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Commutators in Factors of Type III

Published online by Cambridge University Press:  20 November 2018

Arlen Brown
Affiliation:
The University of Michigan, Ann Arbor, Michigan
Carl Pearcy
Affiliation:
The University of Michigan, Ann Arbor, Michigan
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Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Brown, A. and Pearcy, C., Structure of commutators of operators, Ann. Math., 82 (1965), 112—127.CrossRefGoogle Scholar
2. Deckard, D. and Pearcy, C., On continuous matrix-valued functions on a Stonian space, Pacific J. Math., 14(1964), 857869.CrossRefGoogle Scholar
3. Lumer, G. and Rosenbloom, M., Linear operator equations, Proc. Amer. Math. Soc., 10 (1959), 3241.CrossRefGoogle Scholar
4. Pearcy, C., On commutators of operators on Hilbert space, Proc. Amer. Math. Soc, 16 (1965), 5359.CrossRefGoogle Scholar
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