1-bounded entropy conjecture for diffuse subalgebras of free group factors

Establish that every diffuse subalgebra Q of a free group factor L(F_r) with 1-bounded entropy h(Q)=0 is amenable.

Background

The paper invokes the 1-bounded entropy conjecture as a result concerning tracial von Neumann algebras. The conjecture asserts that vanishing 1-bounded entropy forces amenability for diffuse subalgebras of free group factors.

The authors explain that the conjecture was reduced to a strong-convergence question for a tensor random matrix model and state that this strong convergence was subsequently verified by several works. They then apply the resulting conclusion to Hilbert–Schmidt stable, non-amenable groups whose group von Neumann algebras embed into a free group factor, obtaining exponential representation growth.