If one wishes to construct a homomorphism of a group, an obvious approach is to begin with a homomorphism of a subgroup and attempt to extend it in some way. Any such extended homomorphism is determined by its action on elements of the group which, together with the subgroup, generate the whole group. The values assigned to such generators must then satisfy various functional equations. By a governing list for the extension problem, we shall mean a list of functional equations whose solutions describe all possible homomorphisms extending a given subgroup homomorphism.

One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.

Let 1 be a subdivision of [0, 1], and let denote the class of functions whose restriction to each sub-interval is a polynomial of degree at most k. Gaier [1] has shown that for uniform subdivisions △n (that is, subdivisions for which

if and only if f is a polynomial of degree at most k. Here, and subsequently, denotes the usual norm in Lp[0, 1], 1 ≦ p ≦ ∞, and we should emphasize that functions differing only on a set of Lebesgue measure zero are identified.

In all that follows, we let S denote the space {0, 1, 1/2, … , 1/n, …} with the relative usual topology and i : S → S denote the identity map on S. In this note, by a map or mapping we always mean a continuous surjection. A map f : X → Y is said to be hereditarily quotient if y ∊ int f(V) whenever V is open in X and f-1(y) ⊂ V. E. Michael has defined a map f : X → Y to be bi-quotient if whenever is a collection of open sets in X which covers f-1(y), there exists finitely many f(V), with V ∊ , which cover some neighbourhood of y.

H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.

Throughout this paper will denote a C*-algebra and will denote its second dual, which is identified with the enveloping von Neumann algebra of . A projection E in is said to be open if it supports a left ideal in , that is, if E = - for some left ideal in . Here the bar - means the stong closure. When has a unit, this definition coincides with the definition of Akemann [1, Definition II. 1]. In the sequel, we shall solely be concerned with two-sided ideals, and consequently central projections [4, I, § 3, Corollary 3 of Theorem 2]. Our aim is to show that is CCR if and only if the open central projections are strongly dense in the set of central projections on .

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms by

in It is not difficult to see that turns out to be a fundamental space.

It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact that

and

if and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

Suppose 1 ≦ P < ∞, μ is real, and denote by Lμ,p the collection of functions f, measurable on (0, ∞ ), and which satisfy

1.1

Also denote by [X] the collection of bounded operators from a Banach space X to itself. For v > 0, Re α > 0, Re β > 0, let

1.2

and

1.3

where ξ and η are complex numbers. Iv,α,ξ and Jv,β,η, are generalizations of the Riemann-Liouville and Weyl fractional integrals respectively, and consequently we shall refer to them as fractional integrals.

In this paper we continue our investigation of the class of right LCM domains which was introduced in [2]. A right LCM domain is an (not necessarily commutative) integral domain with unity in which the intersection of any two principal right ideals is again principal. In this note we study the right quotient rings of such a ring. In Section 1 we describe some of the characteristic properties of right quotient monoids with respect to which quotient rings are formed. Three particular types of quotient rings are described in Section 2. In Section 3 we relate the right ideals of a ring to those of its quotient ring.

The visibility function assigns to each point x of a fixed measurable set E in a Euclidean space En the Lebesgue outer measure of S(x), the set {y : rx + (1 — r)y ∊ E for every r in [0, 1]}.

The purpose of this paper is to determine sufficient conditions for the continuity of the function on the interor of a starshaped set.