# Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

Open Mathematics (2008)

- Volume: 6, Issue: 3, page 405-421
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topZinaida Lykova. "Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras." Open Mathematics 6.3 (2008): 405-421. <http://eudml.org/doc/269245>.

@article{ZinaidaLykova2008,

abstract = {We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \[ \hat\{\otimes \}\]
-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H n(ϕ): H n(x) → H n (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \[ \hat\{\otimes \}\]
-algebras: the tensor algebra E \[ \hat\{\otimes \}\]
F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε*(G) on a compact Lie group G.},

author = {Zinaida Lykova},

journal = {Open Mathematics},

keywords = {Cyclic cohomology; Hochschild cohomology; nuclear DF-spaces; locally convex algebras; nuclear Fréchet algebra; cyclic cohomology},

language = {eng},

number = {3},

pages = {405-421},

title = {Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras},

url = {http://eudml.org/doc/269245},

volume = {6},

year = {2008},

}

TY - JOUR

AU - Zinaida Lykova

TI - Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

JO - Open Mathematics

PY - 2008

VL - 6

IS - 3

SP - 405

EP - 421

AB - We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \[ \hat{\otimes }\]
-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H n(ϕ): H n(x) → H n (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \[ \hat{\otimes }\]
-algebras: the tensor algebra E \[ \hat{\otimes }\]
F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε*(G) on a compact Lie group G.

LA - eng

KW - Cyclic cohomology; Hochschild cohomology; nuclear DF-spaces; locally convex algebras; nuclear Fréchet algebra; cyclic cohomology

UR - http://eudml.org/doc/269245

ER -

## References

top- [1] Brodzki J., Lykova Z.A., Excision in cyclic type homology of Fréchet algebras, Bull. London Math. Soc., 2001, 33, 283–291 http://dx.doi.org/10.1017/S0024609301007998 Zbl1032.46094
- [2] Connes A., Noncommutative geometry, Academic Press, San Diego, CA, 1994
- [3] Cuntz J., Cyclic theory and the bivariant Chern-Connes character, In: Noncommutative geometry, Lecture Notes in Math., Springer, Berlin, 2004, 1831, 73–135 Zbl1053.46047
- [4] Cuntz J., Quillen D., Operators on noncommutative differential forms and cyclic homology, In: Geometry, topology and physics, Conf. Proc. Lecture Notes Geom. Topology VI, Int. Press, Cambridge, MA, 1995, 77–111 Zbl0865.18009
- [5] Grothendieck A., Sur les espaces (F) et (DF), Summa Brasil. Math., 1954, 3, 57–123 (in French)
- [6] Grothendieck A., Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., 1955, 16
- [7] Helemskii A.Ya., The homology of Banach and topological algebras, Kluwer Academic Publishers Group, Dordrecht, 1989
- [8] Helemskii A.Ya., Banach cyclic (co)homology and the Connes-Tzygan exact sequence, J. London Math. Soc., 1992, 46, 449–462 http://dx.doi.org/10.1112/jlms/s2-46.3.449 Zbl0816.46076
- [9] Helemskii A.Ya., Banach and polynormed algebras. General theory, representations, homology, Oxford University Press, Oxford, 1992
- [10] Husain T., The open mapping and closed graph theorems in topological vector spaces, Friedr. Vieweg and Sohn, Braunschweig, 1965 Zbl0124.06301
- [11] Jarchow H., Locally convex spaces, B.G. Teubner, Stuttgart, 1981
- [12] Johnson B.E., Cohomology in Banach algebras, Mem. Amer. Math. Soc., 1972, 127 Zbl0256.18014
- [13] Kassel C., Cyclic homology, comodules, and mixed complexes, J. Algebra, 1987, 107, 195–216 http://dx.doi.org/10.1016/0021-8693(87)90086-X
- [14] Khalkhali M., Algebraic connections, universal bimodules and entire cyclic cohomology, Comm. Math. Phys., 1994, 161, 433–446 http://dx.doi.org/10.1007/BF02101928 Zbl0792.46034
- [15] Köthe G., Topological vector spaces I, Die Grundlehren der mathematischen Wissenschaften, 159, Springer-Verlag New York Inc., New York, 1969
- [16] Loday J.-L., Cyclic Homology, Springer Verlag, Berlin, 1992
- [17] Lykova Z.A., Cyclic cohomology of projective limits of topological algebras, Proc. Edinburgh Math. Soc., 2006, 49, 173–199 http://dx.doi.org/10.1017/S0013091504000410 Zbl1105.46051
- [18] Lykova Z.A., Cyclic-type cohomology of strict inductive limits of Fréchet algebras, J. Pure Appl. Algebra, 2006, 205, 471–497 http://dx.doi.org/10.1016/j.jpaa.2005.07.014 Zbl1105.46050
- [19] Lykova Z.A., White M.C., Excision in the cohomology of Banach algebras with coefficients in dual bimodules, In: Albrecht E., Mathieu M. (Eds.), Banach Algebras’97, Walter de Gruyter Publishers, Berlin, 1998, 341–361 Zbl0922.46062
- [20] Meise R., Vogt D., Introduction to Functional Analysis, Clarendon Press, Oxford, 1997 Zbl0924.46002
- [21] Meyer R., Comparisons between periodic, analytic and local cyclic cohomology, preprint available at http://arxiv.org/abs/math/0205276 v2
- [22] Pietsch A., Nuclear Locally Convex Spaces, Springer Verlag, Berlin, 1972
- [23] Pirkovskii A.Yu., Biprojective topological algebras of homological bidimension 1, J. Math. Sci., 2001, 111, 3476–3495 http://dx.doi.org/10.1023/A:1016058211668
- [24] Pirkovskii A.Yu., Homological bidimension of biprojective topological algebras and nuclearity, Acta Univ. Oulu. Ser Rerum Natur., 2004, 408, 179–196 Zbl1074.46030
- [25] Pták V., On complete topological linear spaces, Czech. Math. J., 1953, 78, 301–364 (in Russian) Zbl0057.09303
- [26] Puschnigg M., Excision in cyclic homology theories, Invent. Math., 2001, 143, 249–323 http://dx.doi.org/10.1007/s002220000105 Zbl0973.19002
- [27] Robertson A.P., Robertson W., Topological vector spaces, Cambridge Univ. Press, 1973
- [28] Selivanov Yu.V., Biprojective Banach algebras, Izv. Akad. Nauk SSSR Ser. Mat., 1980, 15, 387–399 Zbl0447.46042
- [29] Selivanov Yu.V., Cohomology of biflat Banach algebras with coefficients in dual bimodules, Functional Anal. Appl., 1995, 29, 289–291 http://dx.doi.org/10.1007/BF01077480 Zbl0881.46036
- [30] Selivanov Yu.V., Biprojective topological algebras, preprint Zbl1025.46018
- [31] Taylor J.L., A general framework for a multi-operator functional calculus, Adv. Math., 1972, 9, 183–252 http://dx.doi.org/10.1016/0001-8708(72)90017-5
- [32] Treves F., Topological vector spaces distributions and kernels, Academic Press, New York, London, 1967 Zbl0171.10402
- [33] Wodzicki M., Vanishing of cyclic homology of stable C*-algebras, C. R. Acad. Sci. Paris I, 1988, 307, 329–334 Zbl0652.46052

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.