On spectral gap rigidity and Connes invariant $\chi(M)$

TL;DR

This paper computes Connes' invariant for specific II$_1$ factors derived from subfactors with spectral gap, addressing a long-standing question about the structure of McDuff factors with invariant 1.

Contribution

It provides the first calculation of Connes' invariant for these factors and answers a question posed by Connes in 1975 regarding McDuff factors.

Findings

Calculated Connes' invariant for certain II$_1$ factors

Resolved a 1975 question on McDuff factors with invariant 1

Enhanced understanding of spectral gap rigidity in operator algebras

Abstract

We calculate Connes' invariant $\chi(M)$ for certain II$_1$ factors $M$ that can be obtained as inductive limits of subfactors with spectral gap, then use this to answer a question he posed in 1975, on the structure of McDuff factors $M$ with $\chi(M)=1$.