Infinite order decompositions of C*-algebras

The present paper is devoted to infinite order decompositions of C*-algebras. It is proved that an infinite order decomposition (IOD) of a C*-algebra forms the complexification of an order unit space, and, if the C*-algebra is monotone complete (not necessarily weakly closed) then its IOD is also monotone complete ordered vector space. Also it is established that an IOD of a C*-algebra is a C*-algebra if and only if this C*-algebra is a von Neumann algebra. As a summary we obtain that the norm of an infinite dimensional matrix is equal to the supremum of norms of all finite dimensional main diagonal submatrices of this matrix and an infinite dimensional matrix is positive if and only if all finite dimensional main diagonal submatrices of this matrix are positive.