Abstract
In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology and the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an - or -space all its homologically trivial modules possess the Dunford–Pettis property.
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F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Grad. Texts Math., Vol. 233, Springer (2006).
D. P. Blecher and N. Ozawa, “Real positivity and approximate identities in Banach algebras,” Pacific J. Math., 277, No. 1, 1–59 (2015).
D. P. Blecher and C. J. Read, “Operator algebras with contractive approximate identities,” J. Funct. Anal., 261, No. 1, 188–217 (2011).
J. Bourgain, New Classes of -Spaces, Springer (1981).
J. Bourgain, “On the Dunford–Pettis property,” Proc. Am. Math. Soc., 81, No. 2, 265–272 (1981).
H. G. Dales and M. E. Polyakov, “Homological properties of modules over group algebras,” Proc. London Math. Soc., 89, No. 2, 390–426 (2004).
A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud., Vol. 176, Elsevier (1992).
M. Fabian and P. Habala, Banach Space Theory, Springer (2011).
M. González and J. Gutiérrez, “The Dunford–Pettis property on tensor products,” Math. Proc. Cambridge Philos. Soc., 131, No. 1, 185–192 (2001).
A. W. M. Graven, “Injective and projective Banach modules,” Indag. Math., 82, No. 1, 253–272 (1979).
A. Grothendieck, “Une caractérisation vectorielle-métrique des espaces L 1,” Can. J. Math., 7, 552–561 (1955).
A. Ya. Helemskii, Banach and Polynormed Algebras: General Theory, Representations, Homology [in Russian], Nauka, Moscow (1989).
A. Y. Helemskii, The Homology of Banach and Topological Algebras, Math. Its Appl., Vol. 41, Springer (1989)
A. Ya. Helemskii, “Metric version of flatness and Hahn–Banach type theorems for normed modules over sequence algebras,” Stud. Math., 206, No. 2, 135–160 (2011).
A. Ya. Helemskii, “Metric freeness and projectivity for classical and quantum normed modules,” Sb. Math., 204, No. 7, 1056–1083 (2013).
A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, Transl. Math. Monogr., Vol. 233, Amer. Math. Soc., 2006.
W. B. Johnson and J. Lindenstrauss, Handbook of the Geometry of Banach Spaces, Vol. 2, Elsevier (2001).
G. Köthe, “Hebbare lokalkonvexe Räume,” Math. Ann., 165, No. 3, 181–195 (1966).
H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer (1974).
J. Lindenstrauss and A. Pelczynski, “Absolutely summing operators in -spaces and their applications,” Stud. Math., 29, No. 3, 275–326 (1968).
N. T. Nemesh, “Metrically and topologically projective ideals of Banach algebras,” Math. Notes, 99, No. 4, 524–533 (2016).
J.-P. Pier, Amenable Locally Compact Groups, Wiley–Interscience (1984).
G. Racher, “Injective modules and amenable groups,” Comment. Math. Helv., 88, No. 4, 1023–1031 (2013).
P. Ramsden, Homological Properties of Semigroup Algebras, thesis, University of Leeds (2009).
S. M. Shteiner, “Topological freedom for classical and quantum normed modules,” Vestn. SamGU. Estestvennonauchn. ser., No. 9/1 (110), 49–57 (2013).
C. P. Stegall and J. R. Retherford, “Fully nuclear and completely nuclear operators with applications to - and -spaces,” Trans. Am. Math. Soc., 163, 457–492 (1972).
M. C. White, “Injective modules for uniform algebras,” Proc. London Math. Soc., 3, No. 1, 155–184 (1996).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 3, pp. 161–184, 2016.
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Nemesh, N.T. The Geometry of Projective, Injective, and Flat Banach Modules. J Math Sci 237, 445–459 (2019). https://doi.org/10.1007/s10958-019-04170-8
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DOI: https://doi.org/10.1007/s10958-019-04170-8