The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Recently Passman (attributing the origin of the idea to Zalesskii) has defined the following ideal in a ring, (2).

Definition. For a unitary ring R,

N * R = {α ∈ R | αS is nilpotent for all finitely generated subrings S of R}.

For a group algebra KG over a field K of characteristic p ≠ 0, he has proved the radical property:

In (8), R. L. Goodstein gave necessary and sufficient conditions for the solvability of equations over distributive lattices with 0 and 1 together with an algorithm for computing a solution whenever one exists. In addition, the same problem was considered for a special class of equations over distributive lattices with pseudocomplementation. The validity of several of Goodstein's results for distributive lattices without 0 and 1 was pointed out by Rudeanu in (15) and (16).

Let Γ be an irreducible and non-singular curve in [n] (n ≧ 3) which is the complete intersection of n − 1 primals of order m (m ≧ 2) with a common “self-polar” simplex S: by this I mean that the rth polar of each vertex of S with respect to any one of the defining primals is the opposite face of S counted m−r times, for r = 1, 2, …, m − 1. The various such Γ constitute the curves of the title; they were encountered in (2). When m = 2, Γ is the intersection of n − 1 quadrics with a common self-polar simplex in the familiar classical sense.

A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying

(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,

(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,

(3) (1, k ∈ ℬ, k = 1, 2, ….

Some familiar examples of basic sets are

, where Sk = {(1, k), (k, 1)},

= {(a, b)| a and b are relatively prime positive integers},

ℒ = {(a, b)| a and b are any positive integers}.

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,

with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equation

is oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,

combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the form

and then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form

Throughout the paper, T will be a Markov operator on C(X) (X compact T2), i.e. a continuous positive operator such that Te = e (e the unit function). P will be the set of Borel probability measures on X, which we shall often think of as linear functionals on C(X), and , where T' is the adjoint of T. Let

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressive k-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to the k-set contractions (k < 1). It should be remarked that J. D. Hamilton has extended the same result to A-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whether k-set contractions are A-proper or not.