  We establish a hypertrace characterization of property (T) for
$\mathrm{II}_1$ factors: Given a $\mathrm{II}_1$ factors $M$, $M$ does not have
property (T) if and only if there exists a von Neumann algebra $\mathcal{A}$
with $M\subset \mathcal{A}$ such that $\mathcal{A}$ admits a $M$-hypertrace but
no normal hypertrace. For $M$ without property (T), such an inclusion $M\subset
\mathcal{A}$ also admits almost vanishing Furstenberg entropy. With the same
construction of $M\subset \mathcal{A}$, we also establish similar
characterizations of Haagerup property for $\mathrm{II}_1$ factors.
