## Banach Journal of Mathematical Analysis

### Disjointness preserving linear operators between Banach algebras of vector-valued functions

#### Abstract

We present vector-valued versions of two theorems due to A. Jimenez-Vargas, by showing that, if $B(X,E)$ and $B(Y,F)$ are certain vector-valued Banach algebras of continuous functions and $T:B(X,E)\to B(Y,F)$ is a separating linear operator, then $\widehat{T}:\widehat{B(X,E)}\to \widehat{B(Y,F)}$, defined by $\widehat{T}\hat{f}=\widehat{Tf}$, is a weighted composition operator, where $\widehat{Tf}$ is the Gelfand transform of $Tf$. Furthermore, it is shown that, under some conditions, every bijective separating map $T:B(X,E)\to B(Y,F)$ is biseparating and induces a homeomorphism between the character spaces $M(B(X,E))$ and $M(B(Y,F))$. In particular, a complete description of all biseparating, or disjointness preserving linear operators between certain vector-valued Lipschitz algebras is provided. In fact, under certain conditions, if the bijections $T:Lip^{\alpha}(X,E)\to Lip^{\alpha}(Y,F)$ and $T^{-1}$ are both disjointness preserving, then $T$ is a weighted composition operator in the form $Tf(y)=h(y)(f(\phi(y))),$ where $\phi$ is a homeomorphism from $Y$ onto $X$ and $h$ is a map from $Y$ into the set of all linear bijections from $E$ onto $F$. Moreover, if $T$ is multiplicative then $M(E)$ and $M(F)$ are homeomorphic.

#### Article information

Source
Banach J. Math. Anal., Volume 8, Number 2 (2014), 93-106.

Dates
First available in Project Euclid: 4 April 2014

https://projecteuclid.org/euclid.bjma/1396640054

Digital Object Identifier
doi:10.15352/bjma/1396640054

Mathematical Reviews number (MathSciNet)
MR3189541

Zentralblatt MATH identifier
1308.47047

#### Citation

Ghasemi Honary, Taher; Nikou, Azadeh; Sanatpour, Amir Hossein. Disjointness preserving linear operators between Banach algebras of vector-valued functions. Banach J. Math. Anal. 8 (2014), no. 2, 93--106. doi:10.15352/bjma/1396640054. https://projecteuclid.org/euclid.bjma/1396640054

#### References

• Y.A. Abramovich and A.K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc., 143 (2000), no. 679, 1–162.
• Y.A. Abramovich, A. I. Veksler and A.V. Koldunov, Operators preserving disjointness, Dokl. Akad. Nauk USSR, 248 (1979), 1033–1036.
• J. Araujo and K. Jarosz, Separating maps on spaces of continuous functions, Contemp. Math. 232 (1999), 33–37.
• W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), 199–215.
• E. Beckenstein and L. Narici, A nonarchimedean Stone-Banach theorem, Proc. Amer. Math. Soc. 100 (1987), 242–246.
• E. Beckenstein and L. Narici, Automatic continuity of certain linear isomorphisms, Acad. Roy. Belg. Bull. Cl. Sci. 73 (1987), no. 5, 191–200.
• E. Beckenstein and L. Narici, Automatic continuity of linear maps of spaces of continuous functions, Manuscr. Math. 62 (1988), 257–275.
• H.X. Cao, J. H. Zhang and Z.B. Xu, Characterizations and extensions of Lipschitz-$\alpha$ operators, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 671–678.
• H.G. Dales, Banach Algebras and Automatic Continuity, LMS Monographs 24, Clarendon Press, Oxford, 2000.
• K. Esmaeili and H. Mahyar, Weighted composition operators between vector-valued Lipschitz function spaces, Banach J. Math. Anal. 7 (2013), no. 1, 59-72.
• J.J. Font, Automatic continuity of certain isomorphisms between regular Banach function algebras, Glasg. Math. J. 39 (1997), 333–343.
• J.J. Font and S. Hernandez, On separating maps between locally compact spaces, Arch. Math. (Basel) 63 (1994), 158–165.
• H-L. Gau, J-S. Jeang and N-C. Wong, Biseparating linear maps between continuous vector-valued function spaces, J. Aust. Math. Soc. 74 (2003), 101–109.
• M. Hosseini and F. Sady, Banach function algebras and certain polynomially norm-preserving maps, Banach J. Math. Anal. 6 (2012), no. 2, 1–18.
• K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), 139–144.
• A. Jimenez–Vargas, Disjointness preserving operators between little Lipschitz algebras, J. Math. Anal. Appl. 337 (2008), 984–993.
• A. Jimenez–Vargas and Ya-Shu Wang, Linear biseparating maps between vector-valued little Lipschitz function spaces, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1005–1018.
• E. Kaniuth, A Course in Commutative Banach Algebras, Springer, Graduate Texts in Mathematics 246, 2009.
• J.S. Manhas, Weighted composition operators and dynamical systems on weighted spaces of holomorphic functions on Banach spaces, Ann. Funct. Anal. 4 (2013), no. 2, 58–71.
• A. Nikou and A.G. O'Farrell, Banach algebras of vector-valued functions, Glasgow Math. J., to appear, arXiv:1305.2751.
• D.R. Sherbert, The Structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240–272.
• B.Z. Vulkh, On linear multiplicative operations, Dokl. Akad. Nauk USSR 41 (1943), 148–151.
• B.Z. Vulkh, Multiplication in linear semi-ordered spaces and its application to the theory of operations, Mat. Sbornik 22 (1948), 267–317.