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DERIVATIONS OF FRÉCHET NUCLEAR GB$^{\ast }$-ALGEBRAS

Part of: Topological (rings and) algebras with an involution Topological algebras, normed rings and algebras, Banach algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  04 June 2015

M. WEIGT  and
I. ZARAKAS
M. WEIGT
Affiliation:
Department of Mathematics and Applied Mathematics, Nelson Mandela Metropolitan University, Summerstrand Campus (South), Port Elizabeth 6031, South Africa email weigt.martin@gmail.com, Martin.Weigt@nmmu.ac.za
I. ZARAKAS*
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis, Athens 15784, Greece email gzarak@hotmail.gr
*
✉gzarak@hotmail.gr
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Abstract

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It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

Keywords

GB∗ -algebratopological algebraderivationlocally convex bimodulesmooth locally convex bimodule

MSC classification

Primary: 46H35: Topological algebras of operators
Secondary: 46H05: General theory of topological algebras 46H25: Normed modules and Banach modules, topological modules (if not placed in or ) 46H40: Automatic continuity 46K05: General theory of topological algebras with involution 46L05: General theory of $C^*$-algebras 46L10: General theory of von Neumann algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 290 - 301
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Albeverio, S., Ayupov, Sh. A. and Kudaybergenov, K. K., ‘Non commutative Arens algebras and their derivations’, J. Funct. Anal. 253 (2007), 287–302.CrossRefGoogle Scholar
Albeverio, S., Ayupov, Sh. A. and Kudaybergenov, K. K., ‘Structure of derivations on various algebras of measurable operators for type I von Neumann algebras’, J. Funct. Anal. 256 (2009), 2917–2943.CrossRefGoogle Scholar
Allan, G. R., ‘A spectral theory for locally convex algebras’, Proc. Lond. Math. Soc. (3) 15 (1965), 399–421.CrossRefGoogle Scholar
Allan, G. R., ‘On a class of locally convex algebras’, Proc. Lond. Math. Soc. (3) 17 (1967), 91–114.CrossRefGoogle Scholar
Becker, R., ‘Derivations on LMC∗ -algebras’, Math. Nachr. 155 (1992), 141–149.CrossRefGoogle Scholar
Ber, A. F., Chilin, V. I. and Sukochev, F. A., ‘Non-trivial derivations on commutative regular algebras’, Extracta Math. 21 (2006), 107–147.Google Scholar
Bhatt, S. J., ‘A note on generalized B∗ -algebras’, J. Indian Math. Soc. 43 (1979), 253–257.Google Scholar
Bratelli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics I (Springer, New York, 1979).CrossRefGoogle Scholar
Brödel, C. and Lassner, G., ‘Derivationen auf gewissen Op∗ -algebren’, Math. Nachr. 67 (1975), 53–58.CrossRefGoogle Scholar
Dales, H. G., Banach Algebras and Automatic Continuity (Clarendon Press, Oxford, 2000).Google Scholar
Dixon, P. G., ‘Generalized B∗ -algebras’, Proc. Lond. Math. Soc. (3) 21 (1970), 693–715.CrossRefGoogle Scholar
van Eijndhoven, S. J. L. and Kruszynski, P., ‘GB∗ -algebras associated with inductive limits of Hilbert spaces’, Studia Math. 85 (1987), 107–123.CrossRefGoogle Scholar
Fragoulopoulou, M., Topological Algebras with Involution (North-Holland, Amsterdam, 2005).Google Scholar
Fragoulopoulou, M., Inoue, A. and Kürsten, K.-D., ‘On the completion of a C ∗ -normed algebra under a locally convex algebra topology’, Contemp. Math. 427 (2007), 155–166.CrossRefGoogle Scholar
Fragoulopoulou, M., Inoue, A. and Kürsten, K.-D., ‘Old and new results on Allan’s GB∗ -algebras’, Banach Center Publ. 91 (2010), 169–178.CrossRefGoogle Scholar
Fragoulopoulou, M., Inoue, A. and Weigt, M., ‘Tensor products of GB∗ -algebras’, J. Math. Anal. Appl. 420 (2014), 1787–1802.CrossRefGoogle Scholar
Fragoulopoulou, M., Weigt, M. and Zarakas, I., ‘Derivations of locally convex ∗-algebras’, Extracta Math. 26 (2011), 45–60.Google Scholar
Inoue, A. and Ota, S., ‘Derivations on algebras of unbounded operators’, Trans. Amer. Math. Soc. 261 (1980), 567–577.CrossRefGoogle Scholar
Pirkovskii, A. Yu., ‘Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras’, Trans. Moscow Math. Soc. 69 (2008), 27–104.CrossRefGoogle Scholar
Pirkovskii, A. Yu., ‘Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities’, Homology, Homotopy Appl. 11 (2009), 81–114.CrossRefGoogle Scholar
Podara, C., ‘On smooth modules’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 105 (2011), 173–179.CrossRefGoogle Scholar
Ringrose, J. R., ‘Automatic continuity of derivations of operator algebras’, J. Lond. Math. Soc. (2) 5 (1972), 432–438.CrossRefGoogle Scholar
Sakai, S., Operator Algebras in Dynamical Systems: The Theory of Unbounded Derivations in C∗ -Algebras (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
Takesaki, M., ‘On the crossnorm of the direct product of C∗ -algebras’, Tohoku Math. J. 16 (1964), 111–122.CrossRefGoogle Scholar
Takesaki, M., Theory of Operator Algebras I (Springer, New York, 1979).CrossRefGoogle Scholar
Weigt, M., ‘Derivations of 𝜏-measurable operators’, in: Operator Theory: Advances and Applications, 195 (Birkhäuser, Basel, 2009), 273–286.Google Scholar
Weigt, M. and Zarakas, I., ‘A note on derivations of pro-C∗ -algebras into complete locally convex bimodules’, New Zealand J. Math. 41 (2011), 143–152.Google Scholar
Weigt, M. and Zarakas, I., ‘Derivations of generalized B∗ -algebras’, Extracta Math. 28 (2013), 77–94.Google Scholar
Weigt, M. and Zarakas, I., ‘Unbounded derivations of generalized B$^{\ast }$-algebras’, in: Operator Algebras and Mathematical Physics, T. Bhattacharyya and M. A. Dritschel (eds.), Operator Theory: Adv. and Appl. 47 (Springer, Basel AG, Basel, 2015), to appear.Google Scholar
Zarakas, I., ‘Hilbert pro-C∗ -bimodules and applications’, Rev. Roumaine Math. Pures Appl. 17 (2012), 289–310.Google Scholar