Abstract
In this note we discuss some necessary and some sufficient conditions for the relative injectivity of the \(C_0(S)\)-module \(C_0(S)\), where \(S\) is a locally compact Hausdorff space. We also give a Banach module version of Sobczyk’s theorem. The main result of the paper is as follows: if the \(C_0(S)\)-module \(C_0(S)\) is relatively injective, then \(S=\beta(S\setminus \{s\})\) for any limit point \(s\in S\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Hahn, “Über lineare Gleichungssysteme in linearen Räumen”, J. Reine Angew. Math., 157 (1927), 214–229.
S. Banach, “Sur les fonctionnelles linéaires. I”, Stud. Math., 1:1 (1929), 211–216.
S. Banach, “Sur les fonctionnelles linéaires. II”, Stud. Math., 1:1 (1929), 223–239.
J. L. Blasco and C. Ivorra, “On constructing injective spaces of type \(C(K)\)”, Indag. Math., 9 (1998), 161–172.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
M. Hrušák, A. Tamariz-Mascarúa, and M. Tkachenko, Pseudocompact Topological Spaces, Springer, Cham, 2018.
A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Springer, 1989.
A. Avilés, F. C. Sánchez, J. M. F. Castillo, M. González, and Y. Moreno, Separably injective Banach spaces, Lecture Notes in Math., 2132 Springer International Publishing, Switzerland, 2016.
M. Hasumi, “The extension property of complex Banach spaces”, Tohoku Math. J., Second Ser., 10:2 (1958), 135–142.
S. K. Berberian, Baer \(^*\)-rings, Grundlehren der matematischen Wissenschaften, 195 Springer-Verlag, Berlin–Heidelberg, 1972.
M. Takesaki, “On the Hahn–Banach type theorem and the Jordan decomposition of module linear mapping over some operator algebras”, Kodai Math. Semin. Rep., 12 (1960), 1–10.
M. H. Stone, “Applications of the theory of Boolean rings to general topology”, Trans. Amer. Math. Soc., 41:3 (1937), 375–481.
K. Kunen, Set Theory, Studies in Logic, 34 College Publications, London, 2013.
N. J. Fine and L. Gillman, “Extension of continuous functions in \(\beta {\mathbf{N}}\)”, Bull. Amer. Math. Soc., 66 (1960), 376–381.
E. van Douwen, K. Kunen, and J. van Mill, “There can be \(C^*\)-embedded dense proper subspaces in \(\beta\omega-\omega\)”, Proc. Amer. Math. Soc., 105:2 (1989), 462–470.
P. Ramsden, Homological Properties of Semigroup Algebras, The University of Leeds, 2009.
D. Amir, “Projections onto continuous function spaces”, Proc. Amer. Math. Soc., 15:3 (1964), 396–402.
H. Rosenthal, “On relatively disjoint families of measures, with some applications to Banach space theory”, Stud. Math., 37:1 (1970), 13–36.
A. Sobczyk, “Projection of the space (\(m\)) on its subspace (\(c_0\))”, Bull. Amer. Math. Soc., 47 (1941), 938–947.
M. Zippin, “The separable extension problem”, Israel J. Math., 26:3–4 (1977), 372–387.
M. Fabian, P. Habala, P. Hajek, V. Montesinos, and V. Zizler, Banach Space Theory. The Basis for Linear and Non-Linear Analysis, Springer-Verlag, New York, 2011.
A. Grothendieck, “Sur les applications linéaires faiblement compactes d’espaces du type \(C(K)\)”, Canad. J. Math., 5 (1953), 129–173.
N. Dunford and B. J. Pettis, “Linear operations on summable functions”, Trans. Amer. Math. Soc., 47:3 (1940), 323–392.
Funding
This work was supported by the Russian Foundation for Basic Research (grant no. 19-01-00447).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 55-62 https://doi.org/10.4213/faa3889.
Translated by N. T. Nemesh
Rights and permissions
About this article
Cite this article
Nemesh, N.T. A Note on Relatively Injective \(C_0(S)\)-Modules \(C_0(S)\). Funct Anal Its Appl 55, 298–303 (2021). https://doi.org/10.1134/S0016266321040043
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266321040043