On the theme of Le Page

M. Oudadess

doi:10.15330/ms.48.1.82-96

Some Le Page's commutativity criteria (in complex Banach algebras) are considered. We try to reconstruct a brief history of their evolution. Our recent results, in the real case, are also presented. These propositions are referred in the general framework of not-normalized topological algebras.

1. B. Aupetit, Proprietes Spectrales des Algebres de Banach, Lect. Notes Math. 735, Springer-Verlag, 1979.

2. R. Arens, Linear topological division algebras, Bull. Amer. Math. Soc., 53 (1947), 623-630.

3. J.W. Baker, J.S. Pym, A remark on continuous bilinear mappings, Proc. Edinburgh Math. Soc., 17 (1970-1971), №2, 245.248.

4. V.A. Belfi, R.S. Doran, Norm and spectral characterizations in Banach algebras, Lenseignement mathematique, Tome XXXVI (1977), Fasic, 1-2, 103-130.

5. O.H. Cheikh, M. Oudadess, On a commutativity question in Banach algebras, Arab Gulf J. Sci. Res., A6 (1988), .2, 173-179.

6. R. Choukri, A.El Kinani, M. Oudadess, Etude des algebres A verifiant }$xA=Ax$ ou $xAx=x^{2}Ax^{2}$. General topological algebras (Tartu, 1999), 59-71, Math. Stud. (Tartu), 1, Est. Math. Soc., Tartu, 2001.

7. R. Choukri, A.El Kinani, M. Oudadess, On some von Neumann topological algebras.

8. A.C. Cochran, R. Keown, C.R. Williams, On a class of topological algebras, Pacific J. Math., 34 (1970), 17-25.

9. A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc., 41 (1973), 473-479.

10. J. Duncan, A.W. Tullo, Finite dimensionality, nilpotents and quasi-nilpotents in Banach algebras, Proc. Edinburgh Math. Soc. Ser., 21 (1974), 45-49.

11. P.R. Halmos, A Hilbert Space Problem Book, Sec. Edit., Springer-Verlag, 1980.

12. M. Haralampidou, A local characterization of Le Page Conditon. Dayton algebras, Proc. Ictaa 2001, Acta Univ. Ouluensis, Sci. Rer. Nutur. A 408, 100-106.

13. M. Haralampidou, Classification of locally m-convex algebras through Le Page conditon, Ann. Soc. Math. Polonae, Ser. I: Comm. Math. XLIV (2004), .2, 256-269.

14. R.A. Hirschfeld, W. Zelazko, On spectral norm Banach algebras, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 16 (1968), 195-199.

15. A.El Kinani, A. Najmi, M. Oudadess, Inegalite generalisee de Le Page et commutativite dans les algebres p-Banach, Rev. Acad. Cienc. Zaragoza, 55 (2000), 51-57.

16. A.El Kinani, A. Ifzarne, M. Oudadess, Commutativite de certaines algebres de Banach a automorphisme involutif, Rev. Acad. Ciencias Zaragoza, Serie 2, 53 (1998), 165-173.

17. A.El Kinani, M. Oudadess, Unified inequality and commutativity in Banach algebras with or without involution, Pitman Research Notes Math., 377 (1998), 49-55.

18. A.El Kinani, M. Oudadess, Un Crit`ere generalise de Le Page et commutativite, C. R. Math. Acad. Sci. Canada, XVIII (1996), .2-3, 71-74.

19. A.El Kinani, M.Oudadess, Applications bi-semilineaires et commutativite dans les algebres de Banach involutives, Ann. Math. Blaise Pascal, 6 (1999), .2, 15-20.

20. A.El Kinani, A. Najmi, M. Oudadess, Algebres de Banach bilaterales, Bull. Greek Math. Soc., 45 (2001), 17-29.

21. A.El Kinani, A. Najmi, M. Oudadess, One sided Banach algebras, Turk. J. Math., 26 (2002), 305-316.

22. J. Esterle, M. Oudadess, Structure of Banach algebras satisfing $Ax^{2}=x^{2}A$ for every $x$ in $A$, Proc. Amer. Math. Soc., 96 (1986), №1, 91-94.

23. B. Fuglede, A commutativity theorem for normal operators, Proc. N.A.S., 36 (1950), 35-40.

24. D.C. Kleinke, On operator commutaters, Proc. Amer. Math. Soc., 8 (1957), 535-536.

25. C.Le Page, Sur quelques conditions entranant la commutativite dans les algebres de Banach, C. R. Acad. Sci. Paris Ser A-B, 265 (1967), 235-237.

26. A. Mallios, Topological algebras. Selected topics, North-Holland, Amsterdam, 1986.

27. A. Mallios, Geometry of Vector Shieves. An axiomatique Approach to Differential Geometry, Vols. I (Chapts I-V), II (Chapts VI-XI), Kluwer, Dordrecht (1998).

28. A. Mallios, Moderne Differential Geometry in Gauge Theory: Maxwell Fields, V.I, Yang-Mills Fields, V.II, Birkhauser, Boston (2006-2007).

29. T. Ogasawara, A Theorem on operator algebras, J. Sci. Hiroshima Univ., 18, (1955), №3, 307-309.

30. M. Oudadess, Y. Tsertos, Commutativity results in non unital real topological algebras, Applications and applied Math., 7 (2012), №1, 164-174.

31. M. Oudadess, Commutativity conditions in algebras with $C^*$-equalities, Communications in Mathematics, 3 (2012), №1, 61-73.

32. M. Oudadess, Theorem of Gelfand-Mazur and commutativity in unital real topologial algebras, Mediterranean J. Math. Med. J. Math., 8 (2011), 137.151.

33. M. Oudadess, Commutativite de certaines algebres de Banach, Bol. Soc. Mat. Mexicana, 28 (1983), №1, 9-14.

34. M. Oudadess, Commutativity of some A-convex algebras, J. Univ. Kuwait (Sci.), 14 (1987), 205-210.

35. T. Ogasawara, A Theorem on operator algebras, J. Sci. Hiroshima Univ., 18 (1955), №3, 307-309.

36. C.R. Putnam, On normal operators in Hilbert spaces, Amer. J. Math., 73 (1951), 357-362.

37. M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc., 33 (1958), 376-377.

38. W. Rudin, Functional Analysis, Mc Graw-Hill, 1973.

39. F.V. Shirokov, Proof of a conjecture of Kaplansky, Uspehi Mat. Nauk, 11 (1956), 167-168.

40. A. Srivastav, Commutativity criteria for real Banach algebras, Arch. Math., 54 (1990), 65-72.

	On the theme of Le Page (in French)
Author	M. Oudadess oudadesshha@gmail.com, oudadessm@yahoo.fr A. El Kinani Ecole Normale Superieure Rabat, Morocco
Abstract	Some Le Page's commutativity criteria (in complex Banach algebras) are considered. We try to reconstruct a brief history of their evolution. Our recent results, in the real case, are also presented. These propositions are referred in the general framework of not-normalized topological algebras.
Keywords	commutativity; Le Page's inequality; real algebras; complex algebras; topological algebras; locally $m$-convex algebras
DOI	doi:10.15330/ms.48.1.82-96
Reference	1. B. Aupetit, Proprietes Spectrales des Algebres de Banach, Lect. Notes Math. 735, Springer-Verlag, 1979. 2. R. Arens, Linear topological division algebras, Bull. Amer. Math. Soc., 53 (1947), 623-630. 3. J.W. Baker, J.S. Pym, A remark on continuous bilinear mappings, Proc. Edinburgh Math. Soc., 17 (1970-1971), №2, 245.248. 4. V.A. Belfi, R.S. Doran, Norm and spectral characterizations in Banach algebras, Lenseignement mathematique, Tome XXXVI (1977), Fasic, 1-2, 103-130. 5. O.H. Cheikh, M. Oudadess, On a commutativity question in Banach algebras, Arab Gulf J. Sci. Res., A6 (1988), .2, 173-179. 6. R. Choukri, A.El Kinani, M. Oudadess, Etude des algebres A verifiant }$xA=Ax$ ou $xAx=x^{2}Ax^{2}$. General topological algebras (Tartu, 1999), 59-71, Math. Stud. (Tartu), 1, Est. Math. Soc., Tartu, 2001. 7. R. Choukri, A.El Kinani, M. Oudadess, On some von Neumann topological algebras. 8. A.C. Cochran, R. Keown, C.R. Williams, On a class of topological algebras, Pacific J. Math., 34 (1970), 17-25. 9. A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc., 41 (1973), 473-479. 10. J. Duncan, A.W. Tullo, Finite dimensionality, nilpotents and quasi-nilpotents in Banach algebras, Proc. Edinburgh Math. Soc. Ser., 21 (1974), 45-49. 11. P.R. Halmos, A Hilbert Space Problem Book, Sec. Edit., Springer-Verlag, 1980. 12. M. Haralampidou, A local characterization of Le Page Conditon. Dayton algebras, Proc. Ictaa 2001, Acta Univ. Ouluensis, Sci. Rer. Nutur. A 408, 100-106. 13. M. Haralampidou, Classification of locally m-convex algebras through Le Page conditon, Ann. Soc. Math. Polonae, Ser. I: Comm. Math. XLIV (2004), .2, 256-269. 14. R.A. Hirschfeld, W. Zelazko, On spectral norm Banach algebras, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 16 (1968), 195-199. 15. A.El Kinani, A. Najmi, M. Oudadess, Inegalite generalisee de Le Page et commutativite dans les algebres p-Banach, Rev. Acad. Cienc. Zaragoza, 55 (2000), 51-57. 16. A.El Kinani, A. Ifzarne, M. Oudadess, Commutativite de certaines algebres de Banach a automorphisme involutif, Rev. Acad. Ciencias Zaragoza, Serie 2, 53 (1998), 165-173. 17. A.El Kinani, M. Oudadess, Unified inequality and commutativity in Banach algebras with or without involution, Pitman Research Notes Math., 377 (1998), 49-55. 18. A.El Kinani, M. Oudadess, Un Crit`ere generalise de Le Page et commutativite, C. R. Math. Acad. Sci. Canada, XVIII (1996), .2-3, 71-74. 19. A.El Kinani, M.Oudadess, Applications bi-semilineaires et commutativite dans les algebres de Banach involutives, Ann. Math. Blaise Pascal, 6 (1999), .2, 15-20. 20. A.El Kinani, A. Najmi, M. Oudadess, Algebres de Banach bilaterales, Bull. Greek Math. Soc., 45 (2001), 17-29. 21. A.El Kinani, A. Najmi, M. Oudadess, One sided Banach algebras, Turk. J. Math., 26 (2002), 305-316. 22. J. Esterle, M. Oudadess, Structure of Banach algebras satisfing $Ax^{2}=x^{2}A$ for every $x$ in $A$, Proc. Amer. Math. Soc., 96 (1986), №1, 91-94. 23. B. Fuglede, A commutativity theorem for normal operators, Proc. N.A.S., 36 (1950), 35-40. 24. D.C. Kleinke, On operator commutaters, Proc. Amer. Math. Soc., 8 (1957), 535-536. 25. C.Le Page, Sur quelques conditions entranant la commutativite dans les algebres de Banach, C. R. Acad. Sci. Paris Ser A-B, 265 (1967), 235-237. 26. A. Mallios, Topological algebras. Selected topics, North-Holland, Amsterdam, 1986. 27. A. Mallios, Geometry of Vector Shieves. An axiomatique Approach to Differential Geometry, Vols. I (Chapts I-V), II (Chapts VI-XI), Kluwer, Dordrecht (1998). 28. A. Mallios, Moderne Differential Geometry in Gauge Theory: Maxwell Fields, V.I, Yang-Mills Fields, V.II, Birkhauser, Boston (2006-2007). 29. T. Ogasawara, A Theorem on operator algebras, J. Sci. Hiroshima Univ., 18, (1955), №3, 307-309. 30. M. Oudadess, Y. Tsertos, Commutativity results in non unital real topological algebras, Applications and applied Math., 7 (2012), №1, 164-174. 31. M. Oudadess, Commutativity conditions in algebras with $C^*$-equalities, Communications in Mathematics, 3 (2012), №1, 61-73. 32. M. Oudadess, Theorem of Gelfand-Mazur and commutativity in unital real topologial algebras, Mediterranean J. Math. Med. J. Math., 8 (2011), 137.151. 33. M. Oudadess, Commutativite de certaines algebres de Banach, Bol. Soc. Mat. Mexicana, 28 (1983), №1, 9-14. 34. M. Oudadess, Commutativity of some A-convex algebras, J. Univ. Kuwait (Sci.), 14 (1987), 205-210. 35. T. Ogasawara, A Theorem on operator algebras, J. Sci. Hiroshima Univ., 18 (1955), №3, 307-309. 36. C.R. Putnam, On normal operators in Hilbert spaces, Amer. J. Math., 73 (1951), 357-362. 37. M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc., 33 (1958), 376-377. 38. W. Rudin, Functional Analysis, Mc Graw-Hill, 1973. 39. F.V. Shirokov, Proof of a conjecture of Kaplansky, Uspehi Mat. Nauk, 11 (1956), 167-168. 40. A. Srivastav, Commutativity criteria for real Banach algebras, Arch. Math., 54 (1990), 65-72.
Pages	82-96
Volume	48
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the theme of Le Page (in French)