The 𝐶*-algebra generated by an isometry. II
L. A. Coburn · Transactions of the American Mathematical Society · 1969
I. Introduction.In [2] it was shown that the C*-algebra generated by any nonunitary isometry is *-isomorphic to the C*-algebra ¿rf(S) generated by the unilateral shift of multiplicity one, S. Further, the ideal theory of stf(S) was determined.The main point was that .s/(S)contains the full algebra of compact operators Jf" and si(S)/3f is *-isomorphic to C(T), the algebra of all complexvalued continuous functions on the unit circle, T.Here, I examine the structure of ¿¿(S) and certain related C*-algebras in greater detail.In particular, the irreducible *-subalgebras of ¿¿(S) are characterized and reasonable necessary and sufficient conditions are given for an operator in s/(S) to genérateos').Finally, I provide an example of a C*-algebra which is irreducible and has the same ideal structure as stf(S), but which is not *-isomorphic to stf(S).II.Preliminary results.Henceforth all Hubert spaces are over the complex numbers.If ^f is a Hubert space and F is a bounded operator on Jt then the smallest C*-algebra of operators containing F and 1 is denoted by s/(B).The full algebra of bounded operators on JP with the operator norm topology is called âS(J^) and the subalgebra of all compact operators on Jf is called Jfpf ) (or just Jf ).We let n denote the usual quotient map from S&Ltf) onto 3S(^c°)/c^.In this paper, all ideals are closed and two-sided.We write o(B) for the spectrum of F in @LW).In portions of this paper, we will be concerned with the algebra C(X) of all complex-valued continuous functions on a compact Hausdorff space X with supremum norm.More particularly, let T be the unit circle and let A be the uniformly closed subalgebra of C(T) consisting of those in C(T) which are uniform limits of polynomials in the complex variable z.Further, let p be normalized Haar measure on F and let F2 denote the associated Hubert space of square-integrable functions on T. As usual, we let H2 denote the F2-closure of A. In the following sections we will be interested in the C*-algebras generated by Toeplitz operators [1] on H2 associated with functions in C(T) by the relation T¿f=P( f), where F is the orthogonal projection from F2 onto H2.We will also be concerned with the Laurent operators [1] on F2 associated with in C(T) by the relation Mt,f= f.It should be pointed out that in general [1], [3] \\TA = WA -M^)« = INA/,)||-= u\\.