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Quotient bounded elements in locally convex algebras

Part of: Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  09 April 2009

Subhash J. Bhatt
Subhash J. Bhatt
Affiliation:
Dapartment of Mathematics Sardar Patel UniversityVallabh Vidyanagar-388120 Gujarat, India
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Abstract

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The quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.

MSC classification

Secondary: 46H05: General theory of topological algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 102 - 113
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Allan, G. R. (1965), ‘A spectral theory for locally convex algebras’, Proc. London Math. Soc. (3) 15, 399–421.CrossRefGoogle Scholar
[2]Allan, G. R. (1967), ‘On a class of locally convex algebras’, Proc. London Math. Soc. (3) 17, 91–114.CrossRefGoogle Scholar
[3]Apostol, C. (1971), ‘b*-algebras and their representations’, J. London Math. Soc. (2) 3, 30–38.CrossRefGoogle Scholar
[4]Arens, R. (1946), ‘The space Lω and convex topological rings’, Bull. Amer. Math. Soc. 52, 931–935.CrossRefGoogle Scholar
[5]Bhatt, S. J. (1979), ‘A note on generalized B*-algebras’, J. Indian Math. Soc. 43, 253–257.Google Scholar
[6]Gonsall, 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
[7]Dixon, P. G. (1970), ‘Generalized B*-algebras’, Proc. London Math. Soc. (4) 21, 693–715.CrossRefGoogle Scholar
[8]Feldmann, W. A. and Porter, J. F. (1975), ‘Compact convergence and the bidual of C(X)’, Pacific J. Math. 57 (1), 113–124.CrossRefGoogle Scholar
[9]Giles, J. R. and Koehler, D. O. (1973), ‘On numerical ranges of elements of locally m-convex algebras’, Pacific J. math. 49, 79–91.CrossRefGoogle Scholar
[10]Giles, J. R., Koehler, D. O., Joseph, G. and Sims, B. (1975), ‘On numerical ranges of operators on locally convex spaces’, J. Austral. Math. Soc. 20, 468–482.CrossRefGoogle Scholar
[11]Michael, E. (1952), ‘Locally niultiplicatively convex topological algebras’, Mem. Amer. Math. Soc. 10.Google Scholar
[12]Rudin, W. (1974), Functional analysis (Tata-McGraw Hill, Delhi).Google Scholar
[13]Treves, F. (1967), Topological vector spaces, distributions and kernels (Academic Press, New York).Google Scholar
[14]Wood, A. (1977), ‘Numerical ranges and generalized B*-algebras’, Proc. London Math. Soc. (3) 34, 245–268.CrossRefGoogle Scholar