  We introduce Poisson boundaries of II$_1$ factors with respect to density
operators that give the traces. The Poisson boundary is a von Neumann algebra
that contains the II$_1$ factor and is a particular example of the boundary of
a unital completely positive map as introduced by Izumi. Studying the inclusion
of the II$_1$ factor into its boundary we develop a number of notions, such as
double ergodicity and entropy, that can be seen as natural analogues of results
regarding the Poisson boundaries introduced by Furstenberg. We use the
techniques developed to answer a problem of Popa by showing that all finite
factors satisfy the MV-property. We also extend a result of Nevo by showing
that property (T) factors give rise to an entropy gap.
