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On non-zero values of the central homological dimension of C*-algebras

Published online by Cambridge University Press:  18 May 2009

Z. A. Lykova
Z. A. Lykova
Affiliation:
Algebra and Analysis Department, Moscow Institute of Electronic Machine Building, B. Vuzovskii 3/12, Moscow 109028, Russia
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The paper is related to the area which was recently called topological homology [3, 6, 12, 16, 4]. We consider questions associated with the central Hochschild cohomology of C*-algebras. The study of the latter was begun by J. Phillips and I. Raeburn in [9, 10], when they were investigating some problems of the theory of perturbations of C*-algebras. In [8] we obtained a description of the structure of C*-algebras with central bidimension zero: it was proved that these C*-algebras are unital and have continuous trace. In the special case of separable and a priori unital C*-algebras this statement was proved by J. Phillips and I. Raeburn in [11] with the help of a different approach. The question was raised. Which values can the central bidimension of C*-algebras take? In the present paper it is shown that, for any CCR-algebra A having at least one infinite-dimensional irreducible representation, the central bidimension and the global central homological dimension of A are greater than one. At the same time it is proved that there exist CCR-algebras which are centrally biprojective, but which have both dimensions equal to one. This situation contrasts with the state of affairs in the “traditional” theory of the Banach Hochschild cohomology. Recall [3, Ch. 5] that the bidimension and the global homological dimension of any infinite-dimensional biprojective C*-algebra are equal to two. Besides, there is no CCR-algebra of bidimension one (respectively, global homological dimension one). See [7].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 3 , September 1992 , pp. 369 - 378
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Cartan, H. and Eilenberg, S., Homological algebra. (Princeton University Press, 1956).Google Scholar
2.Dixmier, J., Les C*-algebras et leurs representations, 2nd ed. (Gauthier-Villars, Paris, 1969).Google Scholar
3.Helemskii, A. Ya., Homology in Banach and topological algebras (MSU publishers, Moscow, 1986 (Russian); English (Dordrecht; Cluwer, 1989).Google Scholar
4.Helemskii, A. Ya., Banach and polynormed algebras: general theory, representations, homology (Moscow, Nauka, 1989) (Russian).Google Scholar
5.Helemskii, A. Ya., The periodic product of modules over Banach algebras. Funct. Analysis Appl., 5 (1971), 84–85.CrossRefGoogle Scholar
6.Johnson, B. E., Cohomology in Banach algebras. Mem. Amer. Math. Soc., No 127, (1972).Google Scholar
7.Lykova, Z. A., The lower estimate of the global homological dimension of infinite-dimensional CCR-algebras, Uspekhi Mat. Nauk, 41 (1986), 197–198 (Russian).Google Scholar
8.Lykova, Z. A., Structure of Banach algebras with trivial central cohomology. J. Operator Theory, to appear.Google Scholar
9.Phillips, J. and Raeburn, I., Perturbations of C*-algebras. II. Proc. London Math. Soc. (3), 43 (1981), 46–72.CrossRefGoogle Scholar
10.Phillips, J. and Raeburn, I., Central cohomology of C*-algebras. J. London Math. Soc. (2). 28 (1983), 363–372.CrossRefGoogle Scholar
11.Phillips, J. and Raeburn, I., Voiculescu's double commutant theorem and the cohomology of C*-algebras.—preprint (University of Victoria, Victoria, British Columbia, Canada, 1989).Google Scholar
12.Putinar, M., On analytic modules: softness and quasicoherence, in Proc. 3-rd Conference on Complex Analysis (Varna, 1985).Google Scholar
13.Rieffel, M. A., Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis, 1 (1967) 443–491.CrossRefGoogle Scholar
14.Rickart, C. E., The general theory of Banach algebras (Van Nostrand, 1960).Google Scholar
15.Selivanov, Yu. V., Biprojective Banach algebras, Izv. Akad. Nauk SSSR ser. mat., 43 (1979), 1159–1174 (Russian).Google Scholar
16.Taylor, J. L., Homology and cohomology for topological algebras. Advances in Math., 9 (1972) 137–182.CrossRefGoogle Scholar