Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-07-27T17:22:48.366Z Has data issue: false hasContentIssue false
  • English
  • Français

On commutative V*-algebras

Published online by Cambridge University Press:  20 January 2009

P. G. Spain
P. G. Spain
Affiliation:
University of Glasgow, Glasgow, W.2
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.

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 2 , December 1970 , pp. 173 - 180
Copyright
Copyright © Edinburgh Mathematical Society 1970

References

REFERENCES

(1) Bade, W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345–360.CrossRefGoogle Scholar
(2) Berkson, E., Semi-groups of scalar type operators and a theorem of Stone, Illinois J. Math. 10 (1966) 345–352.Google Scholar
(3) Berkson, E. and Dowson, H. R., Prespectral operators, Illinois J. Math. 13 (1969), 291–315.CrossRefGoogle Scholar
(4) Dunford, N., Spectral Theory II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614.Google Scholar
(5) Dunford, N. and Schwartz, J. T., Linear operators (Interscience Publishers Inc., New York, 1958, 1963).Google Scholar
(6) Edwards, D. A. and Ionescu Tulcea, C. T., Some remarks on commutative algebras of operators on Banach spaces, Trans. Amer. Math. Soc. 93 (1959), 541–551.Google Scholar
(7) Foguel, S. R., The relations between a spectral operator and its scalar part, Pacific J. Math. 8 (1958), 51–65.Google Scholar
(8) Hewitt, E. and Stromberg, K., Real and Abstract Analysis (Springer-Verlag, Berlin-Heidelberg-New York, 1965).Google Scholar
(9) -Nagy, B. SZ., Spektraldarstellung linearer Transfortnationen des Hilbertschen Raumes (Springer-Verlag, Berlin-Heidelberg-New York, 1967).Google Scholar
(10) Palmer, T. W., Characterizations of C*-algebras, Bull. Amer. Math. Soc. 74 (1968), 537–540.CrossRefGoogle Scholar
(11) Palmer, T. W., Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc. 133 (1968), 385–414.CrossRefGoogle Scholar