Infinite order decompositions of C$^*$-algebras

TL;DR

This paper investigates infinite order decompositions of C*-algebras, providing proofs that connect monotone completeness and ultraweak closure to the structure of von Neumann algebras.

Contribution

It establishes conditions under which infinite order decompositions of C*-algebras are themselves C*-algebras or von Neumann algebras, clarifying their structural relationships.

Findings

Monotone complete order unit spaces are C*-algebras.

Monotone complete C*-algebras are von Neumann algebras.

Certain C*-algebra decompositions are von Neumann algebras.

Abstract

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.