We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

Let $I$ be any nonempty set and let $(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$ of (possibly type $\text{III}$ ) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product $(M,\unicode[STIX]{x1D711})=\ast _{i\in I}(M_{i},\unicode[STIX]{x1D711}_{i})$ , we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_{i}$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type $\text{II}_{1}$ factors and is new for free product type $\text{III}$ factors. It moreover provides new rigidity phenomena for type $\text{III}$ factors.

Let (M, ϕ) = (M 1, ϕ1) * (M 2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M 1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M 1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M 1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M 1 ⊂ M in arbitrary free product von Neumann algebras.

The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.