On the K-theory of some C^*-algebras of Toeplitz and singular integral operators

We study the C^*-algebras GJ(X,R) are I(X, R) of singular integral operators and Toeplitz operators (respectively) associated with a strictly ergodic flow (X,R). We show that the commutator ideals of these algebras, CGJ(X,R) and CL(X,R), are simple and are closely related to the transformation group C^*-algebra, C^*(X,R). We calculate the K-theory of GJ(X,R), L(X,R) and their commutator ideals. The main results of this calculation, Corollary 3.8.4 and Theorem 4.1.1, assert that C^*(X,R) is contained in CGJ(X,R) and if j denotes the inclusion map, then j_*: K₀(C^*(X,R)) → K₀(CGJ(X,R)) is an order isomorphism and there is a short exact sequence 0 → K₁(C^*(R)) → i_i K₁C^*(R)) → j_i K₁(CGJ(X,R)) → 0 where i is the canonical imbedding of C^*(R) into C^*(XR). We show also that, up to a change of scale, there is a unique trace on each of the commutator ideals. The key ingredient of our analysis is Theorem 3.1.1 which asserts a bijective correspondence between Silov representations of the algebra of analytic functions on the flow and C^*-representations of GJ(X,R) and L(X,R). This simultaneously generalizes Coburn's theorem on the uniqueness of the C^*-algebra generated by an isometry and Douglas's theorem on the uniqueness of the C^*-algebra generated by an isometric representation of a dense subsemigroup of R₊.