### Topics in Operator Theory, Lecture 6: an overview of noncommutative boundary theory

The purpose of this lecture is to introduce some classical notions in uniform algebras that motivated Arveson’s two seminal papers, “Subalgebras of C*-algebras I + II”, and then to introduce the basic ideas on how to generalize to the noncommutative setting, which were introduced in those papers.

Note: If you are following the notes of this course, please note that the previous lecture has been updated with quite a lot of material.

#### 1. Some notions in the theory of uniform algebras

Definition 1:uniform algebra is a norm closed subalgebra $A \subseteq C(X)$ of the algebra continuous functions on some compact space $X$, which contains the constant functions and separates points.

Sometimes, one uses the terminology function algebra instead. Note that we are talking about algebras of complex valued functions; there is no point in introducing uniform algebras in the case of real valued functions (why?).

The prototypical example of a uniform algebra is the closure of a nice algebra of functions – such as polynomials or rational functions – in the supremum norm of $C(X)$, where $X$ is a subset of $\mathbb{C}^d$ . The Disc Algebra $A(\mathbb{D})$ is defined to be the closure of polynomials in the sup norm on the closed unit disc $\overline{\mathbb{D}} \subset \mathbb{C}$.

We will refer to the example of the Disk Algebra repeatedly in this discussion, to illustrate our ideas in a concrete way. But it is good to keep in mind that every notion that we illustrate in the case of the Disc Algebra, or every question we ask about it, makes sense in general for uniform algebras at large.

The Disc Algebra is also a Banach algebra, so it has an abstract life of its own. It is is clear to us that one would be able to cook up different “representations” of this algebra that will be isometrically isomorphic. Can we represent $A(\mathbb{D})$ as a function algebra on some other topological space $X$?

Consider the unit circle $\mathbb{T} \subset \overline{\mathbb{D}}$, and consider the restriction map $q : C(\overline{\mathbb{D}}) \to C(\mathbb{T})$ given by $q(f) = f \big|_\mathbb{T}$. The kernel of $q$ is quite big, but once it is modded out we have $C(\overline{\mathbb{D}}) / \ker q \cong C(\mathbb{T})$.  Thanks to the maximum modulus principle, the restriction/quotient map $q : f \mapsto f\big|_\mathbb{T}$ is an isometry when restricted to $A(\mathbb{D})$. Said differently, we have a natural isometric inclusion

$A(\mathbb{D}) \hookrightarrow C(\mathbb{T})$,

and we see that a particular Banach algebra can be isometrically isomorphically represented as a uniform algebra of functions living on different topological spaces. The Shilov boundary is introduced as a kind of compass to help us find our way among the different possibilities.

Definition 2: Let $A \subseteq C(X)$ be a uniform algebra. A boundary for $A$ is a closed subset $E \subseteq X$ such that the maximum of every function $f \in A$ is attained on $E$:

$\max_{x \in E}|f(x)| = \max_{x \in X}|f(x)|$.

Proposition 3 (Shilov): The intersection of all boundaries for $A$ is a boundary.

This makes the following definition sound:

Definition 4: Let $A \subseteq C(X)$ be a uniform algebra. The Shilov boundary of $A$ (relative to $X$) is the smallest boundary for $A$. The Shilov boundary is denoted $\partial_S A$.

Thus, the Shilov boundary for $A$ satisfies the following:

1. $\partial_S A \subseteq X$.
2. The map $f \mapsto f\big|_{\partial_S A}$ is isometric on $A$. In particular, $A \hookrightarrow C(\partial_S A)$.
3. $\partial_S A$ is the smallest closed subset of $X$ with this property.

Note that the Shilov boundary of $A$ relative to $\partial_S A$ is simply $\partial_S A$. The Shilov boundary has a deeper property, that it is independent of the original topological space $X$ which we used to represent $A$ as functions on. This deeper property (as well as the previous ones, and in particular the existence) will follow from the work we shall do in the noncommutative setting.

Let us compute the Shilov boundary of $A(\mathbb{D})$ relatie to $\overline{\mathbb{D}}$. On the one hand, we know that $\mathbb{T}$ is a boundary for $A(\mathbb{D})$, so $\partial_S A(\mathbb{D}) \subseteq \mathbb{T}$. On the other hand, it is easy to find, for every $x \in \mathbb{T}$, a function $g_x \in A(\mathbb{D})$ such that $g_x(x) = 1$ and $|g_x(y)|<1$ for all $y \neq x$. Such a point $x \in \overline{\mathbb{D}}$ is said to be a peak point for $A(\mathbb{D})$. It is easy to see that in general, any peak point of a function algebra is contained in its Shilov boundary, hence $\mathbb{T} \subseteq \partial_S A(\mathbb{D})$, and so the circle is indeed the Shilov boundary in this case.

The set of peak points has another characterization which generalizes more readily. In the case of the Disc Algebra, we know that every unital and contractive functional $\phi : A(\mathbb{D}) \to \mathbb{C}$ extends to unital and positive functional $\Phi : C(\overline{\mathbb{D}}) \to \mathbb{C}$. Now, we know from the Riesz representation theorem that there exists a probability measure $\mu$ on $\overline{\mathbb{D}}$ such that $\Phi(f) = \int f d \mu$ for all $f \in C(\overline{\mathbb{D}})$ (and in particular for all $f \in A(\mathbb{D})$). Forgetting for a moment about $C(\overline{\mathbb{D}})$, we find that every unital contractive functional $\phi$ on $A(\mathbb{D})$ has a representing measure,  i.e., a measure $\mu$ on $\overline{\mathbb{D}}$ such that $\phi(f) = \int f d \mu$ for all $f \in A(\mathbb{D})$

Now for any point $x \in \overline{\mathbb{D}}$, the point evaluation $f \mapsto f(x)$ on $A(\mathbb{D})$ is a well defined unital contractive functional, and there is at least one very obvious representing measure: the Dirac measure $\delta_x$. A point in the interior of the disc has many more representing measures; for example,

$f(0) = \frac{1}{2\pi}\int_0^{2 \pi} f(e^{it}) dt$,

for every $f \in A(\mathbb{D})$.

On the other, if $x$ is on the unit circle, then $x$ has only one representing measure – the Dirac measure. It is easy to see that this is so, because this point is a peak point.

We see that in the case of the Disc Algebra, the Shilov boundary is the circle, which is also the set of peak points, which is also precisely the set of points that have aunique representing measure. In general, for a uniform algebra $A \subseteq C(X)$ the set of peak points does not have to equal to the Shilov boundary; however it is a dense subset. Moreover, the set of peak points coincides with the points for which there exists a unique representing measure (equivalently, the set of points that have a unique positive extension to $C(X)$).

Definition 5: Let $A \subseteq C(X)$ be a uniform algebra. The Choquet boundary for $A$ is the set of all $x \in X$ for which the evaluation functional $A \ni f \mapsto f(x)$ has a unique representing measure.

Although the notions of peak points and the Choquet boundary are equivalent in uniform algebras, the definition of Choquet boundary is made in terms of unique representing measures, since this notion generalizes well to the case of unital function spaces

#### 2. Noncommutative versions of the Shilov and Choquet boundaries

In the late 1960s William Arveson initiated a program to study non-selfadjoint operator algebras (here’s something I wrote on Arveson several years ago, which contains some details on his early papers; you might want to read that too because it contains a presentation of the ideas from a slightly different angle). Arveson realized that it would be fruitful to study the relationship between a unital operator algebra $A$ and the C*-algebra $B = C^*(A)$ which it generates. Perhaps his greatest contribution in this context, is the recipe that he came up with for obtaining noncommutative generalizations of the Shilov and Choquet boundaries (which, significantly, he was able to implement in several important applications).

Let us first see how to generalize the notion of a boundary. In the setting of $C(X)$, there is a one-to-one correspondence between closed sets $E \subseteq X$ and closed ideals $I \triangleleft C(X)$, given by

$E \longleftrightarrow I_E$,

where $I_E = \{f \in C(X) : f\big|_E \equiv 0\}$. It is useful to note that the quotient algebra $C(X)/I_E$ is isometrically isomorphic to $C(E)$, and the quotient map can be identified with the restriction map $f \mapsto f\big|_E$.

Now, $E$ is a boundary for a function algebra $A \subseteq C(X)$ precisely when the restriction/quotient map is isometric when restricted to $A$. So if we replace subsets of $X$ with ideals, then we are led to the following definition. In what follows, we will use the word ideal to mean a closed, two-sided ideal.

Definition 6: Let $A \subseteq B = C^*(A)$ be a unital operator space. A ideal $I \triangleleft B$ is said to be a boundary ideal for $A$ if the quotient map $q : B \to B/I$ restricts to a complete isometry $A \hookrightarrow B/I$.

Note that we made the definition for unital operator spaces, to accommodate future applications. Note also that we switched from the quotient map being isometric to it being completely isometric; in the commutative case (that is, when $B$ is commutative) we know that these two notions are equivalent, and so when generalizing to the noncommutative setting one has to make a choice between merely isometric or completely isometric. It turns out that completely isometric is the correct notion to use.

The Shilov boundary was defined as the smallest boundary. In the noncommutative setting we are led to the following definition.

Definition 7: The largest boundary ideal $J \triangleleft B$ is said to be the Shilov boundary ideal (or simply the Shilov ideal) of $A$.

It is not clear that for every unital operator space $A \subseteq B = C^*(A)$ there exists a Shilov boundary ideal; we will prove later that it does indeed exist. After Arveson introduced the Shilov boundary he proved that it exists in certain cases, but its existence in general remained an open problem for about a decade until Hamana proved it using different ideas (in this post that I linked to earlier the history is explained briefly).

I outlined Hamana’s solution in class because I think it is interesting, and makes use of injective envelopes as well as the Choi-Effros product, which are really interesting ideas. See Paulsen’s book for more on that approach. We will continue to follow Arveson’s program.

Recall that in the commutative case, we mentioned (without any proof) that the Shilov boundary of a uniform algebra $A$ does not depend on the particular choice of inclusion $A \subseteq C(X)$. A similar phenomenon happens in the noncommutative setting.

Let $A \subseteq B = C^*(A)$ be a unital operator space, and let $J \triangleleft B$ be its Shilov ideal. Then it is clear that $B/J$ is smallest quotient of $B$ into which $A$ is mapped completely isometrically. Now suppose that we have a surjective unital complete isometry $\phi : A \to A' \subseteq B' = C^*(A')$, and consider the Shilov ideal $J'$ for $A'$ in $B'$. Then we can also form the quotient $B'/J'$, which is the smallest quotient of $B'$ that contains $A$ completely isometrically. It is not obvious, but we will later prove that $B/J \cong B'/J'$. That is, the quotient of $C^*(A)$ by the Shilov ideal is an invariant of $A$. It is perhaps re-emphasizing that in general $B = C^*(A) \ncong C^*(A') = B'$.

Definition 8: Let $A \subseteq B = C^*(A)$ be a unital operator space and let $J$ be the Shilov ideal. The C*-envelope of $A$ is the quotient $C^*_e(A) := B/J$.

Proving that the quotient $B/J$ is invariant to the choice of inclusion $A \hookrightarrow B$ is closely related to showing that it enjoys the following universal property.

Theorem 9: (Universal property of the C*-envelope). Let $A \subseteq B = C^*(A)$ be a unital operator space, and let $J$ be the Shilov ideal for $A$ in $B$. Write $C_e^*(A) = B/J$, and let $q$ be the quotient map. Then for every unital completely isometric map $\phi : A \to B' = C^*(\phi(A))$, there exists a unique surjective *-homomorphism $\pi : B' \to C_e^*(A)$, such that $\pi \circ \phi = q$.

(A commutative diagram should be sketched at this point.)

Finally, we present the noncommutative analogues of Choquet boundary points. Recall that a point $x \in X$ is said to be in the Choquet boundary of a uniform algebra $A \subseteq C(X)$ if the point evaluation $\delta_x\big|_A$ had a unique representing measure; equivalently (since positive unital maps on $C(X)$ are in one-to-one correspondence with probability measures) if the point evaluation $\delta_x \big|_A$ has a unique unital positive extension to $C(X) = C^*(A)$. Now what are the point evaluations $\delta_x$? They are precisely the irreducible representations of $C(X)$.

This leads to the following definition.

Definition 10: Let $A \subseteq B = C^*(A)$ be a unital operator space. A boundary representation for $A$ is an irreducible representation $\pi : B \to B(H)$ such that the only UCP map $B \to B(H)$ that is an extension of $\pi\big|_A : A \to B(H)$ is $\pi$ itself.

Arveson showed that if a unital operator space $A$ has sufficiently many boundary representations, then the intersection of the kernels of all boundary representations is the Shilov ideal, and that the image of $B$ under the direct product of all boundary representations is the C*-envelope. We will explain this later. For the time being, I will just mention that the existence of boundary representations in general was an open problem for forty five years, until it was solved by Davidson and Kennedy not long ago.

Let us see how the above strategy for finding the Shilov boundary works in the simple case of the Disc Algebra. Indeed, we knew that the Shilov boundary was contained in the circle $\mathbb{T}$ thanks to the maximum modulus principle. To see that the circle is the Shilov boundary, we used the fact that every point in $\mathbb{T}$ is a peak point – that is – a point in the Choquet boundary. This is precisely the step of “showing that there are sufficiently many boundary representations”.

In the next lecture we will begin to give proofs for the existence of boundary representations, from which the existence of the Shilov ideal and C*-envelope follows. Later we will see non-commutative examples and applications.