Kubo-Ando means are a rich class of important binary operations on the set of positive elements of an operator algebra. The aim of this article is to understand the algebraic structure of these operations. We generalize a recent result of L. Molnár for C^*-algebras to the class of split C^*-by-nilpotent algebras by using the method of analytic local expansion of binary operations. Eventually, we prove that in any C^*-algebra 𝒜, the identity [x,[y,z]]=0 implies commutativity.
In this paper, we investigate the Dieudonn & eacute; determinant and the definition of a positive semidefinite matrix over a division ring. We establish necessary and sufficient conditions for the decomposition of matrices into a product of a finite number of positive semidefinite matrices over division rings and one diagonalizable matrix, whose Dieudonn & eacute; determinant is the commutator class containing one. Consequently, it is possible to immediately obtain Ballantine's decomposition over the real quaternion division ring and complex number field.
