Narrow Results

In this paper, we answer the Dixmier's question for type II₁-factors with property T in the negative, that is, if G is a discrete i.c.c group with property T of Kazhdan, L(G) is not isomorphic to for any factor of type II₁. To prove this, we study outer automorphism groups on a free product of two finite von Neumann algebras with respect to tracial states.

The classical Airy function has been generalized by Kontsevich to a function of a matrix argument, which is an integral over the space of skew-hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is further generalized to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A very general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases.

We study an abstract class of Hamming spaces (known also as measure algebras) that generalizes standard Hamming spaces ${(\mathbb{Z}/2\mathbb{Z})}^n$. We classify countable locally standard Hamming spaces and show that each of them can be realized as the Boolean algebra of idempotents of a Cartan subalgebra of a locally matrix algebra.

The N-derivation is a natural generalization of derivation and triple derivation. Let be a finitely generated Lie algebra graded by a finite-dimensional Cartan subalgebra. In this paper, a sufficient condition for the Lie N-derivation algebra of coinciding with the Lie derivation algebra of is given. As applications, any N-derivation of the Schrödinger-Virasoro algebra, generalized Witt algebra, Kac-Moody algebra or their Borel subalgebra is a derivation.

We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W*-superrigid) actions that can be completely recovered from their von Neumann algebras and II₁ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.

A degree stable Lie algebra is defined in the paper.¹³ The Lie algebra automorphism group of the Lie algebra W(1, 0, 2) is also found in this paper.² We find the algebra automorphism groups of the Lie algebras W(1³, 1, 1) and W(1³, 2, 0) in this work.