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ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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R. DRNOVŠEK

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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D. KOKOL BUKOVŠEK

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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T. KOŠIR

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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M. OMLADIČ

Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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, and

H. RADJAVI

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, Canada

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https://doi.org/10.1142/S0219498811004616Cited by:4 (Source: Crossref)

Abstract

A set of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are:

(1) Every irreducible semitransitive Jordan algebra is actually transitive.

(2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

Keywords:

AMSC: 17C50, 15A30

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