A correction of the decomposability result in a paper by Meyer–Neutsch

Abstract

In this short note, it is shown that there is a gap in the proof of Theorem 11 in the paper of Meyer and Neutsch (J. of Algebra, 1993). We prove, nevertheless, that the statement of the theorem is true and fix the proof by using a certain extremal property of idempotents which has an independent interest.

Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs

“…Therefore it is important to know whether the set of idempotents in V (u) is empty or not. The next proposition answers this question in positive; its proof can be found in our papers [54], [56]. Proposition 2.5.…”

On an extremal property of Jordan algebras of Clifford type

“…Note also that for all i ≥ 1 (29) e 2 i = 4c − 4c + e = e, and also by virtue of (21) (30) e i e j = 4c i c j − 2c i − 2c j + e = (1 − 2 c i , c j )e, 1 ≤ i, j ≤ n.…”

V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras

“…. If the bilinear form ; is weakly associative and positive definite, one can prove by a variational argument (see [55]) that the algebra A necessarily contains nonzero idempotents. Indeed, any idempotent in such an algebra is (proportional to) a stationary point of the cubic form u(x) on the unit sphere x; x = 1 which is a compact set.…”

“…The interested reader is referred to the recent monograph [31] for unexpected connections of nonassociative algebras to regularity theory of fully nonlinear PDEs. See also the recent papers [52], [55], [56], [18], [58], [57] for further results and basic concepts considered below.…”