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Quaternion normed space with isometry group \(\mathbb Z_{2}\). (English) Zbl 1170.46314

Funct. Anal. Appl. 42, No. 3, 239-241 (2008); translation from Funkts. Anal. Prilozh. 42, No. 3, 90-92 (2008).
In a finite-dimensional linear space over the quaternions, the authors construct a norm with the property that any linear isometry is one of the two transformations \(x\mapsto x\) and \(x\mapsto -x\).

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46B04 Isometric theory of Banach spaces
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References:

[1] Yu. I. Lyubich, Sibirsk. Mat. Zh., 11:2 (1970), 359–369.
[2] Yu. I. Lyubich and O. A. Shatalova, Algebra i Analiz, 16:1 (2004), 15–32.
[3] Yu. I. Lyubich and L. N. Vaserstein, Geom. Dedicata, 47:3 (1993), 327–362. · Zbl 0785.52002 · doi:10.1007/BF01263664
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