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In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, it is shown that every nonlinear bijective map ψ preserving mixed Jordan triple η-products with η≠−1 from a factor von Neumann algebra M with dim M>1 into another factor von Neumann algebra N with dim N>1 is additive. Also, for α∈{−1,1} , we establish that αψ is a linear *-isomorphism, and is a conjugate linear *-isomorphism only when η∈R . [ABSTRACT FROM AUTHOR]