Abstract
For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ G ) is isomorphic toG × {−1, + 1}.
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Jarosz, K. Any Banach space has an equivalent norm with trivial isometries. Israel J. Math. 64, 49–56 (1988). https://doi.org/10.1007/BF02767369
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DOI: https://doi.org/10.1007/BF02767369