Isomorphisms of noncommutative \(\log \)-algebras

This paper establishes a necessary and sufficient condition for the coincidence of non-commutative \(\log \) $log$ -algebras constructed from different faithful normal semifinite traces. Consequently, we provide a criterion for the isomorphism of \(\log \) $log$ -algebras built on non-commutative von Neumann algebras with different faithful normal semifinite traces. Additionally, we demonstrate a connection between the isomorphism of non-commutative \(\log \) $log$ -algebras and the isomorphism of the corresponding \(\log \) $log$ -algebras constructed on the center of these von Neumann algebras. Furthermore, we present a necessary and sufficient condition for the isomorphism of \(\log \) $log$ -algebras derived from different von Neumann algebras of type \(I_n\) $I_n$ .