Research ArticleNo Access

Dual Fréchet algebras: Connes amenability and ( $σ w c$ )-virtual diagonals

Z. Alimohammadi

Department of Mathematics, University of Isfahan, Isfahan, Iran

E-mail Address: z.alimohammadi@sci.ui.ac.ir

Search for more papers by this author

and

A. Rejali

Department of Mathematics, University of Isfahan, Isfahan, Iran

E-mail Address: rejali@sci.ui.ac.ir

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1793557120501545Cited by:0 (Source: Crossref)

Abstract

A relation between Connes amenability of dual Banach algebras and the existence of ( $σ w c$ )-virtual diagonals was investigated by Volker Runde. In this paper, we first introduce and study the notion of dual Fréchet algebras. We then study the weak $^{*}$ -weakly continuous elements of Fréchet bimodules, and the notion of normality for dual spaces of Fréchet bimodules. We obtain some important results about these notions. Finally, we introduce and study amenability for dual Fréchet algebras and investigate the relation between this notion and the existence of ( $σ w c$ )-virtual diagonals.

Communicated by D. Jornet

Keywords:

AMSC: 46H05, 46H25

References

1. Z. Alimohammadi and A. Rejali, Fréchet algebras in abstract harmonic analysis, arXiv:1811.10987v1 [math.FA]. Google Scholar
2. H. Goldmann , Uniform Fréchet Algebras, North-Holland Mathematics Studies, Vol. 162 (North-Holand, Amesterdam-New York, 1990). Google Scholar
3. A. Y. Helemskii , Homology for the algebras of analysis, in Handbook of Algebra, ed. M. Hazewinkel , Vol. 2 (Amesterdam, North-Holand, 2000), pp. 151–274. Crossref, Google Scholar
4. A. Y. Helemskii , The Homology of Banach and Topological Algebras (Moscow University Press, English transl: Kluwer Academic Publishers, Dordrecht, 1989). Crossref, Google Scholar
5. T. G. Honary , Automatic continuity of homomorphisms between Banach algebras and Fréchet algebras, Bull. Iranian Math. Soc. 32(2) (2006) 403–418. Google Scholar
6. H. Jarchow , Locally convex spaces, Mathematische Leitfäden. B.G. Teubner Stuttgart (1981). Crossref, Google Scholar
7. E. Kizgut and M. Yurdakul, The existence of a factorized unbounded operator between Fréchet spaces, to appear in, Asian Eur. J. Math. Google Scholar
8. P. Lawson and C. J. Read , Approximate amenability of Fréchet algebras, Math. Proc. Cambridge Philos. Soc. 145 (2008) 403–418. Crossref, Web of Science, Google Scholar
9. R. Meise and D. Vogt, Introduction to Functional Analysis, Translated from the German by M. S. Ramanujan and revised by the authors (Oxford Graduate Texts Mathematics, Clarendon Press, New York, 1997). Google Scholar
10. A. L. T. Paterson , Amenability (American Mathematical Society, Providence, Rhode island, 1988). Crossref, Google Scholar
11. A. Y. Pirkovskii , Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras, Trans. Moscow Math. Soc. (2008) 27–104. Crossref, Google Scholar
12. A. Y. Pirkovskii , Flat cyclic Fréchet modules, Amenable Fréchet algebras, and approximate identities, Homology, Homotopy Appl. 11/1 (2009) 81–114. Crossref, Web of Science, Google Scholar
13. V. Runde , Amenability for dual Banach algebras, Studia Math. 148 (2001) 47–66. Crossref, Web of Science, Google Scholar
14. V. Runde , Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand. 95 (2004) 124–144. Crossref, Web of Science, Google Scholar
15. H. H. Schaefer , Topological Vector Spaces, Third printing corrected Graduate text in mathematics, Vol. 3 (Springer-Verlag, New York, 1971). Crossref, Google Scholar
16. H. A. Smith , Tensor products of locally convex algebras, Proc. Amer. Math. Soc. 17 (1966) 124–132. Crossref, Web of Science, Google Scholar
17. J. L. Taylor , Homology and cohomology for topological algebras, Adv. Math. 9 (1972) 137–182. Crossref, Web of Science, Google Scholar
18. J. Voigt , Factorization in Fréchet algebras, J. London Math. Soc. 29/2 (1984) 147–152. Crossref, Web of Science, Google Scholar