### Forty five years later, a major open problem in operator algebras is solved

A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras, showing that every operator system has sufficiently many boundary representations.

In 1969, William Arveson published the seminal paper, [“Subalgebras of C*-algebras”, Acta Math. 123, 1969], which is one of the cornerstones, (if not the cornerstone) of the theory of operator spaces and nonself-adjoint operator algebras. In that paper, among other things, Arveson introduced and put to good use the notion of a boundary representation. I wrote on “Subalgebras of C*-algebras” in a previous post dedicated to Arveson, and for some background material the reader is invited to look into that old post. I did not, however, write much about boundary representations (because I was emphasizing his contributions rather what he has left open). Below I wish to explain what are boundary representations, what does it mean that there are sufficiently many of these, and where Davidson and Kennedy’s new results fits in the chain of results leading to the solution of the problem. The paper itself is accessible to anyone who understands the problem, and the main ideas are clearly presented in its introduction.

Let $S$ be an operator system, that is, a self adjoint subspace of a unital C*-algebra $B$ such that the unit of $B$ is contained in $S$. Assume further that $B = C^*(S)$. If $\phi: S \rightarrow B(H)$ is a unital and completely positive map (henceforth UCP) then it extends to a UCP map $\tilde{\phi} : B \rightarrow B(H)$. Indeed, this is the content of a Hahn-Banach type extension theorem — called Arveson’s extensions theorem — which is proved in the “Subalgebras” paper. In general, as in Hahn-Banach type theorems, the extension is highly non-unique. If $\pi : B \rightarrow B(H)$ is a $*$-representation, then obviously $\pi\big|_S$ is a  UCP map, and $\pi$ itself is one of the extensions of $\pi\big|_S$.

Definition: A boundary representation for $S \subseteq B = C^*(S)$ is an irreducible representation $\pi : B \rightarrow B(H)$ such that $\pi$ itself is the unique UCP map from $B$ to $B(H)$ which extends $\pi\big|_S$.

Arveson showed that boundary representations are invariants of $S$, regardless of the particular C*-algebra $B$ into which it is represented. Note that two completely isometric operator systems may generate non-isomorphic C*-algebras; the boundary representations, however, will be the same for these two operator systems. Being the first known invariants of operator systems that do not depend on the C*-algebra into which an operator system is embedded was the first clue that operator systems, operator spaces and operator algebras can be studied abstractly (the abstract theory of operator spaces indeed does exist, and was developed in the 80s by  Blecher, Effros, Hamana, Paulsen, Ruan (among others), without using boundary representations).

Let us say that $S \subset B = C^*(S)$ has sufficiently many boundary representations if for every $n \times n$ operator matrix $X = [x_{ij}]\in M_n(S)$ one has

$\|X \| = \sup \|[\pi (x_{ij})]\|$

where the sup is taken over all boundary representations. In the “Subalgebras” paper Arveson showed that the existence of sufficiently many boundary representations implies the existence of the C*-envelope of $S$, which is the unique smallest C*-algebra into which $S$ can be embedded unitally and  completely isometrically (see here for a few more words about the C*-envelope). For a wide class of examples Arveson proved that there are sufficiently many boundary representations, and in fact he characterized for some C*-algebras when the identity representation is a boundary representation. However, for general operator systems the question of whether or not there exist sufficiently many boundary representations was left open. As I mentioned above, it has been left open until a few days ago, and its new solution is a truly exciting development.

The existence of the C*-envelope was established in 1979 by Hamana by methods quite different from what Arveson proposed. A new proof of the existence of the C*-envelope (which also gives some method for computation of the C*-envelope) which is closer to the original spirit of Arveson was given by Dritschel and McCullough in 2005. What Dritschel and McCullough showed was that there are sufficiently many representations which satisfy all the requirements of a boundary representation except that they might not irreducible. Following Dritschel and McCullough’s breakthrough, and using some of their ideas, Arveson returned to the problem and proved that every separable operator system has sufficiently many boundary representations [“The noncommutative Choquet boundary”, J. Amer. Math. Soc. 21, 2008]. The methods Arveson used in his proof were based on some intricate direct integral analysis, and did not work for non-separable operator systems. It is remarkable that he was able to solve (in the separable case) his own problem forty years after he raised it.

Needless to say, many operator spaces of interest, for example dual algebras, are not separable. However, since the problem resisted solution for so long, people (even Arveson) were quite excited about the solution of separable case. Nevertheless, the solution of the problem in full was still sought after.

Davidson and Kennedy have found a proof that sufficiently many boundary representations exist which does not make use of direct integral techniques. Rather, it emphasizes the role of pure UCP maps, and again uses some of Dritschel and McCullough’s ideas. Even for the separable case it is a simplification, and indeed for the separable case it is somewhat more constructive.

Congratulations!