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 ∗: K 0(C ∗(X,R)) → K 0(CGJ(X,R)) is an order isomorphism and there is a short exact sequence 0 → K 1(C ∗(R)) → i i K 1C ∗(R)) → j i K 1(CGJ(X,R)) → 0 where i is the canonical imbedding of C ∗( R ) into C ∗(X R ). 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 +.