STRUCTURE RESULTS FOR NORMALIZERS OF II1 FACTORS

For an inclusion N ⊆ M of II1 factors, we study the group of normalizers [Formula: see text] and the von Neumann algebra it generates. We first show that [Formula: see text] imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of [Formula: see text], this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of [Formula: see text] in terms of a unique countable subgroup of [Formula: see text]. Implications for inclusions B ⊆ M arising from the crossed product, group von Neumann algebra, and tensor product constructions will also be addressed. Our work leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N ⊆ M is a regular inclusion of II1 factors, then N norms M.

Most finitely generated subgroups of infinite unitriangular matrices are free

In this note we prove that the group G of infinite dimensional upper unitriangular matrices over a finite field contains an explicit countable subgroup ‘full’ of free subgroups. We deduce from this fact that, in a suitable sense, almost all k–generator subgroups of G are free groups of rank k.

Admissible subgroups of full ergodic groups

Let G be a countable group of automorphisms of a Lebesgue space (X, m) and let [G] be the full group of G. For a pair of countable ergodic subgroups H1 and H2 of [G], the following problem is considered: when are the full subgroups [H1] and [H2] conjugate in the normalizer N[G] = {g ∈ Aut X: g[G]g-1 = [G]} of [G]. A complete solution of the problem is given in the case when [G] is an approximately finite group of type II and [H] is admissible, in the sense that there exists an ergodic subgroup [H0] of [G] and a countable subgroup Γ ⊂ N[H0] consisting of automorphisms which are outer for [H0], such that [H0] ⊂ [G] and the full subgroup [Ho, Γ] generated by [H0] and Γ coincides with [G].

Groups with many characteristically simple subgroups

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

Equivalence singularity dichotomies for a class of ergodic measures

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

On the Splitting of Modules and Abelian Groups

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.