Spectral gap characterizations of property (T) for II$_1$ factors

TL;DR

This paper provides new spectral gap characterizations of property (T) for II$_1$ factors, linking it to weak spectral gap conditions and bimodule properties, thereby offering a deeper understanding of property (T) in operator algebras.

Contribution

It introduces novel equivalences and characterizations of property (T) for II$_1$ factors using spectral gap and bimodule conditions, enhancing theoretical understanding.

Findings

Property (T) is equivalent to weak spectral gap in any inclusion into a larger tracial von Neumann algebra.

Lack of non-zero almost central vectors in weakly mixing bimodules characterizes property (T).

A stronger characterization of property (T) requiring only weak spectral gap in irreducible inclusions.

Abstract

For II$_1$ factors, we show that property (T) is equivalent to weak spectral gap in any inclusion into a larger tracial von Neumann algebra. We also show that not having non-zero almost central vectors in weakly mixing bimodules characterizes property (T) for II$_1$ factors, which allows us to obtain a stronger characterization of property (T) where only weak spectral gap in any irreducible inclusion is required.