Isometric embeddings of finite-dimensional ℓ_{𝑝}-spaces over the quaternions

Lyubich, Yu.; Shatalova, O.

doi:10.1090/S1061-0022-04-00842-8

Isometric embeddings of finite-dimensional $\ell _p$-spaces over the quaternions
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by Yu. I. Lyubich and O. A. Shatalova

St. Petersburg Math. J. 16 (2005), 9-24

DOI: https://doi.org/10.1090/S1061-0022-04-00842-8

Published electronically: December 14, 2004

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Abstract:

The nonexistence of isometric embeddings $\ell _{q}^{m}\to \ell _p^{n}$ with $p\neq q$ is proved. The only exception is $q=2$, $p\in 2\mathbb {N}$, in which case an isometric embedding exists if $n$ is sufficiently large, $n\ge N(m,p)$. Some lower bounds for $N(m,p)$ are obtained by using the equivalence between the isometric embeddings in question and the cubature formulas for polynomial functions on projective spaces. Even though only the quaternion case is new, the exposition treats the real, complex, and quaternion cases simultaneously.

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