  We construct a large family of quantum mechanical systems that give rise to
an emergent type III$_1$ von Neumann algebra in the large $N$ limit. Their
partition functions are matrix integrals that appear in the study of various
gauge theories. We calculate the real-time, finite temperature correlation
functions in these systems and show that they are described by an emergent type
III$_1$ von Neumann algebra at large $N$. The spectral density underlying this
algebra is computed in closed form in terms of the eigenvalue density of a
discrete matrix model. Furthermore, we explain how to systematically promote
these theories to systems with a Hagedorn transition, and show that a type
III$_1$ algebra only emerges above the Hagedorn temperature. Finally, we
empirically observe in examples a correspondence between the space of states of
the quantum mechanics and Calabi--Yau manifolds.
