On unitary representations of the Virasoro algebra

Author's comments: This paper, or some aspects of this paper, have been called into question in

https://mathoverflow.net/q/144419

A good many years have passed since I wrote the paper, so that I cannot easily verify the validity of the criticisms in the above reference. They appeared several years ago and are very likely valid, but anyone who is currently concerned with these matters can decide for himself. I thank Anthony Pulido and Yvan Saint-Aubin for drawing my attention to the paper and to the criticisms.