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Symmetry groups on ordered Banach spaces

Published online by Cambridge University Press:  17 April 2009

Sadayuki Yamamuro
Sadayuki Yamamuro
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, A.C.T., Australia.
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Abstract

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A symmetry of an ordered Banach space is an order and norm isomorphism which commutes with its ideal centre. A class of ordered Banach spaces is introduced to show that, for a space in this class, the group of symmetries is trivial if and only if the space is lattice-ordered. When this group becomes larger, the space approaches an antilattice. This phenomenon is also investigated.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 2 , April 1986 , pp. 177 - 187
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Alfsen, E., Compact Convex Sets and Boundary Integrals, (Springer-Verlag, Berlin-Heidelberg-New York, 1971.)Google Scholar
[2]Arendt, W., “Spectral properties of Lamperti operators”, Indiana University Math. J. 32 (1983), 199–215.CrossRefGoogle Scholar
[3]Arendt, W., Chernoff, P. and Kato, T., “A generalization of dissipativity and positive semigroups”, J. Operator Theory 8 (1982), 167–180.Google Scholar
[4]Batty, C.J.K. and Robinson, D.W., “Positive one-parameter semigroups on ordered Banach spaces”, Acta Math. Applicandae 2 (1984), 221–296.CrossRefGoogle Scholar
[5]Bratteli, O., Digernes, T. and Robinson, D.W., “Positive semigroups on ordered Banach spaces”, J. Operator Theory 9 (1983), 371–400.Google Scholar
[6]Bratteli, O. and Robinson, D.W., Operator Algebras and Quantum Statistical Mechanics I, (Springer-Verlag, Berlin-Heidelberg-New York, 1979).CrossRefGoogle Scholar
[7]Bos, W., “A classification for selfdual cones in Hilbert space”, Arch. Math. 30 (1978), 75–82.CrossRefGoogle Scholar
[8]Chu, Cho-ho and Wright, J.D.M., “Une théorie des types pour une classe d'espaces de Banach ordonnés”, C. R. Acad. Sc. Paris, Serie A 281 (1975), 633–636.Google Scholar
[9]Connes, A., “Charactérisation des espaces vectoriels ordonnes sous-jucents aux algèbres de von Neumann”, Ann. Inst. Fourier, Grenoble 24 (1974), 121–155.CrossRefGoogle Scholar
[10]Kadison, R.V., A representation theory for commutative topological algebras, (Memoirs of Amer. Math. Soc., No. 7, 1951).CrossRefGoogle Scholar
[11]Kadison, R. V., “Order properties of bounded selfadjoint operators”, Proc. Amer. Math. Soc. 2 (1951), 505–510.CrossRefGoogle Scholar
[12]Kadison, R. V., “Isometries of operator algebras”, Ann. Math. 54 (1951), 325–338.CrossRefGoogle Scholar
[13]Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I, (North Holland, Amsterdam, 1971).Google Scholar
[14]Robinson, D. W. and Yamamuro, S., “Hereditary cones, order ideals and half-norms”, Pacific J. Math. 110 (1984), 335–343.CrossRefGoogle Scholar
[15]Wils, W., “The ideal centre of partially ordered vector spaces”, Acta Math. 127 (1971), 41–77.CrossRefGoogle Scholar
[16]Yamamuro, S., “On linear operators on ordered Banach Spaces”, Bull. Austral. Math. Soc. 27 (1983), 285–305.CrossRefGoogle Scholar
[17]Yamamuro, S., “On orthogonally decomposable ordered Banach spaces”, Bull. Aust. Math. Soc. 30 (1984), 357–380.CrossRefGoogle Scholar
[18]Yamamuro, S., “Absolute values in orthogonally decomposable spaces”, Bull. Austral. Math. Soc. 31 (1985), 215–233.CrossRefGoogle Scholar