Normal states of type III factors

TL;DR

This paper proves that finite-dimensional C*-subalgebras with specified states can be embedded into type III factors via unitaries, extending to von Neumann algebras and type II_1 factors, with implications for state equivalence.

Contribution

It introduces new embedding theorems for finite-dimensional C*-algebras and von Neumann algebras with faithful states into type III factors, including unitary conjugation results.

Findings

Existence of unitaries aligning states on subalgebras in type III factors

Embedding results for C*-algebras with faithful states into type III factors

Extension of embedding results to type II_1 factors

Abstract

Let M be a factor of type III with separable predual and with normal states phi_1,...,phi_k, omega with omega faithful. Let A be a finite dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u in M such that phi_i Ad u =omega on A for i=1,...,k. This follows from an embedding result of a finite dimensional C*-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C*-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II_1 instead of type III holds.