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By a Banach space $X$ I mean, a complete normed vector space and by a Clifford isometry I mean a surjective isometry $\gamma$ of $X$ such that the distance $d(\gamma x, x)$ is constant on $X$. Inherently, $\gamma$ acts as a "translation" so Clifford isometries are sometimes called Clifford translations. As an example, in Euclidean space any translation "is" a Clifford isometry.

My question is: are there examples of such an $X$ where the set of Clifford isometries consists only of the identity?

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    I don't understand: every translation in a normed space is a Clifford isometry, as $\|(x+v)-x\| = \|v\|$. Are you asking whether there are Clifford isometries that aren't translations?2012-05-15
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    Sure, just consider the vector space with one element $\{ 0 \}$. =)2012-05-15

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