On maximal injective subalgebras in a wΓ factor

Let L(FQ) ×α Z be the crossed product von Neumann algebra of the free group factor L(FQ), associated with the left regular representation λ of the free group FQ with the set {ur : r ∈ Q} of generators, by an automorphism α defined by α(λ(ur)) = exp(2πri)λ(ur), where Q is the rational number set. We show that L(FQ) ×α Z is a wΓ factor, and for each r ∈ Q, the von Neumann subalgebra Ar generated in L(FQ) ×α Z by λ(ur) and v is maximal injective, where v is the unitary implementing the automorphism α. In particular, L(FQ) ×α Z is a wΓ factor with a maximal abelian selfadjoint subalgebra A0 which cannot be contained in any hyperfinite type II1 subfactor of L(FQ) ×α Z. This gives a counterexample of Kadison's problem in the case of wΓ factor.

Let $ \mathcal{L} $ (F ?) × α ? be the crossed product von Neumann algebra of the free group factor $ \mathcal{L} $ (F ?), associated with the left regular representation λ of the free group F ? with the set {u r : r ∈ ?} of generators, by an automorphism α defined by α(λ(u r )) = exp(2πri)λ(u r ), where ? is the rational number set. We show that $ \mathcal{L} $ (F ?) × α ? is a wΓ factor, and for each r ∈ ?, the von Neumann subalgebra $ \mathcal{A}_r $ generated in $ \mathcal{L} $ (F ?) × α ? by λ(u r ) and υ is maximal injective, where υ is the unitary implementing the automorphism α. In particular, $ \mathcal{L} $ (F ?) × 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 ?) × α ?. This gives a counterexample of Kadison’s problem in the case of wΓ factor.