It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘A-indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and ‘proofs’ are morphisms over A. We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A.

We introduce a concept of unique factorization for elements in the context of Noetherian rings which are not necessarily commutative. We will call an element p of such a ring R prime if (i) pR = Rp, (ii) pR is a height-1 prime ideal of R, and (iii) R/pR is an integral domain. We define a Noetherian u.f.d. to be a Noetherian integral domain R such that every height-1 prime P of R is principal and R/P is a domain, or equivalently every non-zero element of R is of the form cq, where q is a product of prime elements of R and c has no prime factors. Examples include the Noetherian u.f.d.'s of commutative algebra and also the universal enveloping algebras of solvable Lie algebras. The latter class provides a rich supply of genuinely non-commutative examples.

The mod two homology of MSpin, the Spin-cobordism Thom spectrum, has a rich algebraic structure. We will describe it explicitly as a comodule algebra, and give some applications to the ring structure of the Spin-cobordism ring.

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

Let A = {Aq}q≥0 be an anti-commutative graded ring with 1. Let X be a space and let G(X; A) be the group of multiplicative units of the ring Πi≥0Hi(X; Ai) of the form l + x1 + x2 +…, where xi∊ Hi (X; Ai), In [6] Segal showed that there is a connective cohomology theory G*(X; A), with G0(X;A) = G(X; A). This was improved by Steiner, who in [7] showed that A→G*(X; A) is a functor from anticommutative graded algebras to cohomology theories.

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:

Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

A classical knot in S3 is said to be slice if it is the boundary of a smooth (or PL locally unknotted) disc in B4. The first obstructions to sliceness were introduced in [3] (the Alexander polynomial is of the form f(t). f(t−1)) and [7] (the signature is zero). Levine defined the notion of algebraically slice knot in [5]. This property implies the first two obstructions vanish.

The technique of introducing singularities of some form to a bordism theory proposed in [7] by Sullivan and developed in [1] has proved itself to be a very efficient method of constructing new homology and cohomology theories with predetermined properties (see, for example [3]–[6]). We propose a generalization of this method taking as our starting point a complex of free U*-modules endowed with an additional ‘geometric’ structure. The Sullivan—Baas construction of killing a regular sequence x1, x2, … of elements of U* fits in our framework as a special case of Koszul resolution of U*/(x1, x2, …) with some canonical structure. Also, when specialized to the U*-complex of the form U* ⊗Z C for a ℤ-complex of length 3, our construction turns out to be equivalent to the ‘bordisms with coefficients’ described in [2].

Let Δl(x, y) be a Alexander polynomial of a link l of two components X and Y in S3. Denote by Arf (Z) the Arf invariant of Z, a knot or a proper link [9]

Throughout this paper all scalars and vector spaces will be assumed to be complex unless there are specific indications to the contrary.

In this paper we examine the existence of extreme points in unbounded, closed and convex subsets K of a Banach space X.

If S is an infinite, discrete, commutative semigroup then the semigroup algebra l1(S) is a commutative Banach algebra. Its dual is l∞(S), which is isometrically iso-morphic to C(βS), the space of continuous functions on the Stone-Čech compactification of S. This fact enables us to identify the second dual of l1(S) with M(βS), the space of bounded regular Borel measures on βS. Endowed with the Arens product the second dual is also a Banach algebra, so it is natural to ask whether a product may be defined in M(βS) without reference to l1(S). In §4 this is shown to be possible even when S is a non-discrete semitopological semigroup, provided that the operation in S may be extended to make βS into a left-topological semigroup in the manner of, for example, [2] where further references may be found. (Note, however, that the construction there is of a right-topological semigroup.) Having done this we may use results on βS to provide information about the measure algebra.

The purpose of this paper is to establish an embedding of QM (A), the Banach space of all quasi-multipliers on A, in the second conjugate space A** of a Banach algebra A, which extends the well-known embedding of the left multipliers (right multipliers) of A in A**. We also prove that, for a dual B*-algebra, QM(A) is isometrically isomorphic to A**.

In this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.

We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

In their paper [5], Klein and Landau prove that given a symmetric ‘local semigroup’ of unbounded operators {T(t); t ≥ 0} on a Hilbert space, there exists a unique selfadjoint operator T such that T(t) is a restriction of e−tT, for each t ≥ 0. A similar representation theorem was proved earlier by Nussbaum [8]. The result of Klein and Landau was recently extended to the setting of reflexive Banach spaces by Kantorovitz ([4], theorem 2–3). More precisely, Kantorovitz presented necessary and sufficient conditions for a local semigroup of unbounded operators {T(t); t ≥ 0} to consist of restrictions of e−tT, t ≥ 0, for some unbounded spectral operator of scalar-type T with real spectrum (cf. [1] for the terminology).

Singer [10] defined a series Σxk in a Banach space X to be weakly p-unconditionally Cauchy if and only if Σλkxk converges in X for all λ∊lp, where 1 < p < ∞. For Banach spaces containing no subspace isomorphic to c0 Singer characterized such series as those for which

where 1/p + 1/q = 1 and X′ is the dual space of X.

Let X1, …, Xn be independent Bernoulli random variables, and let pi = P[Xi = 1], λ = Σi=1n pi and Σi=1n Xi. Successively improved estimates of the total variation distance between the distribution ℒ(W) of W and a Poisson distribution Pλ with mean λ have been obtained by Prohorov[5], Le Cam [4], Kerstan[3], Vervaat[8], Chen [2], Serfling[7] and Romanowska[6]. Prohorov, Vervaat and Romanowska discussed only the case of identically distributed Xi's, whereas Chen and Serfling were primarily interested in more general, dependent sequences. Under the present hypotheses, the following inequalities, here expressed in terms of the total variation distance

were established respectively by Le Cam, Kerstan and Chen:

(Kerstan's published estimate of ([3], p.174, equation (1)) is a misprint for , the constant 2·1 appearing twice on p. 175 of his paper.) Here, we use Chen's [2] elegant adaptation of Stein's method to improve hte estimates given in (1·1), and we complement these estimates with a reverse inequality expressed in similar terms. Second order estimates, and the case of more general non-negative integer valued X's, are also discussed.

Given a compact Hausdorff space X, E and F two Banach spaces, let T: C(X, E) → F denote a bounded linear operator (here C(X, E) stands for the Banach space of all continuous E-valued functions defined on X under supremum norm). It is well known [4] that any such operator T has a finitely additive representing measure G that is defined on the σ–field of Borel subsets of X and that G takes its values in the space of all bounded linear operators from E into the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance [1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that if E* and E** have the Radon-Nikodym property then a bounded linear operator T: C(X, E) → F is weakly compact if and only if the measure G is continuous at Ø (also called strongly bounded), i.e. limn ||G|| (Bn) = 0 for every decreasing sequence Bn ↘ Ø of Borel subsets of X (here ||G|| (B) denotes the semivariation of G at B), and if for every Borel set B the operator G(B) is a weakly compact operator from E to F. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties then E* and E** must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators on C(X, E). It is known [6] that if T: C(X, E) → F is an unconditionally converging operator, then its representing measure G is continuous at 0 and, for every Borel set B, G(B) is an unconditionally converging operator from E to F. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spaces E with the Schur property in terms of properties of Dunford-Pettis operators on C(X, E) spaces.

In an article on the philosophy of chance, Poincaré[7] showed that the distribution of the stopping position of a needle pivoted about its centre tends to the uniform distribution on the circle as the distribution of the initial push becomes spread out along the real line. This result was formalized by Feller [2] and strengthened by Mardia [6] as follows.

We are concerned in this paper with the same general problem as in [4], that is, the relation between the spectral and norm properties of a sub-algebra of M(G) and the corresponding properties of the algebra of its ‘transforms’ in some sense. The spirit of the investigation is akin to that of the earlier paper, and some of the results obtained here directly generalize those of [4]. There are, however, considerable differences in scope and method.