Abstract
In this chapter we fill in details of some of the proofs given in Chapter 2. Also, new results are presented about subspaces and ideals in Jordan algebras that should help further the application of Jordan algebras in statistical problems generally. To assist in this latter task, our treatment of the material is designed to make the chapter largely self-contained, hence many of the definitions and first appearing in Chapter 2 are re-introduced, though more compactly. This chapter is unabashedly much more mathematical than any of the others, and unavoidably so, since the complete proofs of many of the results (previously just stated) are comparatively non-trivial. Many, but not all, of the results appearing here first appeared in Malley [1987].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References for Chapter 3
Braun, H. and Koecher, M. [1966]. Jorden-algebren. Grundlehren der Mathematischen Wissenschaften, 128. Springer-Verlag, Berlin.
Emch, G. [1972]. Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York.
Greub, W. [1978]. Multilinear Algebra. 2nd Edition. Springer-Verlag, New York.
Herstein, I. [1968]. Noncommutative Rings. Carus Mathematical Monograph No. 15, Mathematical Association of America, Wiley, New York.
Jacobson, N. [1968]. Structure and Representations of Jordan algebras. American Mathematical Society, Colloquium Publication Volume XXXIX, Providence, Rhode Island.
Jacobson, N. [1981]. Structure Theory of Jordan Algebras. University of Arkansas Lectures Notes in Mathematics, Vol. 5, Fayettevile, AK.
Jensen, S. T. [1988]. “Covariance hypotheses which are linear in both the covariance and the inverse covariance.” Annals of Statistics, (16), 302–322.
Jordan, P., von Neumann, J., and Wigner, E. [1934]. “On an algebraic generalization of the quantum mechanical formalism.” Annals of Math., (35), 29–64.
Malley, J. [1986]. Optimal Unbiased Estimation of Variance Components. Lecture Notes in Statistics, Vol. 39. Springer-Verlag, New York.
McCrimmon, K. [1969]. “Nondegenerate Jordan rings are von Neumann regular.” Journal of Algebra, (11), 111–115.
Schafer, R. [1966]. An Introduction to Non-Associative Algebras. Academic Press, New York.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Malley, J.D. (1994). Further Technical Results on Jordan Algebras. In: Statistical Applications of Jordan Algebras. Lecture Notes in Statistics, vol 91. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2678-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2678-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94341-1
Online ISBN: 978-1-4612-2678-9
eBook Packages: Springer Book Archive