Topics in Operator Theory, Lecture 10: hyperrigidity
by Orr Shalit
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.
1. Background and definition of hyperrigidity
Recall that the in Bernstein’s proof of the Weierstrass polynomial approximation theorem, one associates with every continuous function a Bernstein polynomial
The operators are clearly linear, positive and unital. It can be shown that and . Therefore
(*) uniformly for every .
To prove Weierstrass’ approximation theorem, one needs to show that uniformly for all . One can give a non-probabilistic proof of this fact, using just that is a sequence of positive and unital maps satisfying (*) (see Chapter 10.3 in Davidson and Donsig’s book “Real Analysis and Applications“).
In fact, Korovkin proved that given a sequence of positive linear and unital operators on such that uniformly for , then uniformly for all . This implies that the generating set has a very “rigid” hold on the (closed) unital algebra that it generates, .
The above discussion should serve as background and motivation for the following definition.
Definition 1: Let be a generating subset of a C*-algebra . We say that is hyperrigid in if for every faithful nondegenerate representation and every sequence of UCP maps,
In the above definition I forced myself to overcome my pedantic self and use Arveson’s notation for different faithful (and nondegenerate) representations of . What we really mean is that is a certain fixed C*-algebra (either abstract or represented faithfully on some space, that’s not the point here) and that for every faithful and nondegenerate representation and sequence of UCP maps , the condition
(*) for all
implies the consequence
(**) for all .
Conventions: Unless emphasized otherwise, our C*-algebras will be unital and all the representations will be nondegenerate (hence unital). In his paper Arveson discussed the nonunital setting to some extent, and time permitting I will touch upon this briefly in class (for a additional discussion of hyperrigidity in the context of nonunital algebras see this paper by Guy Salomon). We also note that a generating set is hyperrigid if and only if the operator system that it generates is. Although it is more precise to say that a set of generators (or the operator algebra that it generates ) is “hyperrigid in B“, we will sometimes just say that it is “hyperrigid“.
In Arveson’s paper it is always assumed that the generating set is finite or countably infinite, and that all Hilbert spaces are separable. I think the reason is that at the time, the existence of boundary representations was known only for separable operator systems. I will not make any countability assumptions here, I think they are not needed now that we know that boundary representations always exist (I will be grateful if an alert reader finds a gap in these notes). On the other hand, we may always stay within the realm of separable Hilbert spaces if our C*-algebras are separable – this is left as an exercise.
2. Characterization of hyperrigidity
The definition of hyperrigidity (Definition 1) has an approximation theoretic flavor. The following theorem connects hyperrigidity to the unique extension property, which we studied in previous lectures.
Theorem 2: Let be an operator system generating a C*-algebra . The following conditions are equivalent.
- is hyperrigid in .
- For every nondegenerate representation and every sequence , if for every , then for all .
- For every nondegenerate representation , the UCP map has the unique extension property.
- For every unital C*-algebra , every unital -homomorphism and every UCP map
for all for all .
Proof: 1 2: Condition 2 is very similar in appearance to the definition of hyperrigidity given by “(*) implies (**)” in the previous section. Indeed, it follows readily by assuming that is represented faithfully as , and then considering the faithful representation
If for all , then for all . By the definition of hyperrigidity (summoned only after invoking Arveson’s extension theorem, so that we will be discussing UCP maps on ), we find that
for all ,
which implies that . Note how we really needed the assumption of hyperrigidity to address every faithful nondegenerate representation of .
2 3: This follows by taking for all .
3 4: Let and be as in Condition 4. Represent faithfully (and non-degenerately) on a Hilbert space as , and extend to a UCP map . Then can be considered as representation of on . If does not fix , then then is an extension of which is different from , in contradiction to Condition 3.
4 1: Suppose that faithfully and non-degenerately, and let be a series of UCP maps. Assume, as in the definition of hyperrigidity, that
for all .
Assuming that Condition 4 holds, we wish to prove that
for all .
Construct the C*-algebra of bounded sequences with values in with the obvious structure, and consider the UCP map given by
If is the ideal of all sequence tending to zero, then . Write and so induces a UCP map . By defining a *-homomorphism by
we are precisely in the situation of Condition 4. It follows that fixes , and this is the same as for all .
That concludes the proof.
The following is a simple corollary that is worth recording:
Corollary 3: Let be a hyperrigid operator system in . Then is the C*-envelope of .
Proof: Since for every representation , the restriction has the UEP, the Shilov ideal – which we know is the intersection of all the kernels of boundary representations – is trivial.
Corollary 4: Let be a hyperrigid operator system in , let be an ideal, and let be the quotient map. Then is hyperrigid in .
Proof: Condition 2 in the theorem is preserved under taking quotients.
It is important to note that the converse to Corollary 3 does not hold: an operator system with trivial Shilov ideal in the C*-algebra it generates need not be hyperrigid. Moreover, in the context of Corollary 4, we note that having trivial Shilov ideal is not preserved by quotients. Here is an example (due to Davidson; personal communication) that illustrates both of these statements.
Example 1: Let be an orthonormal basis for a Hilbert space , and let be a sequence of complex numbers such that and is a dense sequence in . Define and let be the unital (norm closed) operator algebra generated by . (We are suddenly discussing an operator algebra and not an operator system, but we can always pass from to and back).
One can check that is an irreducible operator algebra containing the compacts. The Calkin map is not isometric on , so by the boundary theorem, the Shilov ideal is trivial and .
On the other hand, is a normal operator with spectrum . It follows that and is the disc algebra. Thus, after passing to the quotient, the Shilov boundary ideal is not trivial.
This example shows that – unlike hyperrigidity – trivial Shilov ideal is a property that does not pass to quotients. It also shows that a trivial Shilov boundary does not imply hyperrigidity (why?).
By Theorem 2, if is hyperrigid, then, in particular, every irreducible representation is a boundary representation. Arveson conjectured that the converse also holds true.
Arveson’s Hyperrigidity Conjecture: is hyperrigid in if and only if every irreducible representation of is a boundary representation for .
Example 2: Suppose that is a selfadjoint operator with at least three points in the spectrum, and let be the (unital, as always) C*-algebra generated by . We will show that:
- is hyperrigid in .
- is not hyperrigid in .
(The assumption on the spectrum is no biggy: if the spectrum has two or less points, then is the C*-algebra generated by , and of course it is hyperrigid.)
Note that if we let be the multiplication operator by the identity function on , and if we identify as a C*-subalgebra of , then we obtain a strengthened version of Korovkin’s theorem that we mentioned in the first section. Indeed, in Korovkin’s theorem, the conclusion for all follows from this convergence only of elements only for sequences of (completely) positive maps . Hyperrigidity shows that follows from the same assumption, but now for any sequence of UCP maps .
Let us first show that is not hyperrigid. Suppose that , , and is a point in the spectrum that lies strictly between and . Then point evaluation at is representation of . However, does not have the unique extension property, since is a convex combination of and .
Now let . To show that is hyperrigid, we need to show that for every nondegenerate , the UCP map has the UEP.
To this end, suppose that is a UCP map that extends . This just means that and . We have to show that is multiplicative – this will imply that .
Let be a Stinespring representation of , where is a *-representation and is a space containing . We write , and compute:
This means that , or put differently, that is invariant under . It follows that is a reducing subspace for , and so is multiplicative, as required.
It is worth noting that under the assumption above on , Arveson proved that if is hyperrigid (where is continuous on the spectrum of ) then must be either strictly convex or strictly concave. He also proved that the converse holds, assuming that the Hyperrigidity Conjecture is true.
Example 3: Let be isometries generating a unital C*-algebra . The generating set
is hyperrigid in .
In particular, if are Cuntz isometries (i.e., isometries with pairwise orthogonal ranges, such that ), then it is well known that they generate the Cuntz algebra . Then we have that the operator system generated by is hyperrigid in .
Let’s prove the claimed hyperrigidity in the special case of Cuntz isometries.; the general case is based on the same idea but slightly more tedious. Write for the operator system generated by . Let be a (nondegenerate, as always) representation. Let be a UCP map such that , and let be a representation that is a dilation of . Define for . Then for every ,
is a *-representation, so
Comparing the (1,1)-entry, we find . Since , we must have that , so that . On the other hand
As before, this implies that , which implies that for all .
We conclude that all reduce , and (since these operators generate ) it follows that is a reducing subspace for . As a consequence, is its own minimal Stinespring dilation, so it is multiplicative. Hence , as required.
4. The hyperrigidity conjecture for C*-algebras with countable spectrum
Arveson established his hyperrigidity conjecture for the special case of countable spectrum. Recall that the spectrum of a C*-algebra is the set of all unitary equivalence classes of irreducible representations (if you are the kind of person who -like me – always worries about such things, let me tell you that it is OK to use the word “set” for . Hint: we are speaking about irreducible representations, and the dimension of Hilbert space on which an irreducible representation of can live is restricted by the cardinality of ). A C*-algebra is said to have countable spectrum if is countable, in other words, if only has a countable number of pairwise unitary inequivalent irreducible representations.
Theorem 5: Let be an operator system such that the generated C*-algebra has countable spectrum. If every is a boundary representation, then is hyperrigid in .
Proof: Assume that . To prove that is hyperrigid, we need to prove that for every representation , the UCP map has the UEP. But every representation of is the direct sum of irreducible representations (here we are using the fact that is countable, and some non-trivial facts from the representation theory of (type I) C*-algebras). Since the direct sum of UCP maps with the UEP has the UEP (yet another fact that requires proof, but not as deep as the previous fact we used), it follows that has the UEP.
5. Connection to Arveson’s Essential Normality Conjecture
In this section I wish to discuss the connection between the notion of hyperrigidity and another conjecture of Arveson – the essential normality conjecture.
The problem of essential normality can take place in many spaces, but I like to view in the Drury-Arveson space . See this old post (mostly Section 1) and this old post (mostly Sections 2 and 3) where I already discussed this space. In this old post (Section 1) I discuss the essential normality problem.
In class I plan to discuss the paper “Essential normality, essential norms, and hyperrigidity“ by Kennedy and myself (here is a link to an arxiv version; here is a link to the corrigendum). I wrote about this problem and about this paper a few times before (for example, when announcing the preprint), so I with the above pointers and links in place, we end these notes!
Thanks for listening! You all get a grade 100 in the course!