Abstract: We have two main objectives in this paper. Let R be a unit regular ring (the standard reference is [G1]); R admits a pseudo-metric topology, and the completion, denoted S, is called the N*-completion of R. We study the relations between R and S, complementary to a recent article of Goodearl on N*-complete regular rings. The rings R, S are representable as rings of sections of a sheaf-like object, and we clarify and extend this notion. The collection of pseudo-rank functions of R (terms not defined here, will be found in [G1]), denoted IP(R), is a compact convex set, with extreme boundary denoted either OelP(R) or E. We define N* :R--*[0, 1] via N*(r) =sup{P(r)lPslP(R)}. Then N* induces a pseudo-metric topology on R by means of d(r, s)= N*(r-s). This is a metric precisely when N*(r)=0 implies r = 0 ("R is N*-torsion-free"; in [GH2, p. 208], "R has a Hausdorff family of pseudorank functions"); we shall consider only such rings. Chapter 1 deals with properties of the N*-completion of a general (unit regular) ring R, called S. Our main result here (which takes the bulk of the chapter to establish) is that S possesses no new pseudo-rank functions (i.e., the natural map IP(S)~IP(R) is an affine homeomorphism). (This was claimed in [H1, Proposition 15] but the injectivity part of the proof has a gap.) It follows that S is complete in its intrinsic N*-metric and so the recent results of Goodearl I-G2] apply to S. When the appropriate identifications are made, the sup-norm completion of the image of Ko(R) in AfflP(R) [the affine continuous functions on IP(R)] is just Ko(S ). Some consequences of these results include: (i) If K is a metrizable Choquet simplex, there is an N*-complete regular ring S such that Ko(S ) is order-isomorphic to Aft(K). (ii) If(G, u) is an unperforated interpolation group with order unit (see [EHS]), then the sup-norm closure of its image in its natural representation as a group of affine functions on a simplex, is an interpolation group.

Abstract: . Let F be an algebraic number field and denote by N(a) the absolute norm and by1*?the maximum of the absolute values of the conjugates of the element a of F. Define cF to bethe best possible constant with the property: For every a e F there exists a unit u of F suchthat u3< cFN(a){/iF'(}\ An algorithm for the computation of cF is described and someexamples are given. 1. Introduction. Let F be an algebraic number field of degree d over the field ofrational numbers Q, U the group of units of F, and a,,...,ar a full set ofrepresentatives of nonconjugate embeddings of F into the field of complex numbersC. We denote by cF the best possible constant with the property: For every a E Fthere exists a unit u E U such that max{\ox(ua)\,... ,\or(ua)\] < cF/V(a)1/'/; here N(b) is the absolute value of the usual norm of the element b of F.The existence of and upper bounds for cF are well known (e.g., [6, p. 526]; [7, p. 351]; [9, p. 22]; [10, p. 271]; [13, p. 260]), and it was shown in [3] that the constant cF

Abstract: Let U be an action of a locally compact abelian group G on a Banach lattice E such that each U(t) (t ∈ G) is a lattice isomorphism. We deal with the problem which conditions on G and U will ensure that the spectrum σ(U) equals the dual group G^. The action is said to be non-degenerate if to every compact set K of G not containing the unit e there exists x, O < x ∈ E such that inf (U(t)x,x) = O for all t ∈ K. If U is non-degenerate or ergodic and injective then (under mild additional restrictions) σ(U) = G^. If σ(U) = G^ and G^ is rich, then U is non- de-generate. If G^ is not rich then the latter conclusion fails in general. Applications to G = ℝ (continuous dynamical systems) are given.

Abstract: A group is said to be locally indicable if each of its non-trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image. Such groups were studied by Higman [-9] in connection with the zero-divisor and unit problems for group rings. More recently, they have arisen I-2, 12] in the study of equations over groups. In [12], I proved a Freiheitsatz for locally indicable groups. This has been proved independently by Brodskii [2] and Short [22]. The present paper arises from an investigation of a question put to me by S.J. Pride whether torsionfree 1-relator groups are locally indicable. The question was raised originally by Baumslag (I-1], Problem 19) and an affirmative solution has recently been announced by Brodski~ [2]. As a consequence, the group algebra RG of a torsion-free 1-relator group G over an integral domain R has no non-trivial zero-divisors, and no non-trivial units (using Higman's results [9]). The first fact was also proved by Lewin and Lewin [16], by embedding RG in a division ring. The second appears to be new, as was pointed out to me by K.A. Brown. A second consequence is that no 1-relator group has a non-trivial finitely generated perfect subgroup, which answers [1], Problem 7 and [10], Question 1. In fact, using the Freiheitsatz, and the tower method described in [12], it is possible to prove the following general version of Brodskii's theorem.

Abstract: In Parts I and II we have gone out of our way to stress the enormous difference between the finite procedures of ordinary arithmetic, and those mathematical concepts whose very meaning depends on the introduction and interpretation of infinite processes. In contrast, you have in the past been encouraged to use real numbers (whether rational or irrational) in a naive, unquestioning way—especially in geometry: for example, you have been quietly encouraged to assume that, if we measure the length of a line segment AB in terms of some given unit segment CD, then its length AB/CD can obviously be expressed as a real number. While this is obvious when CD fits into AB a whole number of times leaving no remainder, or when CD and AB have some common measure MN which fits into CD precisely b times with no remainder and into AB precisely a times with no remainder (in which case AB/CD = a/b), it is not at all obvious in general. In Chapter 11.13 we saw one way of justifying the belief that AB can always be measured in terms of CD, but it was not exactly obvious!