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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
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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

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