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An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed., Oberwolfach Preprints;2018,05

We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $\cal K$ of spaces, and suppose that $\phi:K \to G$ is an abstract (i.e. not necessarily continuous) surjective group homomorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $\phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting., Comment: Several corrections in v2. Two references added in v3. To appear in Transformation Groups

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as – decomposing them into local products of their Sylow p- subgroups, – providing new descriptions of periodic abelian torsion groups, and of – periodic abelian divisible groups and their torsion-free and their torsion components, – reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent p-groups, – providing a definition of a general p-rank for all locally compact abelian p-groups.

Completely distributive lattices have attracted attention since the early history of lattice theory, notably through the work of Raney. Indeed, the equivalence operates via the spectral theory for completely distributive lattices and the duality theory of continuous posets. The first section is devoted to a careful analysis of the category CD of completely distributive lattices and their morphisms within the framework of Galois adjunctions. The second section actually provides the basic tools of the free construction and discuss their properties. The chapter provides a sufficient measure of generality to derive the existence of free completely distributive lattices as established in the theorems by Dwinger and Markowsky. It concludes the discussion by introducing a functor from continuous posets to continuous lattices with the aid of the free constructions introduced earlier, and the authors show that this functor is the injective hull of a continuous poset.

We give a detailed description of all totally disconnected locally compact groups with only compact elements in which any two closed subgroups are permutable and their product being a closed subgroup again. This amounts to sharpening the classification results in [2, Section 14] while not being dependent upon them.

Let G be a locally compact topological group, G 0 the connected component of its identity element, and comp ( G ) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB ( G ) of all closed subgroups of G carries a compact Hausdorff topology called the Chabauty topology. Let F 1 ( G ) , respectively, R 1 ( G ) , denote the subspace of all discrete subgroups isomorphic to Z , respectively, all subgroups isomorphic to R . It is shown that a necessary and sufficient condition for G ∈ F 1 ( G ) ‾ to hold is that G is Abelian, and either that G ≅ R × comp ( G ) and G / G 0 is inductively monothetic, or else that G is discrete and isomorphic to a subgroup of Q . It is further shown that a necessary and sufficient condition for G ∈ R 1 ( G ) ‾ to hold is that G ≅ R × C for a compact connected Abelian group C .

If B is a compact connected Lie group and N a finite central subgroup, let $${f\colon B\to B/N}$$ be the associated covering morphism. The mapping cylinder $${{\mathrm{MC}}(f)}$$ is a compact monoid which we call a covering space semigroup. A prominent example is the classical Mobius band $${\mathbb{M}^2}$$ . An (L)-semigroup is a compact n-manifold X with connected boundary B together with a monoid structure on X such that B is a subsemigroup of X. Every covering space semigroup with $${|N|=2}$$ is an (L)-semigroup, and every nonorientable (L)-semigroup is a covering space semigroup. Here $${\mathbb{M}^2}$$ is a guiding example. In general, a covering space semigroup X is not a manifold but does have a well-defined manifold boundary. The study of covering space semigroups leads to the following Theorem. Let B be a compact connected Lie group with a central circle group as a direct factor. Then there exist infinitely many pairwise nonisomorphic covering space semigroups with boundary B, and each such semigroup is a retract of a compact connected Lie group.

Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$., Comment: 38 pages, 1 figure

The definition for an $$n$$ -dimensional Mobius manifold is given; $$n=2$$ yields the classical Mobius band. For $$n=1, 2$$ or $$4$$ , these manifolds are compact topological monoids, for $$n=8$$ , topological Moufang monoids. All of these manifolds are homeomorphic to retracts of topological groups. If $$n \le 4$$ , then any compact $$n$$ -manifold $$X$$ with connected boundary $$B$$ admitting the structure of a topological monoid with $$B$$ being a topological subsemigroup of $$X$$ is a retract of a topological group. The methods proposed here use the monoid structure of mapping cylinders of compact groups and the higher dimensional analogs of the monoid embedding of the classical Mobius band into the solid torus.

Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group., Comment: Minor corrections in v2 an in v3. The proof of Lemma 3.4 was incorrect in v1 - v3. A correct proof is now given in a separate erratum at the end of the article. Both the article and the erratum will appear in Crelle Journal Reine Angew. Mathematik

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

In the book “The Lie Theory of Connected Pro-Lie Groups” the authors proved the local splitting theorem for connected pro-Lie groups. George A. A. Michael subsequently proved this theorem for almost connected pro-Lie groups. Here his result is proved more directly using the machinery of the aforementioned book.

John Dauns died on June 4, 2009 of cancer in New Orleans at the age of 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results he and I had obtained in work we did in the mid-sixties of the last century ([17, 27, 18]) and which became known as the Dauns-Hofmann Theorem in C*-algebra theory. It has been known in the C*-text book and monograph literature up to the recent one under this name ([9, 20, 21, 39, 41]) since it provides apparently a useful tool that continues to be used in current research (see e.g. [7, 8, 25, 38, 40]). The problem with the historical record of the Dauns–Hofmann Theorem is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood for a long time. As John Dauns was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras (and other classes of algebras) by continuous sections in bundles (sometimes called continuous fields) [11–18], and since his work in this area was substantial and contributed much to a local culture of "sectional representation" at Tulane University ([19, 22, 27–34, 43, 44, 48]), I feel that it is justified to attempt a clarification. He can no longer participate himself in such an attempt; nor would he actually protest the occasional lack of acknowledgment were he alive, because that would be contrary to his ever gentle disposition. This small survey is devoted to shedding some additional light on this portion of John Dauns' work in mathematics; it is natural that it should have a personal tenor by a writer remembering his presence and his work as a collaborator.

For upper bounding the word error probability of block codes by application of the union bound, in general the uniform distance property (distance invariance) of the code must be guaranteed. This property says that the distance spectrum of a code is independent of the choice of the reference codeword. In the sequel new sufficient conditions for multilevel block 8-PSK modulation codes to be distance invariant are derived.

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [1] shortened the proof of Theorem 4.3 considerably.

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.

It is shown that the following four conditions are equivalent for a compact connected abelian group G: (i) the exponential function of G is open onto its image; (ii) G has arbitrarily small connected direct summands N such that G/N is a finite dimensional torus; (iii) the arc component G(a) of the identity is locally arcwise connected; (iv) the character group (G) over cap is a torsion free group in which every finite rank pure subgroup is free and is a direct summand.

For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

Are the arc components in a locally compact group Borel subsets? An affirmative answer is provide for locally compact groups satisfying the First Axiom of Count-ability. For general locally compact groups the question is reduced to compact connected Abelian groups. In certain models of set theory the answer is negative.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

and what nowadays, in more general form, is called domains, whileJ. D. LAWSON drew semigroup theoreticians' attention to a very natural class of compact semilattices having enough homomorphisms into the unit interval semilattice. The class of continuous lattices agrees with the class of Lawson semilattices. It generates a network of applications in theoretical computer science under the name "domain theory". - A hundred years after SOPHUS LIE's differentiable groups and semigroups, attention returned back to semigroups and Lie theory. Lie semigroup theory, initiated by E. B. VINBERG, G. I. OLSHANSKY, J. D. LAWSON and the author among others, infused a strong geometric and analytical flavor into topological semigroup theory and generated a new lines of application of semigroup theory such as in geometric control theory, and in the area of unitary representation theory of Lie groups, particulary in the area of holomorphic extensions of unitary representations. A respectable number of mongraphs and collections have been and are being written in this field.

For a compact connected group G, in a compact free G-space M every closed subset is the fixed point set of an autohomeomorphism if the weight of M is not bigger than that of G. In a compact group with nonsingleton components, every closed subset is the fixed point set of an autohomeomorphism if and only if the weight of the whole group is not bigger than that of the component of the identity.