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On Products of Symmetries

Published online by Cambridge University Press:  20 November 2018

Peter A. Fillmore
Peter A. Fillmore*
Affiliation:
University of Chicago and Indiana University
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In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebra A (i.e., one with no finite non-zero central projections), then the symmetries may be chosen from A. A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1 factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 897 - 900
Copyright
Copyright © Canadian Mathematical Society 1966

References

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