Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1. C. 18-28. DOI: 10.26117/2079-6641-2019-26-1-18-28

Содержание

DOI: 10.26117/2079-6641-2019-26-1-18-28

MSC 43A22, 16W20

HYPER-TAUBERIAN ALGEBRAS DEFINED BY A BANACH ALGEBRA HOMOMORPHISM

A. Ebadian¹, A. Jabbari²

¹Department of Mathematics, Urmia University, Iran
²Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran

E-mail: jabbarial@yahoo.com, ali.jabbari@iauardabil.ac.ir

Let A and B be Banach algebras and T: B→A be a continuous homomorphism. We consider left multipliers from A×TB into its the first dual i.e., A*×B* and we show that A×TB is a hyper-Tauberian algebra if and only if A and B are hyper-Tauberian algebras.

Keywords: Local operator, hyper-Tauberian algebra, Tauberian algebra

© Ebadian A., Jabbari A., 2019

УДК 519.53

ГИПЕРТАУБЕРОВЫ АЛГЕБРЫ, ОПРЕДЕЛЕННЫЕ ГОМОМОРФИЗМОМ БАНАХОВОЙ АЛГЕБРЫ

А. Ебадиан¹, A. Джаббари²

¹Математический факультет, Университет Урмия, Иран и Клуб молодых исследователей и элит, Ардебильский филиал
²Исламский университет Азад, Ардебиль, Иран

E-mail: jabbarial@yahoo.com, ali.jabbari@iauardabil.ac.ir

Пусть A и B – банаховы алгебры, а T: B→A – непрерывный гомоморфизм. Мы рассматриваем левые мультипликаторы из A×TB в его первое двойственное, т.е. A*×B*, и показываем, что A×TB является гипертауберовой алгеброй тогда и только тогда, когда A и B являются гипертауберовыми алгебрами.

Ключевые слова: локальный оператор, гипертауберова алгебра, тауберова алгебра.

© Ебадиан А., Жаббари A., 2019

Список литературы/References

  1. Abtahi F., Ghafarpanah A., Rejali A.“Biprojectivity and biflatness of Lau product Banachalgebras defined by a Banach algebra morphism”, Bull. Aust. Math. Soc., 91:1 (2015), 134-144 https://doi.org/10.1017/S0004972714000483.
  2. Abtahi F., Ghafarpanah A., “A note on cyclic amenability of the Lau product Banach algebras defined by a Banach algebra morphism”, Bull. Aust. Math. Soc., 92:2 (2015), 282-289 https://doi.org/10.1017/S0004972715000544.
  3. Bade W. G., Curtis P. C., Dales H. G., “Amenability and weak amenability for Beurling and Lipschitz algebras”, Proc. London Math. Soc., 55:2 (1987), 359-377 https://doi.org/10.1093/plms/s3-55-2.359.
  4. Bagheri A., Haghnejad Azar K., Jabbari A., “Arens regularity of module actions and weak amenability of Banach algebras”, Periodica Math. Hung., 71:2 (2015), 224-235 https://doi.org/10.1007/s10998-015-0103-2.
  5. Bhatt S. J., Dabhi P. A., “Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism”, Bull. Aust. Math. Soc., 2013, №87, 195-206 https://doi.org/10.1017/S000497271200055X.
  6. Dabhi P.A., Jabbari A., Haghnejad Azar K., “Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism”, Acta Math. Sinica, English Series, 31:9 (2015), 1461-1474 https://doi.org/10.1007/s10114-015-4429-8.
  7. Ghaderi E., Nasr-Isfahani R., Nemati M., “Some notions of amenability for certain products of Banach algebras”, Colloquium Math., 130:2 (2013), 147-157 https://doi.org/10.4064/cm130-2-1.
  8. Gourdeau F., “Amenability and the second dual of a Banach algebras”, Studia Math., 125:1 (1997), 75-81 https://doi.org/10.4064/sm-125-1-75-81.
  9. Javanshiri H., Nemati M. On a certain product of Banach algebras and some of its properties, Proc. Rom. Acad. Ser. A, 15:3 (2014), 219-227 https://acad.ro/sectii2002/proceedings/doc2014-3/01-Nemati.pdf.
  10. Johnson B.E., “Cohomology in Banach algebras”, Mem. Amer. Math. Soc., 127 (1972).
  11. Johoson B.E., “Weak amenability of group algebras”, Bull. Lond. Math. Soc., 1991, №23(3), 281–284 https://doi.org/10.1112/blms/23.3.281.
  12. Johnson B. E., “Local derivations on C*-algebras are derivations”, Trans. Amer. Math. Soc., 353:1 (2001), 313-325 https://doi.org/10.1090/S0002-9947-00-02688-X.
  13. Kadison R.V., “Local derivation”, J. Algebra, 130:2 (1990), 494-509 https://doi.org/10.1016/0021-8693(90)90095-6.
  14. Kelly J. L., General topology, American Book, Van Nostrand, Reinhold, 1969.
  15. Lau A. T-M., “Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups”, Fund. Math., 118 (1983), 161–175 https://doi.org/10.4064/fm-118-3-161-175.
  16. Monfared M.S., “On certain products of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups”, Studia Math., 2007, №178(3), 277-294 https://doi.org/10.4064/sm178-3-4.
  17. Nemati M., Javanshiri H., “Some homological and cohomological notions on TLau product of Banach algebras”, Banach J. Math. Anal., 2015, №9(2), 183-195 https://doi.org/10.15352/bjma/09-2-13.
  18. Ramezanpour M., “Weak amenability of the Lau product of Banach algebras defined by a Banach algebra morphism”, Bull. Korean Math. Soc., 2017, №54(6), 1991-1999. https://doi.org/10.4134/BKMS.b160690.
  19. Rickart C. E., General theory of Banach algebra, Van Nostrand, Princeton, 1960.
  20. Samei E. Hyper-Tauberian algebras and weak amenability of Figa-Talamanca-Herz algebras, J. Func. Anal., 2006, №231(1), 195-220 https://doi.org/10.1016/j.jfa.2005.05.005.
  21. Samei E., “Local properties of the Hochschild cohomology of C* -algebras”, J. Aust. Math. Soc., 2008, №84, 117-130. https://doi.org/10.1017/S1446788708000049.
  22. Walter M. E., “W*-algebras and nonabelian harmonic analysis”, J. Func. Anal., 1972, no, 11, 17-38 https://doi.org/10.1016/0022-1236(72)90077-8.
  23. Zhang Y., “Weak amenability of module extensions of Banach algebras Trans.”, Amer. Math. Soc., 354(10) (2002), 4131-4151 https://doi.org/10.1090/S0002-9947-02-03039-8.

Список литературы (ГОСТ)

  1. Abtahi F., Ghafarpanah A., Rejali A. Biprojectivity and biflatness of Lau product Banach algebras defined by a Banach algebra morphism // Bull. Aust. Math. Soc. 2015. vol. 91. no. 1. pp. 134-144. https://doi.org/10.1017/S0004972714000483
  2. Abtahi F., Ghafarpanah A. A note on cyclic amenability of the Lau product Banach algebras defined by a Banach algebra morphism // Bull. Aust. Math. Soc. 2015. vol. 92. no. 2. pp. 282-289. https://doi.org/10.1017/S0004972715000544
  3. Bade W. G., Curtis P. C., Dales H. G. Amenability and weak amenability for Beurling and Lipschitz algebras // Proc. London Math. Soc. 1987. vol. 55. no. 2. pp. 359-377. https://doi.org/10.1093/plms/s3-55-2.359
  4. Bagheri A., Haghnejad Azar K., Jabbari A. Arens regularity of module actions and weak amenability of Banach algebras // Periodica Math. Hung. 2015. vol. 71. no. 2. pp. 224-235. https://doi.org/10.1007/s10998-015-0103-2
  5. Bhatt S. J., Dabhi P. A. Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism // Bull. Aust. Math. Soc. 2013. no. 87. pp. 195-206. https://doi.org/10.1017/S000497271200055X
  6. Dabhi P.A., Jabbari A., Haghnejad Azar K. Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism // Acta Math. Sinica, English Series. 2015. vol. 31. no. 9. pp. 1461-1474. https://doi.org/10.1007/s10114-015-4429-8
  7. Ghaderi E., Nasr-Isfahani R., Nemati M. Some notions of amenability for certain
    products of Banach algebras // Colloquium Math. 2013. vol. 130. no. 2. pp. 147-157. https://doi.org/10.4064/cm130-2-1
  8. Gourdeau F. Amenability and the second dual of a Banach algebras // Studia Math. 1997. vol. 125. no. 1. pp. 75-81. https://doi.org/10.4064/sm-125-1-75-81
  9. Javanshiri H., Nemati M. On a certain product of Banach algebras and some of its properties // Proc. Rom. Acad. Ser. A. 2014. vol. 15. no. 3. pp. 219-227. https://acad.ro/sectii2002/proceedings/doc2014-3/01-Nemati.pdf
  10. Johnson B.E. Cohomology in Banach algebras // Mem. Amer. Math. Soc. vol. 127. 1972.
  11. Johoson B.E. Weak amenability of group algebras // Bull. Lond. Math. Soc. 1991. no. 23(3). pp. 281–284. https://doi.org/10.1112/blms/23.3.281
  12. Johnson B. E. Local derivations on C*-algebras are derivations // Trans. Amer. Math. Soc. 2001. vol. 353. no. 1. pp. 313-325. https://doi.org/10.1090/S0002-9947-00-02688-X
  13. Kadison R.V. Local derivation // J. Algebra. 1990. vol. 130. no. 2. pp. 494-509. https://doi.org/10.1016/0021-8693(90)90095-6
  14. Kelly J. L. General topology. Van Nostrand, Reinhold: American Book, 1969.
  15. Lau A. T-M. Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups // Fund. Math. 1983. no. 118. pp. 161–175. https://doi.org/10.4064/fm-118-3-161-175
  16. Monfared M.S. On certain products of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups // Studia Math. 2007. no. 178(3). pp. 277-294. https://doi.org/10.4064/sm178-3-4
  17. Nemati M., Javanshiri H. Some homological and cohomological notions on T-Lau product of Banach algebras // Banach J. Math. Anal. 2015. no. 9(2). pp. 183-195. https://doi.org/10.15352/bjma/09-2-13
  18. Ramezanpour M. Weak amenability of the Lau product of Banach algebras defined by a Banach algebra morphism // Bull. Korean Math. Soc. 2017. no. 54(6). pp. 1991-1999. https://doi.org/10.4134/BKMS.b160690
  19. Rickart C. E. General theory of Banach algebra. Van Nostrand, Princeton, 1960.
  20. Samei E. Hyper-Tauberian algebras and weak amenability of Figa-Talamanca-Herz algebras // J. Func. Anal. 2006. no. 231(1). pp. 195-220. https://doi.org/10.1016/j.jfa.2005.05.005
  21. Samei E. Local properties of the Hochschild cohomology of C*-algebras // J. Aust. Math. Soc. 2008. no. 84. pp. 117-130. https://doi.org/10.1017/S1446788708000049
  22. Walter M. E. W*-algebras and nonabelian harmonic analysis // J. Func. Anal. 1972. no. 11. pp. 17-38. https://doi.org/10.1016/0022-1236(72)90077-8
  23. Zhang Y.Weak amenability of module extensions of Banach algebras Trans. // Amer. Math. Soc. 2002. no. 354(10). pp. 4131-4151. https://doi.org/10.1090/S0002-9947-02-03039-8

Для цитирования: Ebadian A., Jabbari A. Hyper-Tauberian algebras defined by a Banach algebra homomorphism // Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1. C. 18-28. DOI: 10.26117/2079-6641-2019-26-1-18-28.
For citation: Ebadian A., Jabbari A. Hyper-Tauberian algebras defined by a Banach algebra homomorphism, Vestnik KRAUNC. Fiz.-mat. nauki. 2019, 26: 1, 18-28. DOI: 10.26117/2079-6641-2019-26-1-18-28.

Поступила в редакцию / Original article submitted: 01.03.2019

Ebad

  Ебадиан Али – кандидат физико-математических наук, профессор кафедры математики университета Урмия, Иран.

    Ebadian Ali — PhD. (Phys.& Math), Professor of Department of Mathematics, Urmia University, Iran.

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 ДжаббJabари Али – кандидат физико-математических наук, Клуб молодых исследователей и элит, Ардебильский филиал, Исламский университет Азад, Ардебиль, Иран.
   Jabbari Ali — PhD. (Phys.& Math), Researcher, Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Iran.

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