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

TL;DR

This paper explores the conditions under which non-commutative log-algebras, constructed from different traces and von Neumann algebras, are isomorphic, revealing deep interconnections between their structures.

Contribution

It provides necessary and sufficient conditions for the isomorphism of non-commutative log-algebras based on different traces and von Neumann algebra types.

Findings

Criteria for isomorphism of log-algebras from different traces

Connection between isomorphisms of log-algebras on centers and original algebras

Conditions for isomorphism of log-algebras from type I_n von Neumann algebras

Abstract

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 $I_n$.