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A note on uniquely maximal Banach spaces

Published online by Cambridge University Press:  20 January 2009

E. R. Cowie
E. R. Cowie
Affiliation:
Department of Pure Mathematics, University College of Swansea, Singleton Park, Swansea, SA2 8PP
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Let X be a real or complex Banach space with norm ∥·∥· Let G denote the set of all isometric automorphisms on X. Then G is a bounded subgroup of the group of all invertible operators GL(X) in B(X). We shall call G the group of isometries with respect to the norm ∥·∥· A bounded subgroup of GL(X) is said to be maximal if it is not contained in any larger bounded subgroup. The Banach space X has maximal norm if G is maximal. Hilbert spaces have maximal norm. For the (real or complex) spaces c0, lp (1≦p<∞), Lp[0,1] (1≦p<∞), Pelczynski and Rolewicz have shown that the standard norms are maximal ([3], pp. 252–265). In finite dimensional spaces the only maximal groups of isometries are the groups of orthogonal transformations. Given any bounded group H in B(X), X can be renormed equivalently so that each T∈H is an isometry, by ‖x‖1=sup{|Tx‖; T∈H}. Therefore corresponding to every maximal subgroup G there is at least one maximal norm for which G is the group of isometries. In this paper we shall investigate those maximal groups G for which there is only one maximal norm with G as its group of isometries.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 1 , February 1983 , pp. 85 - 87
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Banach, , Theorie des operations lineaires (Warsaw, 1932).Google Scholar
2.Kalton, N. J. and Wood, G. V., Orthonormal systems in Banach spaces and their applications, Math. Proc. Camb. Phil. Soc. 79 (1976), 493–510.CrossRefGoogle Scholar
3.Rolewicz, , Metric linear spaces (PWN, Warsaw, 1972).Google Scholar
4.Rudin, , Functional analysis (McGraw-Hill, New York, 1973).Google Scholar