Biseparating maps between Lipschitz function spaces. (English) Zbl 1169.47024

Araujo, Jesús; Dubarbie, Luis

J. Math. Anal. Appl. 357, No. 1, 191-200 (2009).

Let \(X,Y\) be bounded complete metric spaces and let \(E,F\) be (real or complex) normed spaces. We write \(\text{Lip}(X,E)= \{\)all bounded \(E\)-valued Lipschitz functions}; \(\text{Lip}(X)= \{\)all bounded Lipschitz functionals}; \(L'(E,F)=\{\)all linear bijections from \(E\) to \(F\}\). A map \(T:\text{Lip}(X,E)\to \text{Lip}(Y,F)\) is said to be separating if \(T\) is linear and \(\|Tf(y)\|\,\|Tg (y)\|=0\) for all \(y\in Y\), whenever \(f,g\in \text{Lip}(X,E)\) satisfy \(\|fx<\|\|g(x) \|=0\) for all \(x\in X\). \(T\) is said to be biseparating if \(T\) is bijective and both \(T\) and \(T^{-1}\) are separating. The authors establish the following results.
Proposition 1. Let \(T:\text{Lip}(X,E)\to \text{Lip}(Y,F)\) be a biseparating map. Then there exists a bi-Lipschitz homeomorphism \(h:Y \to X\) and a map \(J:Y\to L'(E,F)\) such that \(Tf(y)=(Jy) (f(h(y)))\) for all \(f\in \text{Lip}(X,E)\) and \(y\in Y\).
Proposition 2. Let \(T:\text{Lip}(X)\to \text{Lip}(Y)\) be a bijective separating map. If \(Y\) is compact, then \(T\) is biseparating and continuous.

Reviewer: K. Chandrasekhara Rao (Kumbakonam)

MSC:

47B38	Linear operators on function spaces (general)
46E10	Topological linear spaces of continuous, differentiable or analytic functions
54C35	Function spaces in general topology

Keywords:

biseparating map; disjointness preserving map; automatic continuity; Lipschitz function

References:

[1]	Abramovich, Y.A.; Kitover, A.A., Inverses of disjointness preserving operators, Mem. amer. math. soc., 143, (2000) · Zbl 0973.47030
[2]	Araujo, J., Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity, Adv. math., 187, 488-520, (2004) · Zbl 1073.47031
[3]	Araujo, J., Realcompactness and banach – stone theorems, Bull. belg. math. soc. Simon stevin, 11, 247-258, (2004) · Zbl 1077.46029
[4]	Araujo, J., Realcompactness and spaces of vector-valued functions, Fund. math., 172, 27-40, (2002) · Zbl 0997.46028
[5]	Araujo, J., Separating maps and linear isometries between some spaces of continuous functions, J. math. anal. appl., 226, 23-39, (1998) · Zbl 0918.46026
[6]	Araujo, J.; Font, J.J., Linear isometries on subalgebras of uniformly continuous functions, Proc. edinb. math. soc. (2), 43, 139-147, (2000) · Zbl 0945.46032
[7]	Araujo, J.; Jarosz, K., Automatic continuity of biseparating maps, Studia math., 155, 231-239, (2003) · Zbl 1056.46032
[8]	Bustamante, J.; Arrazola, J.R., Homomorphisms on Lipschitz spaces, Monatsh. math., 129, 25-30, (2000) · Zbl 0991.54029
[9]	L. Dubarbie, Separating maps between spaces of vector-valued absolutely continuous functions, Canad. Math. Bull., in press · Zbl 1217.47067
[10]	Font, J.J.; Hernández, S., On separating maps between locally compact spaces, Arch. math. (basel), 63, 158-165, (1994) · Zbl 0805.46049
[11]	Garrido, M.I.; Jaramillo, J.A., Homomorphisms on function lattices, Monatsh. math., 141, 127-146, (2004) · Zbl 1058.54008
[12]	Garrido, M.I.; Jaramillo, J.A., Lipschitz-type functions on metric spaces, J. math. anal. appl., 340, 282-290, (2008) · Zbl 1139.46025
[13]	Gau, H.L.; Jeang, J.S.; Wong, N.C., Biseparating linear maps between continuous vector-valued function spaces, J. aust. math. soc., 74, 101-109, (2003) · Zbl 1052.47017
[14]	Hernández, S.; Beckenstein, E.; Narici, L., Banach – stone theorems and separating maps, Manuscripta math., 86, 409-416, (1995) · Zbl 0827.46032
[15]	Jarosz, K., Automatic continuity of separating linear isomorphisms, Canad. math. bull., 33, 139-144, (1990) · Zbl 0714.46040
[16]	Jeang, J.S.; Wong, N.C., On the banach – stone problem, Studia math., 155, 95-105, (2003) · Zbl 1056.46011
[17]	Jeang, J.S.; Wong, N.C., Weighted composition operators of \(C_0(X)\)’s, J. math. anal. appl., 201, 981-993, (1996) · Zbl 0936.47011
[18]	Jiménez-Vargas, A., Disjointness preserving operators between little Lipschitz algebras, J. math. anal. appl., 337, 984-993, (2008) · Zbl 1159.46030
[19]	Jiménez-Vargas, A.; Villegas-Vallecillos, M.; Wang, Y.S., Banach – stone theorems for vector-valued little Lipschitz functions, Publ. math. debrecen, 74, 1-2, 81-100, (2009) · Zbl 1210.46027
[20]	A. Jiménez-Vargas, Y.S. Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces, preprint · Zbl 1211.46033
[21]	Kantrowitz, R.; Neumann, M., Disjointness preserving and local operators on algebras of differentiable functions, Glasg. math. J., 43, 295-309, (2001) · Zbl 0994.47030
[22]	Miao, X.; Cao, J.; Xiong, H., Banach – stone theorems and Riesz algebras, J. math. anal. appl., 313, 177-183, (2006) · Zbl 1102.46017
[23]	Sherbert, D.R., Banach algebras of Lipschitz functions, Pacific J. math., 13, 1387-1399, (1963) · Zbl 0121.10203
[24]	Su, Li Pi, Algebraic properties of certain rings of continuous functions, Pacific J. math., 27, 175-191, (1968) · Zbl 0165.47403
[25]	Weaver, N., Lipschitz algebras, (1999), World Scientific Publishing · Zbl 0936.46002
[26]	T.C. Wu, Disjointness preserving operators between Lipschitz spaces, Master Thesis, NSYSU, Kaohsiung, Taiwan, 2006