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Locally compact groups. (English) Zbl 1102.22005
EMS Textbooks in Mathematics. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-016-7/hbk). x, 302 p. (2006).
This book is designed to be a student text providing an introduction to the theory of locally compact groups. In fact, this is achieved already with the first hundred pages of the book. The remaining part offers a wealth of material that can be mastered by students but goes way beyond a first introduction. Special features include the Haar integral, Pontryagin duality with many applications, locally compact rings and fields, a treatment of semigroups and a chapter on applications of the approximation by Lie groups. Each section (except those in the last chapter) is accompanied by a fair number of exercises.
The book is at the same time very abstract and very concrete. It maintains a high level of abstraction throughout, in particular by stressing aspects of category theory wherever possible (e.g., in the treatment of limits). On the other hand, this is never done for its own sake, but is always designed to make the treatment of concrete problems more exact, easier, less confusing, and more efficient. The text abounds with concrete information on specific groups, counterexamples, classification and characterization results. It is organized in eight chapters, and we proceed to describe their contents.
Chapter A (Preliminaries) presents necessary material from topology in a concise way. In particular, products are treated emphasising the category aspects, thus paving the way for dealing with limits later in the book. Separation properties, compactness, closed graph theorem and Baire category, completion and Tikhonov’s theorem (the latter without proof) are dealt with. There follows a treatment of quotients. Continuous maps are considered both in terms of nets and of filter bases, and extension of continuous maps from a dense subspace to the whole space is examined. This was Section 1, and Section 2 deals with connectedness, pathwise connectedness, and with the (small inductive) dimension of topological spaces. Here, the central results (the sum theorem and the dimension of Euclidean spaces) are quoted without proof.
Chapter B (Topological Groups) begins (Section 3) with the definition, expressed in the language of universal algebra, but soon translated into the familiar idiom. There follows a number of examples such as discrete groups, vector and matrix groups, topological rings and fields, orthogonal and unitary groups. Next, some simple properties of topological groups are considered; locally compact groups are introduced and it is shown that a locally compact Hausdorff group is discrete or uncountable. A local description of group topologies (in terms of the neighborhood filter of the neutral element) is given which will prove useful for the construction of examples. Homomorphisms and isomorphisms (continuous or not) and their kernels are treated and exact sequences and extensions of topological groups as well as subgroup lattices are introduced. Cartesian products are constructed and their universal property is presented, and centralizers and normalizers are considered. Section 4 deals with subgroups and their closures and with connectedness properties. The identity component is introduced and existence of small compact open (normal) subgroups in totally disconnected locally compact (compact) groups is proved. Divisibility and torsion are considered for abstract groups. Every divisible Abelian group is shown to be a product of a rational vector group with a sum of powers of Prüfer groups \({\mathbb Z}(p^\infty)\), the latter being the \(p\)-torsion groups of the circle group \(\mathbb T\). Section 5 treats linear groups over topological rings and, in particular, the general linear groups over topological fields. In Section 6, quotients (modulo normal subgroups) and homogeneous spaces (modulo arbitrary subgroups) of topological groups are studied systematically. It is shown that the following properties hold for a group if they hold for a normal subgroup and for the quotient group: being connected, totally disconnected, compactly generated and locally compact, \(\sigma\)-compact and locally compact, having no small subgroups. Next, split extensions, the (second) isomorphism theorem, the open mapping theorem and the recognition of internal direct sums are treated, each time with emphasis on examples where the desired isomorphisms of topological groups fail to be open and on conditions that force them to be. A. Weil’s lemma on closures of cyclic subgroups and of one-parameter subgroups in locally compact Hausdorff groups is proved next, and is used to derive that a commutative locally compact Hausdorff group which is generated by a compact neighborhood of the identity is compact modulo some integer lattice \({\mathbb Z}^d\). In Section 7, solvable and nilpotent groups are studied. It is shown that for Hausdorff groups, nothing changes if one uses closed subgroups only in the definition of the relevant normal series. The behaviour of products, quotients and extensions with respect to these properties is examined. Section 8 on completion introduces uniformities, uniform continuity and completeness, the left, right and bilateral uniformities on a topological group, and conditions sufficient for completeness, such as local compactness. The completion of uniform spaces is constructed using minimal Cauchy filter bases rather than equivalence classes of Cauchy filter bases. Completion of Hausdorff groups with respect to the three uniformities mentioned above is studied. Except for the bilateral uniformity, this yields only a topological semigroup in general. Also completion of Hausdorff rings is studied and the ring \({\mathbb Z}_p\) of \(p\)-adic integers is constructed. Section 8 is kept rather independent from the rest of the book, except for the classification of locally compact fields in Section 26, where completion is an essential tool.
Chapter C (Topological Transformation Groups) begins in Section 9 with a thorough study of the compact-open topology and other topologies for sets of mappings; in particular, the modified compact-open topology is introduced, which turns a group of homeomorphisms into a topological group, and the Arzela-Ascoli theorem, characterizing relatively compact sets of continuous mappings as equicontinuous sets with relatively compact point images, is presented. This section will be used later in order to set up Pontryagin duality. In Section 10, topological transformation groups are introduced and important examples are given (homeomorphism groups endowed with the modified compact-open topology, homogeneous spaces, and groups acting on themselves via conjugation). The Baire category argument for identifying orbits with homogeneous spaces under suitable conditions is presented. The results are applied to show that split extensions and semidirect products of topological groups are essentially the same. In Section 11, the results are applied to groups of homomorphisms between topological groups and to the left and right action of a ring on its own additive group. Matrices with homomorphisms as entries are used to describe homomorphisms between Cartesian products, and continuity of matrix operations is shown. This will be needed later (Section 25) in studying automorphisms of locally compact Abelian groups.
So far, the book is a rather elementary introduction to topological groups. The remaining part (roughly two thirds) contains deeper and more involved material. Chapter D (The Haar Integral) starts with Section 12, which contains the construction and the proof of uniqueness up to a constant factor of the Haar integral on a locally compact group (a translation invariant, positive linear form on the space of continuous functions \(G \to {\mathbb R}\) with compact support). The construction uses Tikhonov’s theorem in order to smoothe a rough preliminary version of the integral. The uniqueness is applied in Section 13 to define the module function \(\operatorname{Aut} G\to {\mathbb R}\): any automorphism \(\alpha\) acts on the space of functions on \(G\) and hence changes the Haar integral by a positive factor \(\operatorname{mod}\alpha\). This is later used (Section 26) in order to define valuations on locally compact fields. Criteria for unimodular groups (where the right invariant Haar integral is also left invariant) are given, and the module function of a locally compact, totally disconnected group is computed by counting certain cosets. In Section 14 the Haar integral is complexified; then \(\int_G\psi\overline \varphi\) defines a scalar product on the space of complex valued functions with compact supports, and the Hilbert space \({\text{L}}^2(G)\) is obtained by completion of this function space. Right translation defines a faithful unitary representation of \(G\) on \({\text{ L}}^2(G)\). Now assume that \(G\) is compact. Via restriction to suitable eigenspaces of certain compact operators, one can then produce finite dimensional unitary representations: one obtains the Peter-Weyl theorem which says that every element of \(G\) may be represented by a matrix distinct from the unit matrix. In the abelian case, this means that there are enough characters to distinguish the group elements from one another. Finally, integration is used for Weyl’s trick, which yields a unitary representation from any continuous representation on a Hilbert space by averaging over \(G\).
Chapter E (Categories of Topological Groups) gives first (Section 15) a selection of ten categories of topological groups that are of particular interest, and discusses monics and epics. Section 16 examines the existence and properties of products in each of these categories. Section 17 thoroughly investigates direct and projective limits, after developing these notions from easy examples like rational numbers, Prüfer groups, and \(p\)-adic integers. This provides one of the major tools for a deeper understanding of the structure of locally compact groups. It is applied immediately to re-interpret the Peter-Weyl theorem as saying that every compact Hausdorff group is a projective limit of closed subgroups of unitary groups (of circle groups, in the Abelian case). This is also expressed as a statement about quotients modulo small subgroups. Moreover, compact, totally disconnected groups are described as projective limits of discrete finite groups.
Chapter F (Locally Compact Abelian Groups) treats Pontryagin duality and applies it in several ways. All groups \(A\) here will be locally compact Abelian (LCA). The characters (i.e., the continuous homomorphisms \(A \to {\mathbb T}\)) form again an LCA group \(A^*\) with pointwise addition and with the compact-open topology. There is a natural transformation from double duality \(A \to A^{**}\) to the identity functor, obtained by evaluation of characters at fixed group elements. Pontryagin’s theorem says that this is in fact a natural equivalence. Section 20 formulates this goal and gives preliminary results on how properties of \(A\) translate into properties of \(A^*\); e.g., \(A^*\) is discrete if \(A\) is compact. The dual of a factor group \(A/B\) is shown to be the annihilator of \(B\) in \(A^*\). In Section 21, the duals of \({\mathbb R}, {\mathbb T},{\mathbb Z}\) and \({\mathbb Z}/p{\mathbb Z}\) are computed. The duality theorem is verified in these cases and then obtained for direct products of finitely many such groups, i.e., for compactly generated Abelian Lie groups. It is shown that the groups covered by this result are precisely those which are generated by some compact subset and have no small subgroups. This is obtained by first showing that every compactly generated group \(A\) can be approximated by Lie groups, that is, it has small subgroups with Lie factor groups. (No Lie theory is used at this point.) In Section 22 the duality theorem is proved in full generality; first, it is proved for discrete groups by an argument involving limits, and this is then combined with the result of Section 21. In Section 23, a systematic account is given of how properties of a group or of a homomorphism translate to properties of the dual or adjoint, respectively, and how direct and projective limits are interchanged. Duality is then applied to obtain a splitting theorem for compactly generated (LCA) groups: they are the direct product of their unique maximal compact subgroup with a real vector group and an integer lattice. A similar result is proved in Section 24 for arbitrary LCA groups, assuming that a maximal compact subgroup exists. It is also shown that each LCA group \(A\) has a maximal vector subgroup, unique up to isomorphism of \(A\), and that this subgroup is a direct summand of \(A\). In Section 25, it is observed that the passage to the adjoint defines an isomorphism of topological groups of homomorphisms \(\text{Mor}(A,B)\to\text{Mor}(B^*,A^*)\), and this is exploited in various ways: vector groups are characterized as those LCA groups that admit a connected set of endomorphisms containing the identity such that only the identity fixes a nontrivial group element; moreover, the structure of the automorphism group of compactly generated Abelian Lie groups is studied in detail. Section 26 is devoted to locally compact rings and fields, and contains the classification of the latter. First it is shown by a clever application of duality that every locally compact Hausdorff ring is a finite dimensional \({\mathbb R}\)-algebra (of matrices). For locally compact connected fields, the theorem of Frobenius (quoted without proof) now gives the complete answer. Next, it is shown that compact Hausdorff rings are totally disconnected and fields of this kind are finite. Then a complete description of all finite fields is given. Using the module function in order to obtain a (complete) valuation, it is then shown that every totally disconnected locally compact field contains a \(p\)-adic number field or a field of Laurent series over a finite field. Then it is shown that locally compact vector spaces are of finite dimension, and the complete classification is obtained. Section 27 applies duality to cover a simple part of an area where the author has done much own work, homogeneous topological groups. These are groups whose automorphisms act transitively on the nontrivial group elements. The result of this section is that homogeneous LCA groups are precisely the following: real vector groups, discrete rational vector groups, vector groups over non-discrete locally compact fields, minimal divisible extensions of arbitrary powers \(({\mathbb Z}_p)^d\), and certain locally compact elementary Abelian groups.
Chapter G (Locally Compact Semigroups) is a special feature of this book. After the introductory Section 28, in Section 29 conditions are given which imply that a cancellative (locally compact) topological semigroup embeds in a (locally compact) topological group. Section 30 considers compact semigroups, showing, e.g., that a cancellative compact Hausdorff semigroup \(G\) is a topological group. This is achieved by showing that \(G\) contains an idempotent. In Section 31, it is shown finally that a locally compact Hausdorff group with (jointly) continuous multiplication has continuous inversion. At this point, one could have mentioned the stronger theorem of Ellis (1957; separate continuity of multiplication suffices).
The final Chapter H (Hilbert’s Fifth Problem), based on part of the author’s Habilitationsschrift, aims to carry over as much as possible of the structure theory of Lie groups to locally compact Hausdorff groups (no other groups are considered from now on). It contains several deep results quoted from the literature without giving proofs; yet it contains enough proofs to provide the reader with a coherent picture. Several results here are due to the author. Section 32 presents the general version of the approximation theorem, special cases of which have been proved earlier in the book: if \(G\) is of finite dimension and \(G/G^1\) is compact, then \(G\) has totally disconnected normal subgroups with Lie factor groups. Next, the Malcev-Iwasawa theorem on existence and conjugacy of maximal compact subgroups \(M \leq G\) and the homeomorphism \(G\approx M \times {\mathbb R}^n\) is presented. Section 33 derives a useful dimension formula for transitive group actions. Section 34 shows that the lattice of closed connected subgroups of a group \(G\), endowed with the additional operations of taking commutators, normalizers and centralizers (the rough structure, as the author calls it) is completely determined by the approximating Lie groups. In Section 35, (semi-)simplicity, almost simplicity and nilpotency are considered. For instance, it is shown that a connected (locally compact) almost simple group \(G\) has a totally disconnected centre \(Z\) and that \(G/Z\) is a Lie group having the same (finite) dimension as \(G\). If a group \(G\) contains a closed connected Abelian normal subgroup of finite dimension, then it contains a minimal such subgroup, and either this subgroup is compact and central in \(G^1\), or it is a vector group. In the latter case, the action of \(G\) on the normal subgroup is examined more closely. To some extent, also the Levi decomposition (as a semidirect product of a solvable and a semisimple group) of Lie groups carries over to the general case. Section 36 deals with compact groups. Sections 37 (on countable bases and metrizability) and Section 38 together yield a proof that a locally compact connected group of finite dimension can be obtained as projective limit of a sequence of coverings of a single Lie group. Section 39 proves Goto’s theorem: a locally compact connected group of finite dimension is a Lie group if and only if it is arcwise connected. [This statement is not easily recognized in 39.6d; arcwise connectedness is expressed there as surjectivity of a certain map.] The result implies that a locally Euclidean group is a Lie group; this fact, which is commonly considered as the answer to Hilbert’s fifth problem, is not mentioned, however. Section 40 uses results on algebraic groups in order to show that in a semisimple real Lie group \(G\), the dimensions of closed proper subgroups are bounded by the maximal complex dimension of closed proper subgroups of the complexification of \(G\).
The book is very well suited to give an interested student a good familiarity with the structure of locally compact groups. Researchers using locally compact groups as a tool in other disciplines will find the results of Chapter H particularly enlightening and useful. The book is very well organized, and the style of writing is clear and convenient to read. There is very little to complain about. Even though the book is meant to be a student text, and although there is a bibliography containing 68 items, one might wish to see still more historical notes, acknowledgements and references to original sources. The reviewer learned from the author that he has profited much from lecture courses held by K. H. Hofmann and H. Salzmann; e.g., his approach to the Haar integral draws from these sources. The number of misprints seems to be very small.

MSC:
22D05 General properties and structure of locally compact groups
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
20E18 Limits, profinite groups
22A25 Representations of general topological groups and semigroups
12J10 Valued fields
43A05 Measures on groups and semigroups, etc.
54H15 Transformation groups and semigroups (topological aspects)
22A15 Structure of topological semigroups
22B05 General properties and structure of LCA groups
22C05 Compact groups
22D10 Unitary representations of locally compact groups
22D45 Automorphism groups of locally compact groups
22F05 General theory of group and pseudogroup actions
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