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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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    Sure, just consider the vector space with one element $\{ 0 \}$. =)2012-05-15

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By Mazur-Ulam theorem every surjective isometry is an affine map, i.e. $ \gamma(x)=x_0+T(x) $ for some fixed vector $x_0\in X$ and necessary isometric isomorphism $T\in\mathcal{B}(X)$.

Assume that surjective isometry $\gamma$ is a Clifford isometry, then we have $C\geq 0$ such that for all $x\in X$ we have $ \Vert x_0+T(x)-x\Vert=C $ I think it's quite obvious that in this case $T$ is neccessary the identity map on $X$. Hence, every Clifford isometry is a translation on some vector $x_0\in X$.

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    @Leonid, thanks! It is a bit easier.2012-05-16