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...

The lattice of ordered compactifications of a topological sum of a finite number of totally ordered spaces is investigated. This investigation proceeds by decomposing the lattice into equivalence classes determined by the identification of essential pairs of singularities. This lattice of equivalence classes is isomorphic to a power set lattice. Each of these equivalence classes is further deco...

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a poi...

A pro-C∗-algebra is a (projective) limit of C∗-algebras in the category of topological ∗algebras. From the perspective of non-commutative geometry, pro-C∗-algebras can be seen as non-commutative k-spaces. An element of a pro-C∗-algebra is bounded if there is a uniform bound for the norm of its images under any continuous ∗-homomorphism into a C∗-algebra. The ∗-subalgebra consisting of the bound...

Let a finite partition F of the real interval (0, 1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums and products (and, in the measurable case, all of its infinite sums as well). These results are obtained by using the algebraic structure of the Stone-Čech compactification ...

In this paper we study a notion of preorder that arises in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and its generalization to ultrafilters, which is related to the algebraical and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. In particular, we prove that there exist ultrafilters maximal for...

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)), Homeo(X)), where Exp(X) stands for the hype...