On approximation of flat Banach modules by free modules

O. Yu. Aristov

Sbornik. Mathematics, Tome 196 (2005) no. 11, pp. 1553-1583

Résumé

The local structure of flat Banach modules is considered; in particular, it is shown that if a flat module has the approximation property, then it is freely approximable, that is, the identity operator on it is approximated by operators each of which admits factorization through a free Banach module satisfying a natural finiteness condition. Among the maps involved in the factorization, the first is approximately multiplicative up to $\varepsilon$ on compact sets, and the second is exactly a morphism of modules. The properties of freely approximable and approximately projective modules are studied. It is proved that the standard complex for calculating the derived functor Ext is locally asymptotically exact in the first term for an arbitrary second argument if and only if its first argument is a flat Banach module.

Bibliographie

[1] Khelemskii A. Ya., Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo MGU, M., 1986 | MR

[2] Runde V., Lectures on Amenability, Lecture Notes in Math., 1774, Springer-Verlag, Berlin, 2002 | MR | Zbl

[3] Khelemskii A. Ya., “Ploskie banakhovy moduli i amenabelnye algebry”, Trudy MMO, 47, 1984, 179–218 | MR

[4] Sheinberg M. V., “Gomologicheskie svoistva zamknutykh idealov, obladayuschikh ogranichennoi approksimativnoi edinitsei”, Vestn. MGU. Ser. 1. Matem., mekh., 1972, no. 4, 39–45

[5] Pugach L. I., “Proektivnye i ploskie idealy funktsionalnykh algebr, ikh svyaz s analiticheskoi strukturoi”, Matem. zametki, 31:2 (1982), 223–229 | MR | Zbl

[6] Selivanov Yu. V., “Cohomological characterizations of biprojective and biflat Banach algebras”, Monatsh. Math., 128 (1999), 35–60 | DOI | MR | Zbl

[7] Selivanov Yu. V., Kogomologii banakhovykh i blizkikh k nim algebr, Dis. ... dokt. fiz.-matem. nauk, M., 2002

[8] White M. C., “Injective modules for uniform algebras”, Proc. London Math. Soc. (3), 73 (1996), 155–184 | DOI | MR | Zbl

[9] Golovin Yu. O., “Usloviya prostranstvennykh ploskosti i in'ektivnosti nerazlozhimoi CSL-algebry konechnoi shiriny”, Matem. zametki, 63:1 (1998), 9–20 | MR | Zbl

[10] Blackadar B., Kirchberg E., “Generalized inductive limits of finite-dimensional $C^*$-algebras”, Math. Ann., 307 (1997), 343–380 | DOI | MR | Zbl

[11] Gourdeau F., “Amenability and the second dual of Banach algebras”, Stud. Math., 125:1 (1997), 75–81 | MR | Zbl

[12] Khelemskii A. Ya., Banakhovy i polinormirovannye algebry. Obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989 | MR

[13] Selivanov Yu. V., “Proektivnye moduli Freshe so svoistvom approksimatsii”, UMN, 50:1 (1995), 209–210 | MR | Zbl

[14] Johnson B. E., “Approximate diagonals and cohomology of certain annihilator Banach algebras”, Amer. J. Math., 94 (1972), 685–698 | DOI | MR | Zbl

[15] Grønbaek N., “Morita equivalence for Banach algebras”, J. Pure Appl. Algebra, 99 (1995), 183–219 | DOI | MR

[16] Zhang Y., “Nilpotent ideals in a class of Banach algebras”, Proc. Amer. Math. Soc., 127 (1999), 3237–3242 | DOI | MR | Zbl

[17] Ghahramani F., Loy R. J., “Generalized notions of amenability”, J. Funct. Anal., 208 (2004), 229–260 | DOI | MR | Zbl

[18] Lazard D., “Sur les modules plats”, C. R. Acad. Sci. Paris. Sér. I Math., 258 (1964), 6313–6316 | MR | Zbl

[19] Govorov V. E., “O ploskikh modulyakh”, Sib. matem. zhurn., 6:2 (1965), 300–304 | MR | Zbl

[20] Burbaki N., Algebra. Gl. X. Gomologicheskaya algebra, Nauka, M., 1987 | MR

[21] Pich A., Operatornye idealy, Mir, M., 1982 | MR

[22] Khelemskii A. Ya., “Periodicheskoe proizvedenie modulei nad banakhovymi algebrami”, Funkts. analiz i ego prilozh., 5:1 (1971), 95–96 | MR

[23] Helemskii A. Ya., “The spatial flatness and injectiveness of Connes operator algebras”, Extracta Math., 9 (1994), 75–81 | MR | Zbl

[24] Polyakov M. E., “An example of a spatially non-flat von Neumann algebra”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4:1 (2001), 1–9 | DOI | MR | Zbl

[25] Johnson B. E., “Nonamenability of the Fourier algebra of a compact group”, J. London Math. Soc. (2), 50 (1994), 361–374 | MR | Zbl

[26] Robertson A., Robertson V., Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR | Zbl

[27] Reinov O. I., Approksimatsiya operatorov v normirovannykh operatornykh idealakh, Dis....dokt. fiz.-matem. nauk, S.-Pb., 2003

[28] Zhang Y., “Unbounded approximate identities in algebras of compact operators on Banach spaces”, Proc. Cambridge Philos. Soc., 134 (2002), 187–192 | MR

[29] Zhang Y., “Approximate identities for ideals of Segal algebras on a compact group”, J. Funct. Anal., 191 (2002), 123–131 | DOI | MR | Zbl

[30] Selivanov Yu. V., Superbiprojective and superbiflat Banach algebras, Preprint, Odense, 2001

[31] Grønbaek N., Johnson B. E., Willis G. A., “Amenability of Banach algebras of compact operators”, Israel J. Math., 87 (1994), 289–324 | DOI | MR

[32] Cigler J., Losert V., Michor P., Banach modules and functors on categories of Banach spaces, Marcel Dekker, New York, 1979 | MR | Zbl

[33] Aristov O. Yu., “Amenability and compact type for Hopf-von Neumann algebras from the homological point of view”, Banach algebras and their applications, Proceeding of the 16th International Conference (University of Alberta, Edmonton, Canada, Jule 27–August 9, 2003), Contemp. Math., 363, Amer. Math. Soc., Providence, RI, 2004, 15–37 | MR | Zbl

[34] Baehr H., Stability of continuous cyclic cohomology and operator ideals on Hilbert space, Schriftenreihe des Mathematischen Instituts der Universität Münster, 3, Ser. 30, 2002 | MR

[35] Cuntz J., Quillen D. G., “Algebra extensions and nonsingularity”, J. Amer. Math. Soc., 8 (1995), 251–289 | DOI | MR | Zbl

[36] Lindenstrauss J., Tsafriri L., Classical Banach spaces, Lecture Notes in Math., 338, Springer-Verlag, Berlin, 1973 | MR | Zbl