Introduction

This book contains somewhat expanded versions of ten lectures delivered at

Texas Tech University during the summer of 1983. The Operator algebras of the

title are nonselfadjoint algebras of Operators on Hubert space.

This subject is new, and has shown remarkable growth in the last twenty years.

Indeed, when I was finishing my graduate studies in 1964 I knew of only three

papers that addressed themselves seriously to nonselfadjoint Operator algebras

([42, 61], and a paper of John Schue, The structure of hyperreducible triangulär

algebras, Proc. Amer. Math. Soc. 15 (1964), 766-772). Á few of us believed in the

sixties that this was a promising way to approach the theory of Single Operators,

but we certainly did not see how and were not even sure if that would be

accomplished. What actually happened was that the subject developed in several

directions, and was pursued entirely on its own merits. When the applications to

Single Operator theory did come, they came unexpectedly and in surprising ways

(see Lecture 10). These results are deep and, looking back on it now, I must say

that I cannot conceive of any way that the methodology of single Operator theory

could have produced them.

The subject matter for these lectures has been selected using subjective criteria.

Some of it has historical interest, some of it seems timely or important (at least to

me), some of it seems to suggest new directions, and some of it is just fun to

communicate. I have had to omit several of my favorite topics on which there has

been significant progress, including noncommutative Silov boundaries, abstract

dilation theory, and algebras defined by group actions [7, 8, 73, 74, 50,12, 49].

Some of the material is expository and is presented largely without proofs

(Lectures 1, 2, 5, 6). Some is expository but with complete proofs or complete

ideas of proofs (Lectures 7, 8, 10). Lectures 7 and 8 expand on some notes

distributed to the participants in a seminar at Berkeley during the spring quarter

1983; Lecture 9 is based on a lecture delivered in Busteni, Romania, in September

1983. Lecture 4 contains new material relating to the Feynman-Kac formula. I

have taken some care to present complete proofs, and to develop the background

material from classical mechanics and quantum mechanics in Lecture 3.

Finally, the references are by no means complete. I have referenced only those

items I know about that relate to the subject matter of these lectures. The reader

Vll