The multiplicative property characterizes p and Lp norms
Confluentes Mathematici, Volume 3 (2011) no. 4, pp. 637-647.

We show that ℓp norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the Lp norms are the unique rearrangement-invariant norms on a probability space such that ‖XY‖ = ‖X‖ ⋅ ‖Y‖ for every pair X, Y of independent random variables. Our proof combines the tensor power trick and Cramér's large deviation theorem.

Published online:
DOI: 10.1142/S1793744211000485
Guillaume Aubrun 1; Ion Nechita 1

1
@article{CML_2011__3_4_637_0,
     author = {Guillaume Aubrun and Ion Nechita},
     title = {The multiplicative property characterizes $\ell _p$ and $L_p$ norms},
     journal = {Confluentes Mathematici},
     pages = {637--647},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {3},
     number = {4},
     year = {2011},
     doi = {10.1142/S1793744211000485},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/}
}
TY  - JOUR
AU  - Guillaume Aubrun
AU  - Ion Nechita
TI  - The multiplicative property characterizes $\ell _p$ and $L_p$ norms
JO  - Confluentes Mathematici
PY  - 2011
SP  - 637
EP  - 647
VL  - 3
IS  - 4
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/
DO  - 10.1142/S1793744211000485
LA  - en
ID  - CML_2011__3_4_637_0
ER  - 
%0 Journal Article
%A Guillaume Aubrun
%A Ion Nechita
%T The multiplicative property characterizes $\ell _p$ and $L_p$ norms
%J Confluentes Mathematici
%D 2011
%P 637-647
%V 3
%N 4
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/
%R 10.1142/S1793744211000485
%G en
%F CML_2011__3_4_637_0
Guillaume Aubrun; Ion Nechita. The multiplicative property characterizes $\ell _p$ and $L_p$ norms. Confluentes Mathematici, Volume 3 (2011) no. 4, pp. 637-647. doi : 10.1142/S1793744211000485. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000485/

[1] J. Aczél and Z. Dar´oczy, On Measures of Information and their Characterizations, Mathematics in Science and Engineering, Vol. 115 (Academic Press, 1975), xii+234 pp.

[2] D. Alspach and E. Odell, Lp spaces, in Handbook of the Geometry of Banach Spaces, Vol. I (North-Holland, 2001), pp. 123–159.

[3] G. Aubrun and I. Nechita, Catalytic majorization and lp norms, Comm. Math. Phys. 278(1) (2008) 133–144.

[4] G. Aubrun and I. Nechita, Stochastic domination for iterated convolutions and cat- alytic majorization, Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 611–625.

[5] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Vol. 169 (Springer- Verlag, 1997).

[6] F. Bohnenblust, An axiomatic characterization of Lp-spaces, Duke Math. J. 6 (1940) 627–640.

[7] H. Brézis, Analyse Fonctionnelle: Théorie et Applications (Masson, 1983) (in French).

[8] R. Cerf and P. Petit, A short proof of Cramér’s theorem, to appear in Amer. Math. Monthly (2011).

[9] C. Fern´andez-Gonz´alez, C. Palazuelos and D. Pérez-Garc´ıa, The natural rearrange- ment invariant structure on tensor products, J. Math. Anal. Appl. 343 (2008) 40–47.

[10] E. Howe, A new proof of Erd˝os’ theorem on monotone multiplicative functions, Amer. Math. Monthly 93 (1986) 593–595.

[11] J.-L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976) 1–29.

[12] G. Kuperberg, The capacity of hybrid quantum memory, IEEE Trans. Inform. Th. 49 (2003) 1465–1473.

[13] T. Leinster, A multiplicative characterization of the power means, to appear in Bull. London Math. Soc. (2011).

[14] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces. With an Appendix by M. Gromov, Lecture Notes in Mathematics, Vol. 1200 (Springer-Verlag, 1986).

Cited by Sources: