Tensor products of subnormal operators

We shall use a C*-algebra approach to study operators of the form S x N where S is subnormal and N is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for C*(S x N) to contain a compact operator and a sufficient condition for the algebraic equivalence of S x N and S x M. We also consider the existence of a *-homomorphism phi : C*(S x T) --> C*(S) satisfying phi(S x T) = S. We shall characterize the operators T such that phi exists for every operator S. The problem of when S x N is unitarily equivalent to S x M is considered. Complete results are given when N and M are positive operators with finite multiplicity functions and S has compact self-commutator. Some examples are also given.