On Subalgebras of C*-algebras

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. All C*-algebras are assumed to contain an identity (written as e)y L(H) denotes the algebra of all bounded operators on a Hilbert space H, and C*(S) stands for the C*-algebra generated by 5 and the identity where 5 is either an operator or a subset of a C*-algebra. An operator is irreducible if it commutes with no nontrivial projections.