Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-09T00:36:34.442Z Has data issue: false hasContentIssue false
  • English
  • Français

THE NON-COMMUTATIVE SCHWARTZ SPACE IS WEAKLY AMENABLE

Part of: Topological algebras, normed rings and algebras, Banach algebras Linear spaces and algebras of operators Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  10 June 2016

KRZYSZTOF PISZCZEK
KRZYSZTOF PISZCZEK*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, ul. Umultowska 87, 61-614 Poznań, Poland e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show in a straightforward way that the non-commutative Schwartz space is weakly amenable. At the end, we leave an open problem.

MSC classification

Primary: 46H20: Structure, classification of topological algebras 46K05: General theory of topological algebras with involution 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 46H05: General theory of topological algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 437 - 443
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Bost, J.-B., Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs, Invent. Math. 101 (2) (1990), 261–333.CrossRefGoogle Scholar
2. Ciaś, Tomasz, On the algebra of smooth operators, Studia Math. 218 (2) (2013), 145–166.CrossRefGoogle Scholar
3. Cuntz, J., Cyclic theory and the bivariant Chern-Connes character, in Noncommutative geometry, Lecture Notes in Math., vol. 1831 (Doplicher, S. and Longo, R., Editors) (Springer, Berlin, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence 2004), 73–135.Google Scholar
4. Dales, H. G., Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24 (Katavolos, A., Editor) (The Clarendon Press, Oxford University Press, New York, 2000). Oxford Science Publications.Google Scholar
5. Dubin, D. A. and Hennings, M. A., Quantum mechanics, algebras and distributions, Pitman Research Notes in Mathematics Series, vol. 238 (Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990).Google Scholar
6. Effros, E. G. and Webster, C., Operator analogues of locally convex spaces, in Operator algebras and applications (Samos, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 495 (Katavolos, A., Editor) (Kluwer Academic Publisher, Dordrecht, 1997), 163–207.Google Scholar
7. Effros, E. G. and Winkler, S., Matrix convexity: Operator analogues of the bipolar and Hahn-Banach theorems, J. Funct. Anal. 144 (1) (1997), 117–152.CrossRefGoogle Scholar
8. Elliott, G. A., Natsume, T. and Nest, R., Cyclic cohomology for one-parameter smooth crossed products, Acta Math. 160 (3–4) (1988), 285–305.CrossRefGoogle Scholar
9. Fragoulopoulou, M., Topological algebras with involution, North-Holland Mathematics Studies, vol. 200 (Elsevier Science B.V., Amsterdam, 2005).Google Scholar
10. Meise, R. and Vogt, D., Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2 (The Clarendon Press, Oxford University Press, New York, 1997), Translated from the German by M. S. Ramanujan and revised by the authors.CrossRefGoogle Scholar
11. Christopher Phillips, N., K-theory for Fréchet algebras, Internat. J. Math. 2 (1) (1991), 77–129.CrossRefGoogle Scholar
12. Pirkovskii, A. Yu., Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities, Homology, Homotopy Appl. 11 (1) (2009), 81–114.CrossRefGoogle Scholar
13. Piszczek, K., Automatic continuity and amenability in the noncommutative Schwartz space, J. Math. Anal. Appl. 432 (2) (2015), 954–964.CrossRefGoogle Scholar
14. Schweitzer, L. B., Spectral invariance of dense subalgebras of operator algebras, Internat. J. Math. 4 (2) (1993), 289–317.CrossRefGoogle Scholar
15. Żelazko, W., Selected topics in topological algebras, Lectures 1969/1970, Lecture Notes Series, vol. 31 (Matematisk Institut, Aarhus Universitet, Aarhus, 1971).Google Scholar