Nihonkai Mathematical Journal

Characterization and automatic continuity of separating maps between Banach modules

Lida Mousavi and Fereshteh Sady

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A linear map $T:\mathcal{A}\to \mathcal{B}$ between algebras (or spaces of functions) $\mathcal{A}$ and $\mathcal{B}$ is called separating if $x \cdot y=0$ implies $Tx\cdot Ty=0$ for all $x,y\in \mathcal{A}$. It is well known that a separating map between certain commutative semisimple Banach algebras is very close to being a weighted composition operator on the maximal ideal spaces. In this paper, after introducing the notion of the cozero set for the elements of a Banach module, we first extend the notion of the separating maps to Banach module case. Our approach depends on the notion of point multipliers on a Banach module $\mathcal{X}$ and the relation between hyper maximal submodules of $\mathcal{X}$ and point multipliers on it. Then we generalize some well known results about separating maps between certain subspaces of continuous functions to Banach module case. In particular, we show that, imposing some additional assumptions on Banach modules, such map can be represented as a variation of a weighted composition operator. We also obtain a result concerning the automatic continuity of a bijective separating map whose inverse is also separating.

Article information

Nihonkai Math. J., Volume 23, Number 2 (2012), 75-91.

First available in Project Euclid: 12 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

separating maps Banach modules automatic continuity, cozero set point multipliers


Mousavi, Lida; Sady, Fereshteh. Characterization and automatic continuity of separating maps between Banach modules. Nihonkai Math. J. 23 (2012), no. 2, 75--91.

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  • Y. A. Abramovich, Multiplicative representation of disjointness preserving operators, Indag. Math., 45 (1983), 265-279.
  • J. Araujo and L. Dubarbie, Biseparating maps between Lipschitz function spaces, J. Math. Anal. Appl., 357 (2009), 191-200.
  • J. Araujo and K. Jarosz, Automatic continuity of biseparating maps, Studia Math., 155 (2003), 231-239.
  • W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J., 32 (1983), 199-215.
  • S. Banach, Theorie des Operation Lineaires, Chelsea Publishing Company, New York, 1932.
  • J. Bra$\check{c}$i$\check{c}$, Representations and derivations of modules, Irish Math. Soc. Bull., 47 (2001), 27-39.
  • J. Bra$\check{c}$i$\check{c}$, Simple multipliers on Banach modules, Glasgow Math. J., 45 (2003), 309-322.
  • H. G. Dales, Banach algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.
  • J. J. Font, Automatic Continuity of certain isomorphisms between regular Banach function algebras, Glasgow Math. J., 39 (1997), 333-343.
  • E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag, 1963.
  • K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull., 33 (1990), 139-144.
  • J. S. Jeang and N. C. Wong, Weighted composition operators of $ C_0 (X)$'s, J. Math. Anal. Appl., 201 (1996), 981-993.
  • J. S. Jeang and N. C. Wong, Disjointness preserving Fredholm linear operators of $C_0(X)$, J. Operator Theory, 49 (2003), 61-75.
  • A. Jemenez-Vargas, Disjointness preserving operators between little Lipschitz algebras, J. Math. Anal. Appl., 337 (2008), 984-993.
  • R. Kantrowitz and M. M. Neumann, Disjointness preserving and local operators on algebras of differentiable functions, Glasgow Math. J., 43 (2001), 295-309.
  • J. Lamperti, On the isometries of certain function spaces, Pacific J. Math., 8 (1958), 459-466.
  • K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000.
  • Ch. Leung and N. C. Wong, Zero product preserving linear maps of CCR C$^*$-algebras with Hausdorff spectrum, J. Math. Anal. Appl., 361 (2010), 187-194.
  • L. Mousavi and F. Sady, Banach module valued separating maps and automatic continuity, Bull. Iranian Math. Soc., 37 (2011), 127-139.