Continuity inGLUC

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

Square of countably compact groups without non-trivial convergent sequences

Assessment and treatment of hypertension in a family practice: Max R. Polliack, Department of Family Medicine, Sackler School of Medicine, Tel Aviv University, Israel

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

Notes on compact Discrete Analysis 2017:16, 57 pp. This is the second paper in a two-part series. The first paper, [also published in this journal](http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects), developed the algebraic theory of nilspaces, which are an abstraction both of the notion of a nilmanifold, and of that of an abelian group (both with their attendant parallelogram structures). In this paper, the author applies this algebraic theory to study compact which can be viewed as a generalization of the notion of a compact abelian group. The main motivation for this is that such compact nilspaces arise naturally when trying to attack additive combinatorial problems such as the inverse conjecture for the Gowers norms via the methods of nonstandard analysis (or ultraproducts), as was demonstrated in work of Szegedy ([On higher order Fourier analysis](https://arxiv.org/abs/1203.2260)). The algebraic theory in the first part lets one describe $k$-step compact nilspaces as abelian extensions of $k-1$-step with the extension given by a cocycle. Among other things, this can be used to assign to each nilspace a probability measure that is analogous to the Haar probability measure one can assign to a compact group. However, it is not immediately obvious that this cocycle has any good measurability or continuity properties. Much of the delicate analysis in this paper (particularly those concerning the properties of continuous systems of measures) is devoted to ensuring that such good properties can be imposed (locally at least) on the cocycle, perhaps after modifying the cocycle by a suitable coboundary that does not affect the isomorphism class of the extension. Indeed, one can view the paper as developing some basic for although the formidable machinery of modern cohomology is not really deployed here. As with the first paper in this series, the paper closely follows the previous work of Camarena and Szegedy, but with many more details provided in the arguments. (An alternative approach to this theory has also recently been worked out in a series of papers by Gutman, Manners, and Varju.) A full structural description of compact nilspaces seems to be quite difficult, but in this paper two important partial descriptions are obtained. The first is to describe every compact nilspace as an inverse limit of compact nilspaces of rank. This is somewhat analogous to how the Peter-Weyl theorem can be used to describe a compact group as the inverse limit of finite-dimensional compact Lie groups. The second is to show that every connected compact nilspace of finite rank is isomorphic to a nilmanifold. These results were previously obtained by Camarena and Szegedy (and later used by Szegedy to provide a new proof of the inverse conjecture for the Gowers uniformity norms), but the author here reproves these results with much more detailed proofs, in particular treating with some care certain steps that were only briefly discussed in the prior paper of Camarena and Szegedy.

In the theory of abelian groups, the study of relations between an abelian group and its endomorphism group is an important problem. The question arises on conditions under which an abelian group and its endomorphism group are isomorphic ([1], Problem 31). The answer to this question has been obtained in the classes of completely decomposable abelian groups [2], vector groups [3], in the classes of periodic abelian groups, divisible abelian groups, nonreduced abelian groups, reduced algebraically compact abelian groups [4]. A group A is called an E+-group if it is isomorphic to its endomorphism group End(A). Clearly, if an abelian group A is an E+-group, then its endomorphism group End(A) is also an E+-group. In this paper, we show that the converse statement holds not always. A group A is called an End-group if the following implication takes place:

For a wide class of spaces endowed with symmetries, the maximum size of a one-colour symmetric set existing under any colouring of the space in a given number of colours is determined. Discrete spaces (finite Abelian groups and initial segments of the positive integers) and continuous algebraic and geometric structures (a closed interval on the real line, figures of revolution, and compact Abelian groups) are investigated.

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Kral’, Serra, and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct consequence of our removal result, we obtain the following generalization of Szemeredi’s theorem: for any compact abelian group G, any measurable set \({A \subseteq G}\) with Haar probability \({\mu(A) \geq \alpha > 0}\) satisfies \({\int_{G}\int_{G}1_{A}(x)1_{A}(x+r)...1_{A}(x+(k-1)r)\, {\rm d} \mu(x)\, {\rm d} \mu(r) \geq c}\), where the constant \({c=c(\alpha, k) > 0}\) is valid uniformly for all G. This result is shown to hold more generally for any translationinvariant system of r linear equations given by an integer matrix with coprime \({r \times r}\) minors.

In this paper the structure of all open semigroups in any compact topological group is determined. We can then extend this result to a classification of the angular semigroups of any abelian topological group H which contains a compact open subgroup K. Coupling this with the results for Euclidean n-space, we obtain an essentially complete description of the angular semigroups of an arbitrary locally compact group. For compact groups, the result is remarkably simple.