Isomorphism of amalgamated free-product II₁ factors for varying k

Determine whether the II₁ factors M_k = *_A (N_i)_{i=1}^k, where each N_i is isomorphic to L(Z wr Z) and A = L(Z) denotes the von Neumann algebra of the acting group included in each N_i, are isomorphic for different values of k. Equivalently, ascertain whether there exist k ≠ k' with M_k ≅ M_{k'}, or whether these factors are distinguished by the number of factorial maximal amenable extensions of A or by the invariant θ(M) = sup_{masa A ⊂ M} m(A), where m(A) is the number of factorial maximal amenable extensions of A in M.

Background

The paper constructs, for each integer k ≥ 1, a separable II₁ factor M as an amalgamated free product M = *_A N_i with N_i ≃ L(Z wr Z) and A ≃ L(Z) the von Neumann algebra of the acting group embedded in each N_i. It proves that A has exactly k maximally amenable factorial extensions in M and that the space of maximally amenable extensions of A modulo unitary conjugacy is affinely identified with the k-dimensional real simplex.

These factors are shown to be strongly solid and non-Gamma, and not free group factors (they are strongly 1-bounded with h(M)=0). Motivated by the finiteness and simplex structure of maximally amenable extensions, the authors pose the isomorphism problem for these M as k varies and suggest a potential distinguishing invariant θ(M) based on the maximal number of factorial maximally amenable extensions of a masa in M, but they do not resolve this question.