On maximal injective subalgebras in a wΓ factor

Abstract

Let $\mathcal{L}$(F _ℚ) × _α ℤ be the crossed product von Neumann algebra of the free group factor $\mathcal{L}$(F _ℚ), associated with the left regular representation λ of the free group F _ℚ with the set {u _r : r ∈ ℚ} of generators, by an automorphism α defined by α(λ(u _r )) = exp(2πri)λ(u _r ), where ℚ is the rational number set. We show that $\mathcal{L}$ (F _ℚ) × _α ℤ is a wΓ factor, and for each r ∈ ℚ, the von Neumann subalgebra $\mathcal{A}_r$ generated in $\mathcal{L}$(F _ℚ) × _α ℚ by λ(u _r ) and υ is maximal injective, where υ is the unitary implementing the automorphism α. In particular, $\mathcal{L}$(F _ℚ) × _α ℣ is a wΓ factor with a maximal abelian selfadjoint subalgebra $\mathcal{A}_0$ which cannot be contained in any hyperfinite type II₁ subfactor of $\mathcal{L}$(F _ℚ) × _α ℚ. This gives a counterexample of Kadison’s problem in the case of wΓ factor.