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Abstract
The Cayley-Dickson algebras An are an infinite sequence of (non-associative) algebras beginning with the well-known composition algebras ℝ, ℂ, ℍ, 𝕆. We completely describe all possible dimensions for the alternator Alt(a) ≔ {b ∈ An | a(ab) = (aa)b = 0} of an element a ∈ An, for n ≥ 7. This resolves a conjecture of Biss, Christensen, Dugger, and Isaksen. On the way to obtaining this result, we establish numerous results on the eigentheory of left multiplication operators in An, some of which may be of independent interest.
Received: 2007-11-13
Revised: 2008-02-14
Published Online: 2009-08-31
Published in Print: 2009-September
© de Gruyter 2009
