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

Abstract

For II1 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 II1 factors, which allows us to obtain a stronger characterization of property (T) where only weak spectral gap in any irreducible inclusion is required.

Embedding universality for II$_1$ factors with property (T)

“…As G is embedding universal, one can find Q ∈ G with N ⊂ Q. Since N has property (T), it has w-spectral gap in the sense of [Po09] in any extension (in fact this characterizes property (T), see [Ta22]) and thus N ′ ∩ Q ω = (N ′ ∩ Q) ω . As Q is existentially closed by [Go18, Proposition 5.16] we also have (N ′ ∩ Q) ′ ∩ Q = N .…”

Uniformly Super McDuff II$_1$ Factors

“…In particular, if N ⊆ M is an inclusion of II 1 factors and N has property (T), then L 2 (M ) is a normal N -N -bimodule and therefore N has spectral gap inside M . Moreover, Tan [Tan22] showed the converse implication, namely that if a II 1 factor N has spectral gap in M for every inclusion N ⊆ M , then N has property (T). Hence, it is natural to leverage property (T) subalgebras of M together with our spectral gap criterion for the uniform super McDuff property.…”