Arveson’s hyperrigidity conjecture refuted by Bilich and Dor-On

Boom! This morning Boris Bilich and Adam Dor-On published a short preprint on the arXiv “Arveson’s hyperrigidity conjecture is false” in which they provide a counter example that refutes Arveson’s hyperrigidity conjecture. This is a fantastic achievement! It is one of the most interesting things that happened in my field lately and also somewhat of a surprise, a paper that is sure to make a significant impact on the subject.

(I should say that Adam was kind enough to let me read the manuscript a week ago, so that I had time already to check the details and as far as I can tell it looks correct.)

Let us recall quickly what the conjecture is (for more background see the series of posts that I wrote for the topics course I gave several years ago).

Let be a unital operator algebra generating a C*-algebra .

Topics in Operator Theory, Lecture 10: hyperrigidity

In this lecture we discuss the notion of hyperrigidity, which was introduced by Arveson in his paper “The noncommutative Choquet boundary II: Hyperrigidity“, shortly after he proved the existence of boundary representations (and hence the C*-envelope) for separable operator systems. Most of the results and the examples that we will discuss in this lecture come from that paper, and we will certainly not be able to cover everything in that paper. In the last section of this post I will put some links concerning a result of Kennedy and myself which connects hyperrigidity to the Arveson’s essential normality conjecture.

Topics in Operator Theory, Lecture 7: boundary representations

In this lecture we will present a proof that boundary representations exist in abundance, following Davidson and Kennedy’s breakthrough paper. Davidson and Kennedy’s paper was in the spirit of Arveson’s paper from 1969, and followed Arveson’s solution in the separable case from 2007. (BTW, I wrote about Davidson and Kennedy’s solution in a an old blog post). 

1. The unique extension property and maximal representations

Recall the definition of a boundary representation. 

Our setting will be of an operator system contained in a C*-algebra . Recall that earlier we discussed the situation of a unital operator algebra , and later we extended our attention to unital operator spaces. In this post we will consider only operator systems, but there will be no loss of generality (because every unital completely contractive map extends to a unique unital completely positive map , and vice versa).