The Indecomposability of Free Group Factors over Nonprime Subfactors and Abelian Subalgebras
by
Marius B. Stefan
University of Iowa

We use Voiculescu's free entropy to show that the free group factors L(F_n) cannot be decomposed as

sp   ||·||2   å [(1 <= j1, ... , jt+1 <= f) || (1 <= t <= d)]  Nj1ZNj2Z ... NjtZNjt+1

or as

sp   ||·||2   å [(1 <= j1, ... , jt+1 <= f) || (1 <= t <= d)]  Aj1ZAj2Z ... AjtZAjt+1

if N₁, ... , N_f are nonprime subfactors, A₁, ... , A_f are abelian *-subalgebras, Z is a subset of L(F_n) containing p self-adjoint elements, and n >= p+2f+2. In particular, it follows that L(F_n) =/= [`sp] R<sub>1</sub>R<sub>2</sub> ... R<sub>f</sub> and L(F<sub>n</sub>) =/= [`sp] A₁A₂ ... A_f if R₁, ... , R_f are hyperfinite subalgebras of L(F_n), A₁, ... , A_f are abelian *-subalgebras of L(F_n), and n >= 2f+3, thus settling a conjecture of L. Ge and S. Popa (for n=\infty).

Date received: April 12, 1999