The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $\log$-algebras

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