems like Krein-Milman be extended to the matrix convex setting for operator spaces?
• RQ2 What is the appropriate generalization of compact convex sets in the context of matrix convexity for non-self-adjoint operator spaces?
• RQ3 Do boundary representations of an operator space completely determine its norm, as conjectured by Arveson?
• RQ4 Can the triple envelope of an operator space be canonically reconstructed from its boundary representations?
• RQ5 What is the canonical correspondence between compact rectangular matrix convex sets and operator spaces?

Key findings

• The paper establishes that every compact rectangular matrix convex set is the matrix convex hull of its extreme points, extending the Krein-Milman theorem to the matrix setting.
• It proves the matrix bipolar theorem, showing that a set is a compact rectangular matrix convex set if and only if it is the bipolar of its bounded, matrix-convex subset.
• The authors prove Arveson's conjecture: every operator space is completely normed by its boundary representations, confirming a long-standing open problem.
• A canonical construction of the triple envelope of an operator space is achieved through its boundary representations.
• A one-to-one correspondence is established between compact rectangular matrix convex sets and operator spaces, generalizing the duality between compact convex sets and C*-algebras.
• The theory provides a new framework for understanding operator spaces via matrix convex geometry, with boundary representations playing a role analogous to extreme points in classical convex analysis.