Infinite order decompositions of C${}^{\ast }$-algebras

Abstract

In the given article infinite order decompositions of C${}^{\ast }$-algebras are investigated. We give complete proofs of the following statements: 1) If the order unit space ${\sum }_{\xi ,\eta }^{\oplus }{p}_{\xi }A{p}_{\eta }$ is monotone complete in $B(H)$ (i.e. ultraweakly closed), then ${\sum }_{\xi ,\eta }^{\oplus }{p}_{\xi }A{p}_{\eta }$ is a C${}^{\ast }$-algebra. 2) If $A$ is monotone complete in $B(H)$ (i.e. a von Neumann algebra), then $A={\sum }_{\xi ,\eta }^{\oplus }{p}_{\xi }A{p}_{\eta }$. 3) If ${\sum }_{\xi ,\eta }^{\oplus }{p}_{\xi }A{p}_{\eta }$ is a C${}^{\ast }$-algebra then this algebra is a von Neumann algebra.

arXiv comment11 pages