Independence properties in subalgebras of ultraproduct II1 factors

Let Mn be a sequence of finite factors with dim(Mn)→∞ and denote by M=∏ωMn their ultraproduct over a free ultrafilter ω. We prove that if Q⊂M is either an ultraproduct Q=∏ωQn of subalgebras Qn⊂Mn, with Qn⊀MnQn′∩Mn, ∀n, or the centralizer Q=B′∩M of a separable amenable ⁎-subalgebra B⊂M, then for any separable subspace X⊂M⊖(Q′∩M), there exists a diffuse abelian von Neumann subalgebra in Q which is free independent to X, relative to Q′∩M. Some related independence properties for subalgebras in ultraproduct II1 factors are also discussed.