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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Noncoincidence of the strict and strong operator topologies

Author: Joel H. Shapiro
Journal: Proc. Amer. Math. Soc. 35 (1972), 81-87
MSC: Primary 46E10
MathSciNet review: 0306878
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Abstract: Let E be an infinite-dimensional linear subspace of $ C(S)$, the space of bounded continuous functions on a locally compact Hausdorff space S. If $ \mu $ is a regular Borel measure on S, then each element of E may be regarded as a multiplication operator on $ {L^p}(\mu )(1 \leqq p < \infty )$. Our main result is that the strong operator topology this identification induces on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region G, and $ \mu $ Lebesgue measure on G, this answers negatively a question raised by Rubel and Shields in [9]. In addition, our methods provide information about the absolutely p-summing properties of the strict topology on subspaces of $ C(S)$, and the bounded weak star topology on conjugate Banach spaces.

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Keywords: Strict topology, strong operator topology, absolutely p-summing topology, bounded continuous functions
Article copyright: © Copyright 1972 American Mathematical Society