Hypertrace and entropy gap characterizations of property (T) for II1 factors

We establish a hypertrace characterization of property (T) for II1 factors: Given a II1 factors M, M does not have property (T) if and only if there exists a von Neumann algebra A with M⊂A such that A admits a M-hypertrace but no normal hypertrace. For M without property (T), such an inclusion M⊂A also admits vanishing Furstenberg entropy. With the same construction of M⊂A, we also establish similar characterizations of Haagerup property for II1 factors.