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On a Boolean algebra of projections constructed by Dieudonné

Published online by Cambridge University Press:  20 January 2009

H. R. Dowson
H. R. Dowson
Affiliation:
University of IllinoisUrbana, Illinois and University of Glasgow
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Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra of projections on X such that has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in is EX the direct sum of the cyclic subspaces clm {Ex1:E∈} and clm {Ex2:E∈}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant ′ of is equal to A(), the algebra of operators generated by in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that has the first property.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 16 , Issue 3 , June 1969 , pp. 259 - 262
Copyright
Copyright © Edinburgh Mathematical Society 1969

References

REFERENCES

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