Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-23T17:20:38.760Z Has data issue: false hasContentIssue false
  • English
  • Français

A second anticommutant theorem for symmetric ternary algebras

Published online by Cambridge University Press:  24 October 2008

A. Joseph
A. Joseph
Affiliation:
Mathematical Institute and Corpus Christi College, Oxford
Get access Rights & Permissions [Opens in a new window]

Abstract

It is pointed out that many axiomatic systems include a condition of closure under binary composition. In the present development a study is made of the consequences of the weaker requirement of closure under ternary, or three-fold composition. As a preliminary step in the investigation, subsets of groups closed under n-fold multiplication are examined and characterized abstractly. Attention is then focused upon ternary algebras defined as linear subspaces of ℬ (ℋ) (bounded operators on some Hilbert space ℋ) closed under three-fold multiplication. These are shown to satisfy, under suitable conditions, a secondanticommutanttheorem, the analogue of the second commutant theorem which holds for (binary) algebras. It is a result which provides a useful technique for studying such ternary algebras, the structure of which are examined in some detail.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 1 , January 1971 , pp. 25 - 52
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Green, H. S.Phys. Rev. 90 (1953), 270–273.CrossRefGoogle Scholar
(2)Greenberg, O. W.Mathematical theory of elementary particles, ed. Goodman, R., Segal, I.. Chap. 3, pp. 29–44. (The M.I.T. Press; Cambridge, Massachusetts, 1966.)Google Scholar
(3)Halmos, P. R.Introduction to Hilbert space and the theory of spectral multiplicity (Chelsea Publ. Co.; New York, 1951).Google Scholar
(4)Hestenes, M. R.Arch. rat. Mech. Anal. 11 (1962), 138–194.CrossRefGoogle Scholar
(5)Jacobson, N.Lie algebras, Chap. V, pp. 163–167. (Interscience; New York, 1962.)Google Scholar
(6)Jauch, J. M.Foundations of quantum mechanics, Chap. 5, pp. 83–87. (Addison-Wesley Publ. Co.; London, 1968.)Google Scholar
(7)Lister, W. G.Trans. Amer. Math. Soc. 72 (1952), 217–242.CrossRefGoogle Scholar
(8)Loos, O.Symmetric spaces vol. I, Chap. II, pp. 78–82, 116. (W. A. Benjamin, Inc.; New York, 1969.)Google Scholar
(9)Naimark, M. A.Normed rings (P. Noordhoff Ltd.; Groningen, 1964).Google Scholar
(10)Penrose, R.Proc. Cambridge Philos. Soc. 51 (1955), 406–413.CrossRefGoogle Scholar
(11)Schwartz, J. T.W*-Algebras, Notes on mathematics and its applications. (T. Nelson and Sons Ltd.; London, 1967.)Google Scholar
(12)Sullivan, W. G. and Joseph, A.Proc. Cambridge Philos. Soc. (in press).Google Scholar