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.
HOU ChengJun School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ξ given by a vector ξ in H. Assume that ξ is a separating vector for N , the core of the nest algebra Alg(N ). We show that L is a Kadison-Singer lattice, and hence the corresponding algebra Alg(L) is a Kadison-Singer algebra. We also describe the center of Alg(L) and its commutator modulo itself, and show that every bounded derivation from Alg(L) into itself is inner, and all n-th bounded cohomology groups H n (Alg(L), B(H)) of Alg(L) with coefficients in B(H) are trivial for all n≥1.
HOU ChengJun Institute of Operations Research, Qufu Normal University, Rizhao 276826, China
