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Boundedness and completeness in locally convex spaces and algerbras

Published online by Cambridge University Press:  09 April 2009

Gerard A. Joseph
Gerard A. Joseph
Affiliation:
Department of Mathematics, University of Newcastle, New South Wales, 2308, Australia
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Abstract

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The set of bounded elements of a unital l.m.c. algebra is characterised as the union of certain naturally defined normed subalgebras, and an analogous characterisation is given for algebras of quotient-bounded operators on a locally convex space. Pseudocomplete l.m.c. algebras are characterised in terms of the completeness of these subalgebras, and the equivalence of this condition with the pseudocompleteness of the quotient-bounded operator algebras established. The scalar multiples of the identity in a unital l.m.c. algebra are characterised in terms of certain boundedness conditions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 24 , Issue 1 , August 1977 , pp. 50 - 63
Copyright
Copyright © Australian Mathematical Society 1977

References

Allan, G. R. (1965), ‘A spectral theory for locally convex algebras’, Proc. London Math. Soc. (3) 15, 399–421.Google Scholar
Bonsall, F. F. and Duncan, J. (1971), Numerical ranges of operators on normed spaces and of elements of normed algebras (London Math. Soc. Lecture Note Series, 2, Cambridge).CrossRefGoogle Scholar
Chilana, A. K. (1970). ‘Invariant subspaccs for linear operators in locally convex spaces’. J. London Math. Soc. (2) 2, 493–503.CrossRefGoogle Scholar
Chilana, A. K. (1972). ‘Some special operators and new classes of locally convex spaces’, Proc. Cambridge Philos. Soc. 71, 475–490.CrossRefGoogle Scholar
Giles, J. R., Joseph, G., Koehler, D. O. and Sims, B. (1975), ‘On numerical ranges of operators on locally convex spaces’, J. Austral. Math. Soc. 20, 468–482.CrossRefGoogle Scholar
Joseph, G. A. (1975). ‘Numerical range and operators on locally convex spaces’. Ph.D. Thesis, University of Newcastle, N.S.W.Google Scholar
Michael, E. A. (1952). ‘Locally multiplicatively-convex topological algebras’, Mem. Amer. Math. Soc. 11.Google Scholar
Moore, R. T. (1967). ‘Completeness, equicontinuity and hypocontinuity in operator algebras’, J. Functional Analysis 1, 419–442.Google Scholar
Moore, R. T. (1968). ‘Measurable, continuous and smooth vectors for semigroups and group representations’, Mem. Amer. Math. Soc. 78.Google Scholar
Moore, R. T. (1969), ‘Banach algebras of operators on locally convex spaces’. Bull. Amer. Math. Soc. 75. 68–73.CrossRefGoogle Scholar