Narrow Results

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

Given a division ring D with center F, the structure of maximal subgroups M of GL_n(D) is investigated. Suppose D ≠ F or n > 1. It is shown that if M/(M ∩ F*) is locally finite, then char F=p > 0 and either n=1, [D:F]=p² and M ∪ {0} is a maximal subfield of D, or D=F, n=p, and M ∪ {0} is a maximal subfield of M_p(F), or D=F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M^(r) of M is locally finite, then either M^(r) is abelian or F is algebraic over its prime subfield.

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.