Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T10:53:54.543Z Has data issue: false hasContentIssue false
  • English
  • Français

On Boolean algebras of projections

Published online by Cambridge University Press:  18 May 2009

J. A. Erdos
J. A. Erdos
Affiliation:
King's College, London
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In spectral theory on Banach spaces, certain more incisive results hold when the underlying space is weakly complete (that is, weakly sequentially complete). The standard proofs rely on the following deep theorem: any bounded linear map from the algebra of all complex continuous functions on a compact Hausdorff space to a weakly complete Banach space is weakly compact. The proof of this result depends in turn on a considerable amount of measure-theoretic machinery (see [4, Section VI.7]). We present here some alternative methods which avoid these technicalities. The results are then used to give an example of a set of projections, each having unit norm, which generate an unbounded Boolean algebra.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 69 - 72
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

1.Akemann, C. A., The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286–302.CrossRefGoogle Scholar
2.Barry, J. Y., On the convergence of ordered sets of projections, Proc. Amer. Math. Soc. 5 (1954), 313–314.CrossRefGoogle Scholar
3.Dowson, H. R. and Spain, P. G., An example in the theory of well-bounded operators, Proc. Amer. Math. Soc. 32 (1972), 205–208.CrossRefGoogle Scholar
4.Dunford, N. and Schwartz, J. T., Linear operators (Interscience, 1958, 1963, 1971).Google Scholar
5.Halmos, P. R., Measure theory (Van Nostrand, 1950).CrossRefGoogle Scholar