Tensor products of subnormal operators

Abstract

We shall use a C ∗ C^* –algebra approach to study operators of the form S ⊗ N S \otimes N where S S is subnormal and N 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 ⊗ N ) C^*(S \otimes N) to contain a compact operator and a sufficient condition for the algebraic equivalence of S ⊗ N S \otimes N and S ⊗ M S \otimes M . We also consider the existence of a ∗ − *- homomorphism ϕ : C ∗ ( S ⊗ T ) → C ∗ ( S ) \phi :C^*(S \otimes T) \to C^*(S) satisfying ϕ ( S ⊗ T ) = S \phi (S \otimes T) = S . We shall characterize the operators T T such that ϕ \phi exists for every operator S S . The problem of when S ⊗ N S \otimes N is unitarily equivalent to S ⊗ M S \otimes M is considered. Complete results are given when N N and M M are positive operators with finite multiplicity functions and S S has compact self–commutator. Some examples are also given.