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.References
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Bibliographic Information
- Yu. I. Lyubich
- Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel
- Email: lyubich@tx.technion.ac.il
- O. A. Shatalova
- Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel
- Email: oksana@tx.technion.ac.il
- Received by editor(s): October 31, 2003
- Published electronically: December 14, 2004
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 9-24
- MSC (2000): Primary 46B04
- DOI: https://doi.org/10.1090/S1061-0022-04-00842-8
- MathSciNet review: 2068351
Dedicated: Dedicated to M. Sh. Birman on the occasion of his 75th birthday