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

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.