Tensor products of subnormal operators

We shall use a $C^*$-algebra approach to study operators of the form $S \otimes N$ 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 $C^*(S \otimes N)$ to contain a compact operator and a sufficient condition for the algebraic equivalence of $S \otimes N$ and $S \otimes M$.

We also consider the existence of a $*-$homomorphism $\phi:C^*(S \otimes T) \to C^*(S)$ satisfying $\phi(S \otimes T) = S$. We shall characterize the operators $T$ such that $\phi$ exists for every operator $S$.

The problem of when $S \otimes N$ is unitarily equivalent to $S \otimes M$ 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.