Let me begin by expressing my sincere gratitude to the Japanese Association of Mathematical Sciences for inviting me to present this lecture and for giving me the JAMS International Prize for 2003 . I am deeply honored. This lecture is not a survey, but simply a discussion of some topics that I find interesting. For the most recent surveys of this subject area in which I have participated see [...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

Recently [4] we have obtained some new algebraic results about βN, the Stone-Čech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a nonempty alphabet with infinitely many variables adjoined. (The results about βW extend the Graham-Rothschild Parameter Sets Theorem.) In this paper we derive some Ramsey Theoretic consequences of these re...

It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Čech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups. In [KW], Kirchberg and Wassermann discussed exactness for groups. A discrete group G is said to be exact if its reduced group C-algebra C λ(G) is exact. Th...

Pointfree topology is, as the name suggests, a way of studying spaces without (mentioning) points. This idea is more natural than one might initially think. For example, when drawing a point on paper, we do no draw an actual point, but a collection of points somewhere near the desired one. We drew a “spot”, which can be reduced in size if that would be required to serve our purposes. Hence it m...

It has been observed that constructions so apparently different as Kronecker products, extension of the ring of operators of a module, field of quotients of an integral domain, free groups, free topological groups, completion of a uniform space, Cech compactification enter in the same frame. We intend in this paper to explain a rather general process of construction which may be applied to most...

0. Introduction. M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534]. Knowing this we now ask to what extent the invariant means on a given abelian group are alike. In this paper we shall show that they all have the same "measure-theoretic character." An invariant mean on an abelian group G is a normalized, positive, ...

Coxeter groups are Higson-Roe amenable, i.e. they admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension. This answers affirmatively a question from [H-R]. 1. Higson-Roe amenability. An action of a discrete group G on a compact space X is topologically amenable [AD-R] if there is a sequence of continuous maps b n : X → P (G) to the space of probability measure...

A twistor Cech cohomological characterization of mixed frequency zero rest-mass elds is developed. Using a Cech form of the cup product we derive an expression for the symplectic product of two real mixed frequency scalar elds in terms of a single vertex diagram. An expression for the frequency splitting complex structure of quantum eld theory is derived, and presented by its action on a mixed ...