Perturbations of von Neumann subalgebras with finite index

TL;DR

This paper investigates how small perturbations affect von Neumann subalgebras with finite index, showing that close subalgebras are unitarily equivalent with near-identity unitaries.

Contribution

It establishes conditions under which perturbations of finite index von Neumann subalgebras preserve unitary equivalence with minimal deviation.

Findings

Close subalgebras are unitarily equivalent

Implementing unitaries can be chosen close to identity

Perturbations preserve algebraic structure under certain conditions

Abstract

In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let N and M be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner-Popa. If N and M are sufficiently close, then N and M are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.