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M-structure in Banach spaces

Published online by Cambridge University Press:  24 October 2008

F. Cunningham Jr
F. Cunningham Jr
Affiliation:
Bryn Mawr College, Pennsylvania
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Extract

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfying

for all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 3 , July 1967 , pp. 613 - 629
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

(1)Bohnenblust, H. F. and Kakutani, S.Concrete representation of (M)-spaces. Ann. of Math. 42 (1941), 1025–1028.CrossRefGoogle Scholar
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