Commutators with finite spectrum
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.
Locally quasi-nilpotent elementary operators
Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $\phi=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(\phi)=\frac{n(n-1)}{2} $, then there exists a representation of $\phi$ as $\phi=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.
A Hopewellian Burial Site in the Lower Republican Valley, Kansas
The senior author has for a number of years worked on the problem of the numerous and virtually unknown burial sites on the bluffs bordering the valley of the Republican River in Geary and Clay Counties, Kansas. This work has been a matter of some urgency, owing to the prevalence of unsystematic amateur excavation on the more prominent stone cairns found in the area. Fortunately, the complete excavation of the larger cairns involves too much labor for the merely curious, and it has often been possible to salvage material and information despite disturbance.