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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Tensor products of subnormal operators


Author: Nathan S. Feldman
Journal: Proc. Amer. Math. Soc. 127 (1999), 2685-2695
MSC (1991): Primary 47B20; Secondary 47A80
Published electronically: April 9, 1999
MathSciNet review: 1625745
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Abstract: 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.


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Additional Information

Nathan S. Feldman
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: feldman@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05054-6
Keywords: Tensor product, subnormal operator, dual operator
Received by editor(s): November 20, 1997
Published electronically: April 9, 1999
Additional Notes: This paper was written while the author was a graduate student at the University of Tennessee working under the direction of Professor John B. Conway. He received partial support from the NSF grant DMS–9401027.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society