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

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