# EXTENSION OF PHASE-ISOMETRIES BETWEEN THE UNIT SPHERES OF ATOMIC Lp-SPACES FOR p > 0

• Huang, Xujian (Department of Mathematics Tianjin University of Technology) ;
• Jin, Xihong (Department of Mathematics Tianjin University of Technology)
• Received : 2018.06.09
• Accepted : 2019.04.01
• Published : 2019.11.30

#### Abstract

In this paper, we prove that for every surjective phase-isometry between the unit spheres of real atomic $L_p$-spaces for p > 0, its positive homogeneous extension is a phase-isometry which is phase equivalent to a linear isometry.

#### Acknowledgement

Supported by : Natural Science Foundation of China

#### References

1. D. F. Almeida and C. S. Sharma, The first mathematical proof of Wigner's theorem, J. Natur. Geom. 2 (1992), no. 2, 113-123.
2. S. Banach, Theorie des operations lineaires, reprint of the 1932 original, Editions Jacques Gabay, Sceaux, 1993.
3. G. Ding, On isometric extension problem between two unit spheres, Sci. China Ser. A 52 (2009), no. 10, 2069-2083. https://doi.org/10.1007/s11425-009-0156-x
4. Gy. P. Geher, An elementary proof for the non-bijective version of Wigner's theorem, Phys. Lett. A 378 (2014), no. 30-31, 2054-2057. https://doi.org/10.1016/j.physleta.2014.05.039 https://doi.org/10.1016/j.physleta.2014.05.039
5. M. Gyory, A new proof of Wigner's theorem, Rep. Math. Phys. 54 (2004), no. 2, 159-167. https://doi.org/10.1016/S0034-4877(04)80012-0 https://doi.org/10.1016/S0034-4877(04)80012-0
6. X. Huang and D. Tan, Wigner's theorem in atomic Lp-spaces (p > 0), Publ. Math. Debrecen 92 (2018), no. 3-4, 411-418. https://doi.org/10.5486/pmd.2018.8005 https://doi.org/10.5486/PMD.2018.8005
7. J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459-466. http://projecteuclid.org/euclid.pjm/1103039892 https://doi.org/10.2140/pjm.1958.8.459
8. L. Molnar, Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn's version of Wigner's theorem, J. Funct. Anal. 194 (2002), no. 2, 248-262. https://doi.org/10.1006/jfan.2002.3970 https://doi.org/10.1006/jfan.2002.3970
9. A. M. Peralta and M. Cueto-Avellaneda, The Mazur-Ulam property for commutative von Neumann algebras, Linear and Multilinear A.; arXiv:1803.00604, 2018.
10. J. Ratz, On Wigner's theorem: remarks, complements, comments, and corollaries, Aequationes Math. 52 (1996), no. 1-2, 1-9. https://doi.org/10.1007/BF01818323 https://doi.org/10.1007/BF01818323
11. D. Tingley, Isometries of the unit sphere, Geom. Dedicata 22 (1987), no. 3, 371-378. https://doi.org/10.1007/BF00147942 https://doi.org/10.1007/BF00147942
12. A. Turnsek, A variant of Wigner's functional equation, Aequationes Math. 89 (2015), no. 4, 1-8, https://doi.org/10.1007/s00010-015-0343-5
13. J. Wang, On extension of isometries between unit spheres of $AL^{p}$-spaces (0 < p < ${\infty}$), Proc. Amer. Math. Soc. 132 (2004), no. 10, 2899-2909. https://doi.org/10.1090/S0002-9939-04-07482-9