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 a...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.展开更多
基金the National Natural Science Foundation of China (Grant Nos. 10201007, A0324614)the Natural Science Foundation of Shandong Province (Grant No. Y2006A03)
文摘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.
