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Complete Sets of Observables and Pure States

Published online by Cambridge University Press:  20 November 2018

Stanley P. Gudder
Stanley P. Gudder*
Affiliation:
University of Wisconsin, Madison, Wisconsin
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It was shown in (1) that a complete set of bounded observables is metrically complete. However, an extra axiom was needed to prove this result (1, footnote, p. 436). In this note we prove the above-mentioned result without the extra axiom. We also show that there is an abundance of pure states if M is closed in the weak topology and give a necessary and sufficient condition for the latter to be the case.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1276 - 1280
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

*

The author is indebted to Harry Mullikin for the proof of part of this theorem.

References

1. Gudder, S. P., Spectral methods for a generalized probability theory, Trans. Amer. Math. Soc 119 (1965), 428–442.Google Scholar
2. Ramsey, A., A theorem on two commuting observables, J. Math. Mech. 15 (1966), 227–234.Google Scholar
3. Segal, I., Postulates for general quantum mechanics, Ann. of Math. (2) 1+8 (1947), 930–948.Google Scholar
4. Varadarajan, V., Probability in physics and a theorem on simultaneous observability, Comm. Pure Appl. Math. 15 (1962), 189–217.Google Scholar