Maps Preserving Zero Lie Brackets on a Maximal Nilpotent Subalgebra of the Symplectic Algebra

Let F be a field with char F = 2, l a maximal nilpotent subalgebra of the symplectic algebra sp（2m,F）. In this paper, we characterize linear maps of l which preserve zero Lie brackets in both directions. It is shown that for m ≥ 4, a map φ of l preserves zero Lie brackets in both directions if and only if φ = ψcσT0λαφdηf, where ψc,σT0,λα,φd,ηf are the standard maps preserving zero Lie brackets in both directions.

Maximal Abelian Subalgebras of the Hyperfinite Factor,Entropy and Ergodic Theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, 7), the crossed product LX (X, 7)p G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra.

Maximal von Neumann subalgebras arising from maximal subgroups

Ge(2003)asked the question whether LF∞can be embedded into LF2 as a maximal subfactor.We answer it affirmatively with three different approaches,all containing the same key ingredient:the existence of maximal subgroups with infinite index.We also show that point stabilizer subgroups for every faithful,4-transitive action on an infinite set give rise to maximal von Neumann subalgebras.By combining this with the known results on constructing faithful,highly transitive actions,we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.

On Maximal Abelian Self-adjoint Subalgebras of Factors of Type II_1

In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.