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.
HOU ChengJun School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
We introduce two notions of the pressure in operator algebras, one is the pressure Pα（π, T） for an automorphism α of a unital exact C^＊-algebra A at a self-adjoint element T in A with respect to a faithful unital ＊-representation π the other is the pressure Pτ,α（T） for an automorphism α of a hyperfinite von Neumann algebra M at a self-adjoint element T in M with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invaxiant, and also prove that the pressure of the implementing inner automorphism of a crossed product A×α Z at a self-adjoint operator T in A equals that of α at T.
We show that every local 3-cocycle of a von Neumann algebra $\mathcal{R}$ into an arbitrary unital dual $\mathcal{R}$ -bimodule $\mathcal{S}$ is a 3-cocycle.
Cheng-jun HOU~(1+) Ben-yin FU~2 1 School of Mathematical Sciences,Qufu Normal University,Qufu 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
