# Journal of Operator Theory

Volume 79, Issue 1, Winter 2018 pp. 139-172.

Boundary representations of operator spaces, and compact
rectangular matrix convex sets

**Authors**:
Adam H. Fuller (1), Michael Hartz (2), and Martino Lupini (3)

**Author institution:** (1) Department of Mathematics, Ohio University, Athens,
OH 45701, U.S.A.

(2) Department of Mathematics, Washington University in St Louis, One
Brookings Drive, St. Louis, MO 63130, U.S.A.

(3) Mathematics Department, California Institute of Technology, 1200 E. California Blvd, MC 253-37,
Pasadena, CA 91125, U.S.A.

**Summary: **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.

**DOI: **http://dx.doi.org/10.7900/jot.2017jan28.2165

**Keywords: **operator space, operator system, boundary representation, compact
matrix convex set, matrix-gauged space

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