The central result is the existence and uniqueness for an arbitrary l-group G of two hulls, Ḡ and Ḡω, which in the representable case coincide with the orthocompletion and strongly protectable hull of G. This is done by introducing two notions of extension, written ≼ and ≼ω, and proving that each G has a maximal ≼ extension Ḡ and a maximal ≼ω extension Ḡω. Two constructions of Ḡ and Ḡω are-given: an l-permutation construction leads to descriptive structural information, and a construction by “consistent maps” leads to a strong universal mapping property.

The notion of a strongly projectable hull has a lengthy history. The concept of an orthocompletion, together with the first proof of its existence and uniqueness, is due to Bernau [4]. Conrad summarized and extended all these results in an important paper [10].

A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

The theory of measure on topological spaces has in recent years found its most natural setting in the study of pavings and measures on such pavings (see e.g. [1-3; 5; 6; 10; 19; 22; 32; 33]. In this setting the relationship between measure and topology crystallizes since one concentrates primarily on the simpler internal lattice structure associated with sublattices of the topology rather than on the more complex topological structure.

In an interesting, but apparently largely unknown, paper [1] Halin has introduced the concept of a primitive set of vertices of a graph. This concept, or rather a slight modification of it, seems to provide the key to a new approach to the wrell-known and important class of graph-theoretical problems centering on the notion of separation. As is well known, if A, B, C are sets of vertices of a (non-oriented) graph X, C is said to separate A and B if and only if every AB-path in X contains a vertex of C.

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.

In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

In this paper we give a characterization of the extreme positive contractions on finite dimensional lp-spaces for 1 < p < ∞. This is related to the characterization of the extreme doubly stochastic operators. In Section 2 we present the basic properties of the facial structure of the set of doubly stochastic n × m matrices. In Section 3 we use these facts for description of the facial structure of the set of positive contractions on finite dimensional lp-space. Next is considered stability of the positive part of the unit ball of operators (Section 5). In Section 7 we prove that extreme positive contractions on are strongly exposed.

1. Terminology and notation. Let (X, , m) be a a-finite measure space. As usual, we denote by LP(m), 1 < p < ∞, the Banach lattice of all p-summable real-valued functions on X with standard norm and order. If X = {1, 2 , …, n) n < ∞, and m is a counting measure we write instead of LP(m). If X = [0, 1] and m is Lebesgue measure we write briefly LP.

In the study of compact semigroups the constructive method rather than the representational method is usually the better plan of attack. As it was pointed out by Hofmann and Mostert in the introduction to their book [10] this method is more productive than searching for a representation theory. Hofmann and Mostert described a constructive method called the Hormos and showed that any irreducible compact semigroup is obtained by the Hormos construction. Many of the important examples of irreducible semigroups which motivated their work were obtained by Hunter [11; 12; 13; 14].

In this paper, we apply the constructive method of ideal extensions [5] in algebraic semigroups to topological semigroups which are not necessarily compact. Many of Hunter's examples and examples of the Hormos technique can also be obtained by our method of topological ideal extensions. The topological ideal extension method, however, is, in general, a different type of construction technique.

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.

The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),

then

1.1

The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.

The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.

We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

In this paper we shall prove the following two theorems (the terminology is explained in § 2 below; all rings are assumed to be associative).

THEOREM 1. Suppose that is a division ring of finite order m over its centre Z and let μ(m) denote the factor sequence 1, m, m2, … , mn, … . Then the rings μ(w) and Zμ(m) are isomorphic.

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.

In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital ${{\text{C}}^{*}}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear ${{\text{C}}^{*}}$-algebra is less than or equal to 1. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital ${{\text{C}}^{*}}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital $\text{MF}$ algebras is also an $\text{MF}$ algebra. As an application, we obtain that $\text{Ext(}C_{r}^{*}\text{(}{{F}_{2}}\text{)}{{\text{*}}_{\mathbb{C}}}C_{r}^{*}\text{(}{{F}_{2}}\text{))}$ is not a group.

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the form

where the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].

A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).

In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).

Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).

The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).

In a series of fundamental papers [20], [21], [22], [23], K. Krickeberg introduced 'Vitali’ conditions on σ-algebras and showed that they are sufficient for convergence of properly bounded martingales, and supermartingales. It is now known that the conditions V∞ (= V), and V′ are both sufficient and necessary for convergence of L1-bounded amarts, and ordered amarts (Astbury [1]; [24], [25]); an amart (ordered amart) is a process (Xt) such that the net (EXτ)τ∈T* converges, where T* is the net of simple (ordered) stopping times. We undertake here to similarly characterize the Vitali conditions Vp, 1 ≦ p < ∞, in terms of convergence of properly defined classes of amarts. (In terms of convergence of L∞-bounded martingales, Krickeberg himself [22] was able to characterize V1.)

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

A strongly maximal triangular AF algebra which is defined by a realvalued cocycle is said to be analytic. Formulas for generic cocycles are given separately for both the integer-valued case and the real-valued coboundary case, and also for certain nest algebras. In the case of an integer-valued cocycle, there is an associated partial homeomorphismof the maximal ideal space of the diagonal. If the partial homeomorphism extends to a homeomorphism, then the algebra embeds in a crossed product. This occurs for a large class of subalgebras of UHF algebras, but an example shows that this does not always occur. An example is given of a triangular AF algebra which is analytic via a coboundary but is not a nest algebra; also, it is shown that a nest algebra need not be analytic

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $g$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $g$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $g$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.