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

by Orr Shalit

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 .

A unital completely contractive homomorphism is said to have the unique extension property (UEP, for short) if is the restriction for some -homomorphism , and if this -homomorphism is the unique unital completely positive map that extends .

The algebra is said to be hyperrigid in if for every unital representation , the restriction has the UEP. There are other, equivalent definitions, this one is shortest. Hyperrigidity is an interesting notion because it implies a very strong connection between an operator algebra and the C*-algebra it generates, thereby providing a framework for studying certain noncommutative approximation theorems.

Arveson conjectured that hyperrigidity can be detected by testing just irreducible representations. That is, he conjectured that if all irreducible representations of satisfy the condition that their restriction to has the UEP, then is hyperrigid in ; see Conjecture 4.3 in this paper of Arveson, where the notion of hyperrigidity is introduced in somewhat different form from (but proved to be equivalent to) how it was defined above. Respect to Arveson, such a cool guy, who didn’t hide his gut feeling and ceremoniously formulated this problem as an official Conjecture, thereby making things a little more dramatic and fun for all of us.

Boris and Adam’s counterexample is remarkably simple, and you can see their elegant short paper (with great introduction) for more details. In fact, the operator algebra arising in their example is commutative, and the C*-algebra is type I.

I wrote above that this result came as some kind of surprise. Indeed, I don’t recall knowing anyone who was working to disprove this conjecture, while I do know several people who tried hard to prove it (there are also two or three cases that I know of that people were already convinced that they proved it in special cases – this happens with tricky conjectures). There were definitely some people who believed that the hyperrigidty conjecture is true for the case of type I C*-algebras. Perhaps the fact that Arveson conjectured it led many of us to believe that it is true without seriously questioning it. Luckily, Boris and Adam were open minded enough to see outside of this box.

Personally, I was interested in hyperrigidity for a class of special cases – in which the conjecture is not refuted yet! In fact, Matt Kennedy and I once proved that certain operator algebras are hyperrigid if and only if they satisfy another conjecture of Arveson’s: his essential normality conjecture. Boris and Adam’s result shows that, in a sense, operator algebras are less likely to be hyperrigid than what Arveson thought (it also serves as an example of a false conjecture by Arveson). One cannot help but wonder now whether the new development will lead to an example of an operator algebra of the kind that arises in the essential normality conjecture that is not hyperrigid, thereby disproving Arveson’s essential normality conjecture, as well.

It will be very interesting to see what new developments will come up now!