On subalgebras of $C^*$-algebras

Abstract

In this note we announce some new methods and results in the theory of nonnormal Hilbert space operators and nonself ad joint operator algebras. A main difficulty in the subject has been the apparent absence of relations between, say, a nonself ad joint algebra of operators and its generated C*-algebra. For example, given full information about the norm-closed algebra P(T) generated by all polynomials in a given (nonnormal) operator T, what can one say about the C*-algebra C*(T) generated by T and the identity? While one cannot expect much of an answer in general, we will describe here a class of operators and operator algebras for which these relations are as simple as one could hope for.