Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On extension of isometries between unit spheres of $AL_p$-spaces $(0<p<\infty)$

Author(s): Wang Jian
Journal: Proc. Amer. Math. Soc. 132 (2004), 2899-2909.
MSC (2000): Primary 46B04; Secondary 46A40
Posted: May 20, 2004
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: In this paper, we study the extension of isometries between unit spheres of atomic $AL_p$-spaces $(0<p<\infty, p\neq2)$. We find a condition under which an isometry $T$ between unit spheres can be linearly isometrically extended. Moreover, we prove that every onto isometry between unit spheres of atomic $AL_p$-spaces $(0<p<\infty, p\neq2)$ can be linearly isometrically extended to the whole space.

References:

[1]
C. D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Pure and Applied Math., Vol. 76, Academic Press, New York-San Francisco-London, 1978. MR 58:12271

[2]
C. D. Aliprantis and O. Burkinshaw, Positive operators, Pure and Applied Mathematics, Vol. 119, Academic Press, Orlando, FL, 1985. MR 87h:47086

[3]
Ding Guanggui, On the extension of isometries between unit spheres of $E$ and $C(\Omega)$, Acta Math. Sinica (to appear).

[4]
Ding Guanggui, The $1$-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space, Science in China, Ser. A, 45(4) (2002), 479--483. (English, English summary) MR 2003d:46029

[5]
Ding Guanggui, The isometric extension problem in the unit spheres of $l^{p}(\Gamma),(p>1)$ type spaces, Science in China, Ser. A, 46(3) (2003), 333--338. (English, English summary)

[6]
Ding Guanggui, On perturbations and extensions of isometric operators, Taiwanese J. Math., 5 (2001), 109--115. (English, English summary) MR 2002f:47031

[7]
H. E. Lacey, The isometric theory of classical Banach spaces, Springer-Verlag, New York, 1974. MR 58:12308

[8]
J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math., 8 (1958), 459--466. MR 21:3764

[9]
P. Mankiewicz, Fat equicontinuous groups of homeomorphisms of linear topological spaces and their application to the problem of isometries in linear metric spaces, Studia Math., 64 (1979), 13--23. MR 80g:46004

[10]
Y.-M. Ma, Isometries of the unit spheres, Acta. Math. Sci., 12 (1992), 366--373. (English, English summary) MR 94c:46021

[11]
H. L. Royden, Real Analysis, Macmillan, New York, 1968. MR 27:1540

[12]
H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974. MR 54:11023

[13]
D. Tingley, Isometries of the unit spheres, Geometriae Dedicata, 22 (1987), 371--378. MR 88f:46047

[14]
R.-S. Wang, Isometries between the unit spheres of $C_{0}(\Omega)$ type spaces, Acta Math. Sci., 14(1) (1994), 82--89. (English, English summary) MR 95b:46041

[15]
R.-S. Wang, Isometries on the $C_{0}(\Omega,E)$ type spaces, J. Math. Sci. Univ. Tokyo, 2 (1995), 117--130. (English, English summary) MR 96h:46047

[16]
R.-S. Wang, Isometries of $C_{0}(\Omega,\sigma)$ type spaces, Kobe J. Math., 12 (1995), 31--43. (English, English summary) MR 97e:46024

[17]
R.-S. Wang, Isometries of $C_{0}^{(n)}(X)$, Hokkaido Math. J., 25(3) (1996), 465--519. (English, English summary) MR 97m:46033

[18]
R.-S. Wang and A. Orihara, Isometries on the $l^1$-sum of $C_{0}(\Omega,E)$ type spaces, J. Math. Sci. Univ. Tokyo, 2 (1995), 131--154. (English, English summary) MR 96h:46048

[19]
R.-S. Wang, Isometries on the $l^p$-sum of $C_{0}(\Omega,E)$ type spaces, J. Math. Sci. Univ. Tokyo, 3 (1996), 471--493. (English, English summary) MR 97i:46071

[20]
R.-S. Wang and A. Orihara, Isometries between the unit spheres on the $l^1$-sum of strictly convex spaces, (to appear).

[21]
Y.-H. Xiao, On extension of isometries between unit spheres of abstract $L_p$-spaces, Acta. Sci. Nat. Univ. Nankai, 28 (1995), 11--14. (English, English summary)

[22]
Y.-H. Xiao and R.-S. Wang, On extension of isometries between unit spheres of abstract $AM$-spaces, Chinese Ann. Math. (A), 22(1) (2001), 97--104. (Chinese, English summary); English transl., Chinese J. Contemp. Math. 22 (2001), 79--88. MR 2002a:46002

[23]
D.-P. Zhan, Extensions of isometries between unit spheres, Acta. Mathematica Sinica, 41(2) (1998), 275--280. (Chinese, English summary) MR 99m:46046

[24]
D.-P. Zhan, On extension of isometries between unit spheres on the $L^{p}(\Omega,H)$ type spaces, Acta. Sci. Nat. Univ. Nankai, 30(4) (1997), 31--35. (Chinese, English summary)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B04, 46A40

Retrieve articles in all Journals with MSC (2000): 46B04, 46A40

Additional Information:

Wang Jian
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People's Republic of China - Department of Mathematics, Fujian Normal University, Fuzhou 350007, People's Republic of China
Email: wjmath@nju.edu.cn

DOI: 10.1090/S0002-9939-04-07482-9
PII: S 0002-9939(04)07482-9
Keywords: Atomic $AL_p$-space, isometric extension, atom
Received by editor(s): April 15, 2002
Received by editor(s) in revised form: June 24, 2002
Posted: May 20, 2004
Additional Notes: This work belongs to the Doctoral Programme Foundation of the Institution of Higher Education (20010055013) and the Programme of National Science Foundation of China (10271060). It was supported by the National Science Foundation of China (10171014) and the Foundation of Fujian Educational Committee (JA02166).
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2004, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google