Skip to main content

Multiplication Algebras: Algebraic and Analytic Aspects

  • Conference paper
  • First Online:
Associative and Non-Associative Algebras and Applications (MAMAA 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 311))

Included in the following conference series:

Abstract

Applications of multiplication algebras to the algebraic and analytic strengthenings of primeness and semiprimeness of (possibly non-associative) algebras are fully surveyed, and complete normed complex algebras whose closed multiplication algebras satisfy the von Neumann inequality are studied in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Albert, A.A.: The radical of a non-associative algebra. Bull. Am. Math. Soc. 48, 891–897 (1942)

    Article  MathSciNet  Google Scholar 

  2. Ara, P.: The extended centroid of \(C^{*}\)-algebras. Archiv. Math. 54, 358–364 (1990)

    Article  MathSciNet  Google Scholar 

  3. Ara, P., Mathieu, M.: Local multipliers of \(C^*\)-algebras. Springer Monographs in Mathematics. Springer, London (2003)

    Book  Google Scholar 

  4. Arazy, J.: Isometries of Banach algebras satisfying the von Neumann inequality. Math. Scand. 74, 137–151 (1994)

    Article  MathSciNet  Google Scholar 

  5. Baxter, W.E., Martindale 3rd, W.S.: Central closure of semiprime nonassociative rings. Commun. Algebra 7, 1103–1132 (1979)

    Article  MathSciNet  Google Scholar 

  6. Beidar, K.I., Mikhalev, A.V., Slinko, A.M.: A criterion for primeness of nondegenerate alternative and Jordan algebras. Trudy Moskov. Mat. Obshch. 50, 130–137 (1988). (Translation in Trans. Moscow Math. Soc. 50, 129–137 (1987))

    Google Scholar 

  7. Bunce, L.J., Chu, C.-H., Stachó, L.L., Zalar, B.: On prime \(JB^{*}\)-triples. Quart. J. Math. Oxford 49, 279–290 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Cabello, J.C., Cabrera, M.: Structure theory for multiplicatively semiprime algebras. J. Algebra 282, 386–421 (2004)

    Article  MathSciNet  Google Scholar 

  9. Cabello, J.C., Cabrera, M., Fernández, A.: \(\pi \)-complemented algebras through pseudocomplemented lattices. Order 29, 463–479 (2012)

    Article  MathSciNet  Google Scholar 

  10. Cabello, J.C., Cabrera, M., López, G., Martindale 3rd, W.S.: Multiplicative semiprimeness of skew Lie algebras. Commun. Algebra 32, 3487–3501 (2004)

    Article  MathSciNet  Google Scholar 

  11. Cabello, J.C., Cabrera, M., Rodríguez, Á., Roura, R.: A characterization of \(\pi \)-complemented algebras. Commun. Algebra 41, 3067–3079 (2013)

    Article  MathSciNet  Google Scholar 

  12. Cabello, J.C., Cabrera, M., Roura, R.: A note on the multiplicative primeness of degenerate Jordan algebras. Sib. Math. J. 51, 818–823 (2010)

    Article  MathSciNet  Google Scholar 

  13. Cabello, J.C., Cabrera, M., Roura, R.: \(\pi \)-complementation in the unitisation and multiplication algebras of a semiprime algebra. Commun. Algebra 40, 3507–3531 (2012)

    Article  MathSciNet  Google Scholar 

  14. Cabrera, A.M., Cabrera, M.: Multiplicative primeness of strongly prime non-commutative Jordan algebras. J. Algebra 538, 253–260 (2019)

    Google Scholar 

  15. Cabrera, M., Fernández, A., Golubkov, A.Yu., Moreno, A.: Algebras whose multiplication algebra is PI or GPI. J. Algebra 459, 213–237 (2016)

    Article  MathSciNet  Google Scholar 

  16. Cabrera, M., Mohammed, A.A.: Extended centroid and central closure of the multiplication algebra. Commun. Algebra 27, 5723–5736 (1999)

    Article  MathSciNet  Google Scholar 

  17. Cabrera, M., Mohammed, A.A.: Extended centroid and central closure of multiplicatively semiprime algebras. Commun. Algebra 29, 1215–1233 (2001)

    Article  MathSciNet  Google Scholar 

  18. Cabrera, M., Mohammed, A.A.: Totally multiplicatively prime algebras. Proc. R. Soc. Edinb. Sect. A 132, 1145–1162 (2002)

    Article  MathSciNet  Google Scholar 

  19. Cabrera, M., Rodríguez, Á.: Extended centroid and central closure of semiprime normed algebras: a first approach. Commun. Algebra 18, 2293–2326 (1990)

    Article  MathSciNet  Google Scholar 

  20. Cabrera, M., Rodríguez, Á.: Nonassociative ultraprime normed algebras. Q. J. Math. Oxford 43, 1–7 (1992)

    Article  MathSciNet  Google Scholar 

  21. Cabrera, M., Rodríguez, Á.: Non-degenerately ultraprime Jordan-Banach algebras: a Zel’manovian treatment. Proc. Lond. Math. Soc. 69, 576–604 (1994)

    MATH  Google Scholar 

  22. Cabrera, M., Rodríguez, Á.: The von Neumann inequality in complete normed non-associative complex algebras. Math. Proc. R. Irish Acad. 118A, 83–125 (2018)

    Article  MathSciNet  Google Scholar 

  23. Cabrera, M., Rodríguez, Á.: Non-associative normed algebras. Volume 1: The Vidav-Palmer and Gelfand-Naimark Theorems, and Volume 2: Representation Theory and the Zel’manov Approach, Encyclopedia of Mathematics and Its Applications 154 and 167. Cambridge University Press, Cambridge, 2014 and 2018

    Google Scholar 

  24. Cabrera, M., Villena, A.R.: Multiplicative-semiprimeness of nondegenerate Jordan algebras. Commun. Algebra 32, 3995–4003 (2004)

    Article  MathSciNet  Google Scholar 

  25. Civin, P., Yood, B.: Lie and Jordan structures in Banach algebras. Pac. J. Math. 15, 775–797 (1965)

    Article  MathSciNet  Google Scholar 

  26. Erickson, T.S., Martindale III, W.S., Osborn, J.M.: Prime nonassociative algebras. Pac. J. Math. 60, 49–63 (1975)

    Article  MathSciNet  Google Scholar 

  27. Fernández, A., García, E., Rodríguez, Á.: A Zel’manov prime theorem for \(JB^{*}\)-algebras. J. Lond. Math. Soc. 46, 319–335 (1992)

    MATH  Google Scholar 

  28. Fernández, A., Rodríguez, Á.: A Wedderburn theorem for nonassociative complete normed algebras. J. Lond. Math. Soc. 33, 328–338 (1986)

    MathSciNet  MATH  Google Scholar 

  29. Foias, C.: Sur certains théorèmes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. Szeged. 18, 15–20 (1957)

    Google Scholar 

  30. Jacobson, N.: A note on non-associative algebras. Duke Math. J. 3, 544–548 (1937)

    Article  MathSciNet  Google Scholar 

  31. Jacobson, N.: Abraham Adrian Albert 1905–1972. Bull. Am. Math. Soc. 80, 1075–1100 (1974)

    Article  MathSciNet  Google Scholar 

  32. Mathieu, M.: Applications of ultraprime Banach algebras in the theory of elementary operators. Ph.D. thesis, Universität Tübingen, Tübingen (1986)

    Google Scholar 

  33. Mathieu, M.: Rings of quotients of ultraprime Banach algebras, with applications to elementary operators. In: Loy, R.J. (ed.) Conference on Automatic Continuity and Banach Algebras (Canberra, 1989). Proceedings of the Centre for Mathematical Analysis, Australian National University. vol. 21, pp. 297–317. Australian National University, Canberra (1989)

    Google Scholar 

  34. Mathieu, M.: Elementary operators on prime \(C^{*}\)-algebras, I. Math. Ann. 284, 223–244 (1989)

    Article  MathSciNet  Google Scholar 

  35. Mohammed, A.A.: Álgebras multiplicativamente primas: visión algebraica y analítica. Ph.D. thesis, Granada (2000)

    Google Scholar 

  36. Regev, A.: Existence of identities in \(A \otimes _F B\). Israel J. Math. 11, 131–152 (1972)

    Article  MathSciNet  Google Scholar 

  37. Riesz, F., Sz-Nagy, B.: Leçons d’analyse fonctionelle. Cinquième édition. Gauthier-Villars, Paris; Akadémiai Kiadó, Budapest (1968)

    Google Scholar 

  38. Rodríguez, Á.: Números hipercomplejos en dimensión infinita. Discurso de ingreso en la Academia de Ciencias Matemáticas, Físico-Químicas y Naturales de Granada, Granada (1993)

    Google Scholar 

  39. Rodríguez, Á., Villena, A.R.: Centroid and extended centroid of \(JB^{*}\)-algebras. In: González, S., Myung, HCh. (eds.) Nonassociative algebraic models. Proceedings of the workshop held at the Universidad de Zaragoza, Zaragoza, April 1989, pp. 223–232. Nova Science Publishers Inc, Commack, NY (1992)

    Google Scholar 

  40. Schatten, R.: Norm ideals of completely continuous operators. Second printing. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 27. Springer, Berlin (1970)

    Google Scholar 

  41. Skosyrskii, V.G.: Strongly prime noncommutative Jordan algebras. Trudy Inst. Mat. (Novosibirsk) 16, 131–164 (1989)

    Google Scholar 

  42. Villena, A.R.: Continuity of derivations on \(H^{*}\)-algebras. Proc. Am. Math. Soc. 122, 821–826 (1994)

    MathSciNet  MATH  Google Scholar 

  43. von Neumann, J.: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nach. 4, 258–281 (1951)

    Article  Google Scholar 

  44. Yood, B.: Closed prime ideals in topological rings. Proc. Lond. Math. Soc. 24, 307–323 (1972)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the organizers of the 3rd Moroccan Andalusian Meeting on Algebras and their Applications for inviting them to attend it and to deliver a lecture.

This chapter has been partially supported by the Junta de Andalucía and Spanish government grants FQM199 and MTMT2016-76327-C3-2-P.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miguel Cabrera García .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cabrera García, M., Rodríguez Palacios, Á. (2020). Multiplication Algebras: Algebraic and Analytic Aspects. In: Siles Molina, M., El Kaoutit, L., Louzari, M., Ben Yakoub, L., Benslimane, M. (eds) Associative and Non-Associative Algebras and Applications. MAMAA 2018. Springer Proceedings in Mathematics & Statistics, vol 311. Springer, Cham. https://doi.org/10.1007/978-3-030-35256-1_6

Download citation

Publish with us

Policies and ethics