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Extreme Points of Positive Functionals and Spectral States on Real Banach Algebras

Published online by Cambridge University Press:  20 November 2018

Anand Srivastav
Anand Srivastav*
Affiliation:
Research Institute of Discrete Mathematics, University of Bonn, Nassestr. 2, W-5 300 Bonn 1, Germany
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Abstract

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Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

Keywords

Primary: 46J0546J20Real Banach algebrapositive functionalsmaximal idealsShilov boundary
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 856 - 866
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Ailing, N.L., Real Banach algebras and non-orientable Klein surfaces I, Jour. Reine Angew. Math. 241 (1970), 200–208.Google Scholar
2. Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements ofnormed algebras, Cambridge University Press, Lond. Math. Soc. Lect. Note Ser. 2, Cambridge University Press (1971).Google Scholar
3. Bonsall, F. F. and Duncan, J., Complete normed algebras, Springer Verlag, Berlin, Heidelberg, 1973.Google Scholar
4. Bucy, R.S. and Maltese, G., A representation theorem for positive Junctionals oninvolutive algebras, Math. Ann. 162 (1966), 364–367.Google Scholar
5. Bucy, R.S. and Maltese, G., Extreme positive definite functions and Choquet's representation theorem, Jour. Math. Anal. Appl. 12 (1965), 371–377.Google Scholar
6. Choquet, G., Lectures on analysis II, Cummings, B. Publ. Comp., 1969.Google Scholar
7. Dixmier, J., C* -algebras, North-Holland, Amsterdam, 1966.Google Scholar
8. Fialkowski, P., Representation theorem for real B*-algebras, Bull. Soc. Sci. Lett. Lodz (2) 34 (1984).Google Scholar
9. Goodearl, K.R., Notes on real and complex (T-algebras, Shiva Publ. Ltd., Cheshire (England) and Birkhäuser, Boston, 1982.Google Scholar
10. Ingelstam, L., Real Banach algebras, Ark. Math. 5 (1964), 239–279.Google Scholar
11. Kulkarni, S.H. and Limaye, B.V., Gelfand-Naimark theorems for real Banach* algebras, Math. Jap. 25 (1980), 545–558.Google Scholar
12. Limaye, B.V. and Simha, R.R., Deficiencies of certain real uniform algebras, Canad. Jour. Math. 27 (1975), 121–132.Google Scholar
13. Maltese, G., Multiplicative extensions of multiplicative Junctionals in Banach algebras, Arch. Math. 21 (1970), 502–505.Google Scholar
14. Niestegge, G., Extreme spectral states of commutative Banach algebras, Math. Ann. 264 (1983), 179–188.Google Scholar
15. Rickart, C.E., General theory of Banach algebras, Van Nostrand, Princeton, 1960.Google Scholar
16. Rudin, W., Functional analysis, McGraw-Hill, New York, 1973.Google Scholar
17. Srivastav, A., Commutativity criteria for real Banach algebras, Arch. Math. 54 (1990), 65–72.Google Scholar
18. Srivastav, A., A generalization of the Gleason-Kahane-Zelazko theorem for real Banach algebras, Indian Jour. Math. (3) 32 (1990), 217–221.Google Scholar