Fitting classes and their associated injectors have been considered in a number of classes of locally finite groups [9, 8, 4, 5]. In particular, Dixon has considered the class [Lfr ] of countable locally finite groups with min −p for all primes p. He has shown that if the [Lfr ]-group G is radical (that is, it has an ascending locally nilpotent series) then G has locally nilpotent injectors, any two of which are isomorphic. Question 7.4.7 in [5] asks whether the restriction of being radical can be removed in this result.

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebra which is isomorphic to a unital subalgebra of the centre of M(A), the multiplier algebra of A. Letting Ω = Ĉ, so that we may write C = C(Ω), we call A a C(Ω)-algebra (following Blanchard [7]). Suppose that B is another C(Ω)-algebra, then we form A[otimes ]CB, the algebraic tensor product of A with B over C as follows: A [otimes ] B is the algebraic tensor product over [Copf ], IC = {[sum ]ni=1 (fi [otimes ] 1−1[otimes ] fi)x| fi∈C, x∈A[otimes ]B} is the ideal in A[otimes ]B generated by 〈f[otimes ]1−1[otimes ] f|f∈C〉, and A [otimes ]CB = A[otimes ]B/IC. Then A[otimes ]CB is an involutive algebra over [Copf ], and we shall be interested in deciding when A[otimes ]CB is a pre-C*-algebra; that is, when is there a C*-norm on A[otimes ]CB? There is a C*-semi-norm, which we denote by ‖·‖C-min, which is minimal in the sense that it is dominated by any semi-norm whose kernel contains the kernel of ‖·‖C-min. Moreover, if A [otimes ]CB has a C*-norm, then ‖·‖C-min is a C*-norm on A[otimes ]CB. The problem is to decide when ‖·‖C-min is a norm. It was shown by Blanchard [7, Proposition 3.1] that when A and B are continuous fields and C is separable, then ‖·‖C-min is a norm. In this paper we show that ‖·‖C-min is a norm when C is a von Neumann algebra, and then we examine some consequences.

There is extensive literature concerning approximately finite-dimensional (AF) C*-algebras. For example see [1, 5]. In recent years study has focused on non-self-adjoint subalgebras of AF C*-algebras ([4, 8, 9], for example) and this paper continues that theme. We define T(m, n) to be the block upper triangular subalgebra of Mm+n with self-adjoint part equal to Mm [oplus ] Mn. The class of algebras that are considered here are norm closures of ascending chains of algebras of the form T(m, n) and are referred to as uniformly T2 algebras. This class of algebras properly contains the matroid C*-algebras of Dixmier.

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning λ-harmonic functions on the hyperbolic spaces ℍn and contained in the paper [9].

B. Hartley in [4] introduced the class of barely transitive groups. By definition, a group of permutations G on an infinite set X is called barely transitive if G itself is transitive on X while every orbit of every proper subgroup of G is finite. If G is locally finite and G′≠G then the theorem of B. Love, proved in [4], shows that G is a locally nilpotent p-group of Heineken–Mohamed type. However, it is not known if perfect barely transitive locally nilpotent p-groups exist. Obviously this is a more general question than the corresponding one about perfect minimal non FC-groups (see below for definition). In this work it will be shown that a barely transitive locally nilpotent p-group cannot be perfect if the stabilizer of a point is hypercentral and solvable.

1. Some groups with the small index property

In this paper we shall use the methods of the paper [6] by Hodges, Hodkinson, Lascar and Shelah to show that the free group of countably infinite rank and certain relatively free groups of countably infinite rank have the small index property.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p with p[ges ]0. Then G contains finitely many conjugacy classes of unipotent elements. There is a unique class of unipotent elements of largest dimension – the class of regular unipotent elements; the centralizer of any such element is a unipotent group of dimension equal to the rank of G. For example, if G is the linear group SL (V) on the finite dimensional vector space V over K, then the regular unipotent elements are precisely the unipotent matrices consisting of a single Jordan block.

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n[ges ]3. Let Φ be the root system associated with G(k) and let Π={α1, α2, [eDDot ], αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=[sum ]α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=[sum ]α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.

A 4n-dimensional Riemannian manifold (M, g) is hyperkähler if it possesses three anti-commuting complex structures I, J, K such that the metric g is Kähler with respect to each of them. The reduced holonomy group of such a manifold is necessarily a subgroup of Sp(n) so the Ricci tensor of g vanishes and (M, g) can be regarded as a positive definite solution to Einstein's equations in vacuum.