Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-01T11:16:30.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Tensor Products of Jordan Algebras

Published online by Cambridge University Press:  20 November 2018

Aubrey Wulfsohn
Aubrey Wulfsohn*
Affiliation:
Centre de recherches mathématiques, Université de Montreal, Montreal, Québec; Bar I Ian University, Ramat Gan, Israel
Rights & Permissions [Opens in a new window]

Extract

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.

Let J1 and J2 be two Jordan algebras with unit elements. We define various tensor products of J1 and J2. The first, which we call the Kronecker product, is the most obvious and is based on the tensor product of the vector spaces. We find conditions sufficient for its existence and for its non-existence. Motivated by the universal mapping property for the tensor product of associative algebras we define, in Section 2, tensor products of J1 and J2 by means of a universal mapping property. The tensor products always exist for special Jordan algebras and need not coincide with the Kronecker product when the latter exists. In Section 3 we construct a more concrete tensor product for special Jordan algebras. Here the tensor product of a special Jordan algebra and an associative Jordan algebra coincides with the Kronecker product of these algebras. We show that this "special" tensor product is the natural tensor product for some Jordan matrix algebras.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 60 - 74
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Cohn, P. M., On homomorphic images of special Jordan algebras, Can. J. Math. 6 (1954), 253–264.Google Scholar
2. Cohn, P. M., Universal algebra (Harper and Rowe, New York, 1965).Google Scholar
3. Dixmier, J., Les algèbres d'opérateurs dans l'espace hilbertien, (Gauthier-Villars, Paris, 1957).Google Scholar
4. Guichardet, A., Sur la catégorie des algèbres de von Neumann, Bull. Sci. Math. 90 (1966), 41–64.Google Scholar
5. Jacobson, N., A Kronecker factorisation theorem for Caley algebras and the exceptional simple Jordan algebra, Amer. J. Math. 76 (1954), 447–452. ,£# Structure and representations of Jordan algebras Amer. Math. Soc. Publ. 89 (1968).Google Scholar
6. Jacobson, N., Structure and representations of Jordan algebras, Amer. Math. Soc. Publ. 39 (1968).Google Scholar
7. Sikorsky, R., Products of abstract algebras, Fund. Math. 39 (1952), 211–228.Google Scholar
6. Wulfsohn, A., Von Neumann factors are perfect, Proc. Amer. Math. Soc. 41 (1973), 189–192.Google Scholar