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Isometries on conservative subalgebras of bounded sequences

Authors: J. Connor and I. Loomis
Journal: Proc. Amer. Math. Soc. 107 (1989), 743-749
MSC: Primary 47B37; Secondary 46B25
MathSciNet review: 986647
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Abstract: Using extreme point techniques, we show that if $ A$ is a closed subalgebra of the bounded sequences which contain $ c$, then any linear isometry of $ A$ onto itself is a permutation up to a modulus one multiplication. If the subalgebra $ A$ is generated by an ideal, then a permutation $ P$ maps $ A$ onto itself if and only if $ P$ maps $ \mu $-null sets to $ \mu $-null sets where $ \mu $ is a 0,1-valued finitely additive measure associated with the ideal. In particular, if $ T$ is a nonnegative regular summability method, we characterize the isometries which map the bounded strongly $ T$-summable sequences onto themselves and give a concrete sufficient condition for a permutation to map the bounded strongly Cesaro summable sequences onto themselves.

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Keywords: Linear isometry, subalgebras of bounded sequences, strong matrix summability
Article copyright: © Copyright 1989 American Mathematical Society

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