Proc. Amer. Math. Soc. **127** (1999), no. 9, pp. 2685-2695.

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We use a C*-algebra approach to study operators of the form S tensor N where S is subnormal and N is normal. We determine the spectral properties for these operators, including the normal spectrum. We also find the minimal normal extension and the dual operator and give a necessary condition for the C*-algebra, C*(S tensor N), generated by S tensor N to contain a compact operator.

We also characterize the operators T such that there exists a *-homomorphism from C*(S tensor T) to C*(S) mapping S tensor T to S for any operator S. The class of operators T for which this works is precisely the operators T such that there exists a non-zero multiplicative linear functional on C*(T); this includes the hyponormal operators. This allows one to easily transfer C*-properties (such as subnormality, hyponormality, etc) from the tensor product to the individual factors.

The problem of when S tensor N is unitarily equivalent to S tensor M is considered. Here 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.