Infinite order decompositions of C$^*$-algebras
  In the given article infinite order decompositions of C$^*$-algebras are investigated. We give complete proofs of the following statements:   1) If the order unit space $\sum_{\xi,\eta}^\oplus p_\xi Ap_\eta$ is monotone complete in $B(H)$ (i.e. ultraweakly closed), then $\sum_{\xi,\eta}^\oplus p_\xi Ap_\eta$ is a C$^*$-algebra.   2) If $A$ is monotone complete in $B(H)$ (i.e. a von Neumann algebra), then $A=\sum_{\xi,\eta}^\oplus p_\xi Ap_\eta$.   3) If $\sum_{\xi,\eta}^\oplus p_\xi Ap_\eta$ is a C$^*$-algebra then this algebra is a von Neumann algebra. 
F.N. Arzikulov
math.OA