William Arveson

William B. Arveson was born in 1934 and died last year on November 15, 2011. He was my mathematical hero; his written mathematics has influenced me more than anybody else’s. Of course, he has been much more than just my hero, his work has had deep and wide influence on the entire operator theory and operator algebras communities. Let me quickly give an example that everyone can appreciate: Arveson proved what may be considered as the “Hahn-Banach Theorem” appropriate for operator algebras. He did much more than that, and I will expand below on some of his early contributions, but I want to say something before that on what he was to me.

When I was a PhD student I worked in noncommutative dynamics. Briefly, this is the study of actions of (one-parameter) semigroups of *-endomorphisms on von Neumann algebras (in short E-semigroups). The definitive book on this subject is Arveson’s monograph “Noncommutative Dynamics and E-Semigroups”. So, naturally, I would carry this book around with me, and I would read it forwards and backwards. The wonderful thing about this book was that it made me feel as if all my dreams have come true! I mean my dreams about mathematics: as a graduate student you dream of working on something grand, something important, something beautiful, something elegant, brilliant and deep. You want your problem to be a focal point where different ideas, different fields, different techniques, in short, all things, meet.

When reading Arveson there was no doubt in my heart that, e.g., the problem classifying E-semigroups of type I was a grand problem. And I was blown away by the fact that the solution was so beautiful. He introduced product systems with such elegance and completeness that one would think that this subject has been studied for the last 50 years. These product systems were measurable bundles of operator spaces – which turn out to be Hilbert spaces! – that have a group like structure with respect to tensor multiplication. And they turn out to be complete invariants of E-semigroups on $B(H)$. The theory set down used ideas and techniques from Hilbert space theory, operator space theory, C*-algebras, group representation theory, measure theory, functional equations, and many new ideas – what more could you ask for? Well, you could ask that the new theory also contribute to the solution of the original problem.

It turned out that the introduction of product systems immensely advanced the understanding of E-semigroups, and in particular it led to the full classification of type I ones.

So Arveson became my hero because he has made my dreams come true. And more than once: when reading another book by him, or one of his great papers, I always had a very strong feeling: this is what I want to do. And when I felt that I gave a certain problem all I thought I had in me, and decided to move on to a new problem, it happened that he was waiting for me there too.

I wish to bring here below a little piece that I wrote after he passed away, which explains from my point of view what was one of his greatest ideas.

For a (by far) more authoritative and complete review of Arveson’s contributions, see the two recent surveys by Davidson (link) and Izumi (link).

Subalgebras of C*-algebras I and II

Arveson was one of the first to realize the importance of non-selfadjoint operator algebras as well as the role of operator spaces, and he developed a theory which had both immediate applications (to problems operator theory, including “single operator theory”), as well as long term, far reaching implications. In his paper [“Analyticity in Operator Algebras”, Amer. J. Math. 89, 1967] he wrote:

In the study of families of operators on Hilbert space, the self-adjoint algebras have occupied a preeminent position.

By “self-adjoint” he means, of course, C*-algebras and von Neumann algebras. He continues:

Nevertheless, many problems in operator theory lead obstinately toward questions about algebras that are not necessarily self-adjoint. Indeed, the general theory of a single operator on a finite-dimensional space rests on an analysis of the polynomials in that operator, and has nothing at all to do with the *-operation.

Arveson’s great insight was that if C*-algebras are a non-commutative analogue of algebras of continuous functions, then the theory of non-selfadjoint algebras should be considered as a non-commutative version of the theory of of function algebras. His vision was expounded in two ground breaking papers ([“Subalgebras of C*-algebras”, Acta Math. 123, 1969], and [“Subalgebras of C*-algebras II”, Acta Math. 128, 1972]), where he laid the foundations to the theory of (not necessarily self-adjoint) operator algebras and of operator spaces.

It was probably in these papers where the notions of complete positivity, complete boundedness and complete isometry were first shown to play such a fundamental role in operator theory.

Digression: You may wonder: “what is complete this and complete that“. Let me briefly explain what is a complete isometry, and the rest will be clear. An operator algebra $A$ is just a subalgebra of $B(H)$, and it inherits the operator norm from $B(H)$. A linear map $\phi : A \rightarrow B$ (where $B$ is another operator algebra) is said to be an isometry if $\|\phi(a)\| = \|a\|$ for all $a \in A$. This is a very familiar notion, the existence of such a $\phi$ means that $A$ and $B$ have the same Banach space structure. It turns out that the Banach space structure of an operator algebra is far from determining its other properties. Arveson somehow understood that the complete version of isometry is what one is required to look at. Here is what this means.

One can form the algebra $M_n(A)$ of $n \times n$ matrices over $A$. This is a subalgebra of

$M_n(B(H)) = B(\underbrace{H \oplus H \oplus \cdots \oplus H}_{n \textrm{ times}})$,

so it is again an operator algebra. If $\phi : A \rightarrow B$ is a linear map, we define $\phi_n : M_n(A) \rightarrow M_n(B)$ by letting $\phi$ act entrywise, that is, if  $[a_{ij}] \in M_n(A)$ then $\phi_n ([a_{ij}])$ is the matrix $[\phi(a_{ij})]$. Finally, the map $\phi$ is said to be a complete isometry if $\phi_n$ is an isometry for all $n$. The same notions can be defined when $A$ and $B$ are operator spaces, that is, simply subpaces of $B(H)$End of digression.

The main issue confronted in the two “Subalgebras” papers mentioned above is this: given an algebra $A$ of operators on a Hilbert space, one can form the C*-algebra $C^*(A)$ which it generates. However, if $\tilde{A}$ is a completely isometrically isomorphic (i.e., indistinguishable) copy of $A$, it might be the case that $C^*(\tilde{A})$ is different than $C^*(A)$. What is the precise relationship between an operator algebra and the C*-algebra that it generates? What is the “correct” C*-algebra in which one should study $A$? How can one study operator algebras abstractly?

Arveson’s approach was the following: if $I$ is an ideal in $C^*(A)$ such that the quotient map $C^*(A) \rightarrow C^*(A)/I$ is completely isometric when restricted to $A$, then the quotient $C^*(A)/I$ is another C*-algebra that contains $A$ completely isometrically isomorphically, and this C*-algebra is better than $C^*(A)$ because it is smaller. Such an ideal $I$ is said to be a boundary ideal. Now suppose that there exists an ideal $J$ which is the largest boundary ideal — $J$ is said to be the Shilov boundary ideal and in the commutative case (when $A$ is a function algebra) it is precisely the ideal of functions which vanish on the Shilov boundary of $A$. Then $C^*(A)/J$ is the smallest such quotient that can be formed. Arveson proved that $C^*(A)/J$ is a canonical invariant of $A$ and that it is in fact the smallest C*-algebra that contains $A$ completely isometrically isomorphically. This canonical C*-algebra later became to be known as the C*-envelope of $A$.

To be precise, it was shown that if $J$ is the Shilov boundary ideal for $A$ in $C^*(A)$, and if $\tilde{J}$ is the Shilov boundary ideal for $\tilde{A}$ in $C^*(\tilde{A})$, and if $\phi : A \rightarrow \tilde{A}$ is a complete isometry, then $\phi$ extends to a *-isomorphism between $C^*(A)/I$ and $C^*(\tilde{A})/J$.

As a beautiful and very simple example of the kind of result that flowed from this theory let me bring the following:

Theorem: Let $A$ and $B$ be two irreducible compact operators on $H$. Then $A$ and $B$ are unitarily equivalent if and only if the two dimensional operator spaces $\textrm{span}\{ I, A\}$ and $\textrm{span}\{ I, B\}$ are completely isometric via a unital map that takes $A$ to $B$

The urgent question now becomes: does the Shilov boundary ideal exist?.

To prove the existence of the Shilov boundary ideal, Arveson introduced boundary representations. Although boundary representations were central to his investigations, and although he proved that they exist in some special cases, the question of whether boundary representations exist in general was left open (consequently, the existence of the C*-envelope was also left open). The existence of boundary representations in general (for separable operator spaces) was established by Arveson four decades later (!) in [“The noncommutative Choquet boundary”, J. Amer. Math. Soc. 21, 2008] following important works of Dritschel and McCullough and of Muhly and Solel (the existence of the C*-envelope was established by Hamana in 1979 by a different approach).

One of the immediate applications of this theory that Arveson presented was a dilation theorem that unified and clarified the various bits and pieces of dilation theory (initiated by Sz.-Nagy) that were present at the time:

Arveson’s Dilation Theorem (1972). Let $T$ be a $k$-tuple of commuting operators and let $X \subset \mathbb{C}^k$ be a compact set. Then $T$ has a normal dilation $N$ with  $\textrm{sp}(N) \subseteq \partial X$ if and only if $X$ is a complete spectral set for $T$

I do not want to go into the definitions, the reader is referred to this exposition by Arveson for details.

Many of Arveson’s results in the two “Subalgebras” papers mentioned above relied on a Hahn–Banach type extension theorem for completely positive maps which he proved in the first section of the first paper. Let me state a version for completely contractive maps which is perhaps clearer to the newcomer:

Arveson’s Extension Theorem (1969). Let $B$ be a C*-algebra, let $S \subseteq B$ be an linear subspace containing the identity and let $\phi : S \rightarrow B(H)$ be a unital, completely contractive map. Then $\phi$ has an extension to a unital, completely contractive and completely positive map $\Phi : B \rightarrow B(H)$.

Anyone with some experience in functional analysis can appreciate the utility of this result. Indeed, this theorem is used by operator algebraists on a daily basis, to this day.

The ideas and results of these two papers of Arveson which we mentioned have had a profound impact on operator algebras and operator theory. Without them, the study of abstract operator algebras, the solution of the Sz.-Nagy–Halmos problem (is every polynomially bounded operator similar to a contraction?) and the numerous applications of operator space techniques to operator algebras, would have all been unthinkable.