Topologia 2, 2017/18 Topological Groups Versione 26.02.2018 Introduction to Topological Groups Dikran Dikranjan To the memory of Ivan Prodanov (1935 { 1985) Abstract These notes provide a brief introduction to topological groups with a special emphasis on Pontryagin-van Kam- pen's duality theorem for locally compact abelian groups. We give a completely self-contained elementary proof of the theorem following the line from [57, 67]. According to the classical tradition, the structure theory of the locally compact abelian groups is built parallelly. 1 Introduction Let L denote the category of locally compact abelian groups and continuous homomorphisms and let T = R=Z be the unit circle group. For G 2 L denote by Gb the group of continuous homomorphisms (characters) G ! T equipped with the compact-open topology. Then the assignment G 7! Gb is a contravariant endofunctor b: L!L. The celebrated Pontryagin-van Kampen duality theorem ([122]) says that this functor is, up to natural equivalence, an involution i.e., Gb =∼ G and this isomorphism is \well behaved" (see Theorem 12.5.4 for more detail). Moreover, this functor sends compact groups to discrete ones and viceversa, i.e., it defines a duality between the subcategory C of compact abelian groups and the subcategory D of discrete abelian groups. This allows for a very efficient and fruitful tool for the study of compact abelian groups, reducing many problems related to topological properties of these group to the related problems concerning algebraic properties in the category of discrete groups. The reader is advised to give a look at the Mackey's beautiful survey [114] for the connection of charactres and Pontryagin-van Kampen duality to number theory, physics and elsewhere. This duality inspired a huge amount of related research also in category theory, a brief comment on a specific categorical aspect (uniqueness and representability) can be found in x8.1 of the Appendix. The aim of these notes is to provide a self-contained proof of this remarkable duality theorem, providing all necessary steps, including basic background on topological groups and the structure theory of locally compact abelian groups. Peter-Weyl's theorem asserting that the continuous characters of the compact abelian groups separate the points of the groups (see Theorem 11.2.1) is certainly the most important tool in proving the duality theorem. The Peter-Weyl's theorem is valid for arbitrary compact groups, but then continuous characters must be replaced by finite-dimensional unitary representations and the usual proof of the theorem in this general case involves Haar integration. In the case of abelian groups the irreducible ones turn out the be one-dimensional, i.e., continuous characters. We prefer here a different approach. Namely, Peter-Weyl's theorem in the abelian case can be obtained as an immediate corollary of a theorem of Følner (Theorem 10.3.5) whose elementary proof uses nothing beyond elementary properties of the finite abelian groups, a local version of the Stone-Weierstraß approximation theorem proved in x2 and the Stone-Cechˇ compactification of discrete spaces. As another application of Følner’s theorem we describe the precompact groups (i.e., the subgroups of the compact groups) as having a topology generated by continuous characters. As a third application of Følner’s theorem one can obtain the existence of the Haar integral on locally compact abelian groups for free (see Theorem 11.4.21, the proof follows [57, x2.4, Theorem 2.4.5]). The notes are organized as follows. In Section 2 we recall basic results and notions on abelian groups and general topology, which will be used in the rest of the paper. Section 3 contains background on topological groups, starting from scratch. Various ways of introducing a group topology are considered (x3.2), of which the prominent one is by means of characters (x3.2.3). In x4.3 we recall the construction of Protasov and Zelenyuk [131] of topologies arising from a given sequence that is required to be convergent to 0. In x4.1 we discuss separation axioms and metrizability of topological groups. Connectedness and related properties in topological groups are discussed in x4.2. In x5 the Markov's problems on the existence of non-discrete Hausdorff group topologies is discussed. In x5.1 we introduce two topologies, the Markov topology and the Zariski topology, that allow for an easier understanding of i Markov's problems. In x5.2 we describe the Markov topology of the infinite permutation groups, while x5.3 contains the first two examples of non-topologizable groups, given by Shelah and Ol0shanskii, respectively. The problems arising in extension of group topologies are the topic of x5.4. Several cardinal invariants (weight, character and density character) are introduced in x6.1, whereas x6.2 discuses completeness and completions. Further general information on topological groups can be found in the monographs or surveys [4, 36, 37, 38, 57, 106, 119, 122]. Section 7 is dedicated to specific properties of the (locally) compact groups used essentially in these notes. The most important property of locally compact group we recall in x7.3 is the open mapping theorem. x7.4 is dedicated to the minimal and the totally minimal groups, which need not be locally compact , yet satisfy the open mapping theorem. In x8 we recall (with complete proofs) the structure of the closed subgroups of Rn as well as the description of the closure of an arbitrary subgroup of Rn. These groups play an important role in the whole theory of locally compact abelian groups. Section 9 starts with x9.1 dedicated to big (large) and small subsets of abstract groups. In x9.2.1 we give an internal description of the precompact groups using the notion of a big set of a group and we show that these are precisely the subgroups of the compact groups. Moreover, we define a precompact group G+ that \best approximates" G. Its completion bG, the Bohr compactification of G, is the compact group that \best approximates" G. Here we introduce almost periodic functions and briefly comment their connection to the Bohr compactification of G. In x9.2.2 we establish the precompactness of the topologies generated by characters. In x9.3 we recall (without proofs) some relevant notions in the non-abelian case, as Haar integral, unitary representation, etc., that play a prominent role in the general theory of topological groups, but are not used in this exposition. The Haar integral in locally compact abelian groups is built in x11.4.3. In x10 prepares all ingredients for the proof of Følner’s theorem (see Theorem 10.3.5). This proof, follows the line of [57]. An important feature of the proof is the crucial idea, due to Prodanov, to eliminate all discontinuous characters in the uniform approximation of continuous functions via linear combinations of characters obtained by means of Stone- Weierstraß approximation theorem. This step is ensured by Prodanov's lemma 10.3.1, which has many other relevant applications towards independence of characters and the construction of the Haar intergral for LCA groups. The last subsection contains the final stage of the proof of Følner’s theorem. In Section 11 gives various applications of Følner’s theorem. The first one is a description of the precompact group topologies of the abelian groups (x10.1). The main application of Følner’s theorem is an immediate proof of Peter- Weyl's theorem (in x11.2). To the structure of the compactly generated locally compact abelian groups is dedicated x11.3. Applications of these structure theorems are given in x11.4.3. In x11.3 we provide useful information on the dual of a locally compact abelian group, to be used in Section 12. x11.4 is dedicated to the almost periodic functions of the abelian group. As another application of Følner's theorem we give a proof of Bohr - von Neumann's theorem describing the almost periodic functions as uniform limits of linear combinations of characters. Among other things, we obtain as a by-product of Prodanov's approach an easy construction of the Haar integral for almost periodic functions on abelian groups, in particular for all continuous functions on a compact abelian group (x11.4.2). In x11.4.3 we build a Haar integral on arbitrary locally compact abelian groups, using the construction from x11.4.2 in the compact case. In x11.5 we consider a precompact version of the construction form x4.3 of topologies making a fixed sequence converge to 0. Section 12 is dedicated to Pontryagin-van Kampen duality. In xx12.1-12.3 we construct all tools for proving the duality theorem 12.5.4. More specifically, xx12.1 and 12.2 contain various properties of the dual groups that allow for an easier computation of the dual in many cases. Using further the properties of the dual, we see in x12.3 that many specific groups G satisfy the duality theorem, i.e., G =∼ Gb. In x12.4 we stress the fact that the isomorphism G =∼ Gb is natural by studying in detail the natural transformation !G : G ! Gb connecting the group with its bidual. It is shown in several steps that !G is an isomorphism, considering larger and larger classes of locally compact abelian groups G where the duality theorem holds (elementary locally compact abelian groups, compact abelian groups, discrete abelian groups, compactly generated locally compact abelian groups). The last step uses the fact that the duality functor is exact, this permits us to use all previous steps in the general case. As an immediate application of the duality theorem we obtain the main structure theorem for the locally compact abelian groups, a complete description of the monothetic compact groups, the torsion compact abelian groups, the connected compact abelian groups with dense torsion subgroup, etc.