Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators

Pirkovskii, A. Yu.

Canadian mathematical bulletin, Tome 47 (2004) no. 3, pp. 445-455

Résumé

For a locally compact group $G$ , the convolution product on the space $N({{L}^{p}}\ (G))$ of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra $N({{L}^{p}}\ (G))$ and relate them to some properties of the group $G$ , such as compactness, finiteness, discreteness, and amenability.

Détail

DOI : 10.4153/CMB-2004-044-6

Mots-clés : 46M10, 46H25, 43A20, 16E65

Bibliographie

[1] [1] Dales, H. G., Banach algebras and automatic continuity. Oxford Univ. Press, New York, 2000. Google Scholar

[2] [2] Diestel, J. and Faires, B., On vector measures. Trans. Amer.Math. Soc. 198 (1974), 253–271. Google Scholar

[3] [3] Diestel, J. and Uhl, J. J. Jr., Vector measures. American Mathematical Society, Providence, RI., 1977. Google Scholar

[4] [4] Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. II. Springer-Verlag, Berlin, 1970. Google Scholar

[5] [5] Helemskii, A. Ya., On a method for calculating and estimating the global homological dimension of Banach algebras. Mat. Sbornik 87 (1972), 122–135 (Russian); English transl.: Math. USSR Sb. 16 (1972), 125–138. Google Scholar

[6] [6] Helemskii, A. Ya., Flat Banach modules and amenable algebras. Trudy MMO 47 (1984), 179–218 (Russian); English transl.: Trans. Moscow Math. Soc. 47 (1985), 199–244. Google Scholar

[7] [7] Helemskii, A. Ya., The homology of Banach and topological algebras. Moscow University Press, 1986 (Russian); English transl.: Kluwer Academic Publishers, Dordrecht, 1989. Google Scholar

[8] [8] Helemskii, A. Ya., Banach and polynormed algebras: general theory, representations, homology. Nauka, Moscow, 1989 (Russian); English transl.: Oxford University Press, 1993. Google Scholar

[9] [9] Johnson, B. E., Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127(1972). Google Scholar

[10] [10] Johnson, W. B. and Lindenstrauss, J., Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, 1–84, North-Holland, Amsterdam, 2001. Google Scholar

[11] [11] Neufang, M., Abstrakte harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren. Dissertation zur Erlangung des Grades des Doktors des Naturwissenschaften, Saarbrücken, 2000. Google Scholar

[12] [12] Runde, V., Lectures on amenability. Lecture Notes in Math. 1774, Springer-Verlag, Berlin, 2002. Google Scholar

[13] [13] Selivanov, Yu. V., Biprojective Banach algebras. Izvestia. Akad. Nauk SSSR ser. mat. 43 (1979), 1159–1174; English transl.: Math. USSR Izvestija 15 (1980), 387–399. Google Scholar

[14] [14] Selivanov, Yu. V.,Weak homological bidimension and its values in the class of biflat Banach algebras. Extracta Math. 11 (1996), 348–365. Google Scholar

[15] [15] Selivanov, Yu. V., Superbiprojective and superbiflat Banach algebras. Unpublished manuscript, Odense, 2001. Google Scholar

[16] [16] Wendel, J. G., Left centralizers and isomorphisms of group algebras. Pacific J.Math. 2 (1952), 251–261. Google Scholar

[17] [17] Wojtaszczyk, P., Banach spaces for analysts. Cambridge University Press, Cambridge, 1991. Google Scholar