×

Dual Fréchet algebras: Connes amenability and \((\sigma wc)\)-virtual diagonals. (English) Zbl 1471.46045

The authors study dual Fréchet algebras. They obtain some elementary results concerning Connes amenability in the Fréchet algebra setting. Connes amenability for von Neumann algebras was studied earlier by B. E. Johnson et al. [Bull. Soc. Math. Fr. 100, 73–96 (1972; Zbl 0234.46066)]. For general dual Banach algebras the theory was set up by V. Runde [Stud. Math. 148, No. 1, 47–66 (2001; Zbl 1003.46028); Math. Scand. 95, No. 1, 124–144 (2004; Zbl 1087.46035)].

MSC:

46H05 General theory of topological algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Z. Alimohammadi and A. Rejali, Fréchet algebras in abstract harmonic analysis, arXiv:1811.10987v1 [math.FA].
[2] Goldmann, H., Uniform Fréchet Algebras, , Vol. 162 (North-Holand, Amesterdam-New York, 1990). · Zbl 0718.46017
[3] Helemskii, A. Y., Homology for the algebras of analysis, in Handbook of Algebra, ed. Hazewinkel, M., Vol. 2 (Amesterdam, North-Holand, 2000), pp. 151-274. · Zbl 0968.46061
[4] Helemskii, A. Y., The Homology of Banach and Topological Algebras (Moscow University Press, English transl: Kluwer Academic Publishers, Dordrecht, 1989).
[5] Honary, T. G., Automatic continuity of homomorphisms between Banach algebras and Fréchet algebras, Bull. Iranian Math. Soc.32(2) (2006) 403-418. · Zbl 1140.46023
[6] Jarchow, H., Locally convex spaces, . B.G. TeubnerStuttgart (1981). · Zbl 0466.46001
[7] E. Kizgut and M. Yurdakul, The existence of a factorized unbounded operator between Fréchet spaces, to appear in, Asian Eur. J. Math. · Zbl 1451.46002
[8] Lawson, P. and Read, C. J., Approximate amenability of Fréchet algebras, Math. Proc. Cambridge Philos. Soc.145 (2008) 403-418. · Zbl 1160.46030
[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). · Zbl 0924.46002
[10] Paterson, A. L. T., Amenability (American Mathematical Society, Providence, Rhode island, 1988).
[11] Pirkovskii, A. Y., Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras, Trans. Moscow Math. Soc. (2008) 27-104. · Zbl 1207.46064
[12] Pirkovskii, A. Y., Flat cyclic Fréchet modules, Amenable Fréchet algebras, and approximate identities, Homology, Homotopy Appl.11/1 (2009) 81-114. · Zbl 1180.46039
[13] Runde, V., Amenability for dual Banach algebras, Studia Math.148 (2001) 47-66. · Zbl 1003.46028
[14] Runde, V., Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand.95 (2004) 124-144. · Zbl 1087.46035
[15] Schaefer, H. H., Topological Vector Spaces, , Vol. 3 (Springer-Verlag, New York, 1971). · Zbl 0217.16002
[16] Smith, H. A., Tensor products of locally convex algebras, Proc. Amer. Math. Soc.17 (1966) 124-132. · Zbl 0151.18503
[17] Taylor, J. L., Homology and cohomology for topological algebras, Adv. Math.9 (1972) 137-182. · Zbl 0271.46040
[18] Voigt, J., Factorization in Fréchet algebras, J. London Math. Soc.29/2 (1984) 147-152. · Zbl 0569.46028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.