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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.

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.

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.

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.

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.

For each pair (𝔤,𝔞) consisting of a real Lie algebra 𝔤 and a subalgebra a of some Cartan subalgebra 𝔥 of 𝔤 such that [𝔞, 𝔥]∪ [𝔞, 𝔞] we define a Weyl group W(𝔤, 𝔞) and show that it is finite. In particular, W(𝔤, 𝔥,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra 𝔤, the normalizer N(𝔥, G) acts on the finite set Λ of roots of the complexification 𝔤c with respect to hc, giving a representation π : N(𝔥, G) S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(𝔥) of G with respect to h; the image is isomorphic to W(𝔤, 𝔥), and C(𝔥)= {gG : Ad(g)(h)— h e [h,h] for all h e h }. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ 𝔥 the set 𝔥⊂ ϕ(b) remains finite as ϕ ranges through the full group of inner automorphisms of 𝔤.

A compact p-group G (p prime) is called near abelian if it contains an abelian normal subgroup A such that G/A has a dense cyclic subgroup and that every closed subgroup of A is normal in G. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular) if every pair of compact subgroups commutes setwise. We show that for p ≠ 2 a compact p-group G is near abelian if and only if it is quasihamiltonian. The case p = 2 is discussed separately.

We show that the set of compact subgroups in a connected Lie group is inductive. In fact, a locally compact group G G has the inductivity property for compact subgroups if and only if the factor group G / G 0 G/{G_0} modulo the identity component has it.

BY KARL H. HOFMANNFachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7,Z)-6100 Darmstadt, GermanyAND SIDNEY A. MORRISFaculty of Informatics, University Wollongong, of Wollongong, NSW 2522, Australia(Received 6 July 1992)IntroductionIt is well-known that a compact connected abelian group G has weight w(G) lessthan or equal to the cardinality c of the continuum if and only if it is monothetic;that is, if and only if it can be topologically generated by one element. Hofmann andMorris [2] extended this by showing that a compact connected (not necessarilyabelian) group can be topologically generated by two elements if and only ifw(G) c, the compact connectedgroup G is not topologically generated by any finite set. In this case we look fortopological generating sets which are, in some sense, 'thin'. A subset GX i os callef dsuitable if it topologically generates G, is discrete and (?\{1} is close wher, d ien 1 isthe identity o G.f If X has the smallest cardinality of any suitabl G thee subsen t ofG is called a special subset and its cardinality is denoted b s(G).y In [2] it was provedthat if G is a connected locally compact group w(G) with > c, then s(G)^° = w(Cr)

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and references to the history of the discussion are given at the end of the paper., 17 pages; we have cut some points ; to appear in Math. Proc. Cambridge Phil. Soc

ZusammenfassungWir skizzieren die Konturen einer Theorie der Halbgruppen, also eines Zweiges der Mathematik, der sich ganz im Gegensatz zu der im 19. Jahrhundert geschaffenen Gruppentheorie im 20. Jahrhundert entwickelte. Als Arbeitshypothese schlagen wir vor, daß die Anfänge der Halbgruppentheorie bereits im Werk von Sophus Lie angelegt sind. Über eine präzise Definition des Gruppenbegriffs war man sich im 19. Jahrhundert noch keineswegs im Klaren. Objekte, die man später als Halbgruppen bezeichnen wird, heißen damals Gruppen. Lie ist sich des Unterschieds wohl bewußt und versucht, sich mit diesem Problem auseinan-derzusetzen. Wir schließen mit einem Ausblick auf die gegenwärtige Forschung über die Lieschen Fundamentalsätze im Rahmen der Halbgruppentheorie.AbstractAfter framing the outlines of a theory of semigroups as a mathematical domain developed in the 20th century in contrast with group theory as an achievement of the 19th, we propose, as a working hypothesis, the claim that the beginning of semigroup theory is indicated in the work of Sophus Lie. Insecurity on the axiomatic definition of a group is still widespread in the nineteenth century; objects which later will be recognized in their own right as semigroups are called groups at that period. Lie is conscious of the difference and attempts to deal with it. We give an outlook on contemporary research on Lie's Fundamental Theorems in the context of semigroups.

Let G denote a Lie group with Lie algebra g and with a subsemigroup S whose infinitesimal generators generate g . We construct real analytic curves y: R+ S such that y(O) is a preassigned tangent vector of S at the origin and that y(t) is in the interior of S for all positive t . Among the consequences, we find that the boundary of S has to be reasonably well behaved. Our procedure involves the construction of certain linear generating sets from a given Lie algebra generating set, and this may be of independent interest.

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Introduction The basic theory of exponential semigroups in Lie groups Weyl groups and finiteness properties of Cartan subalgebras Lie semialgebras More examples Test algebras and groups Groups supporting reduced weakly exponential semigroups Roots and root spaces Appendix: The hyperspace of a locally compact space References Index.

We answer a question raised by V.G.Pestov in the affirmative., Comment: 2 pages

Every continuous idempotent multiplication on a space induces an idempotent comultiplication on its cohomology algebra over a commutative ring and a homomorphic idempotent multiplication on each homotopy group. We classify all idempotent comultiplications on any graded anticommutative algebra A∗ over a principal ideal domain K up to degree 2 provided the degree 1 component A1 is torsion free and the degree 2 component A2 is of rank 1. All algebraic possibilities can be topologically realized. We also describe all homomorphic idempotent multiplications on arbitrary groups. This allows a complete classification up to homotopy of all idempotent multiplications on aspherical CW-complexes. For surfaces we obtain an explicit list. Notably, the Klein bottle allows infinitely many nonhomotopic idempotent multiplications, but all other surfaces with nonabelian fundamental group have only the projections as idempotent multiplications (up to homotopy). Introduction. Idempotent multiplications on sets and topological spaces have been considered by many authors, for instance as an axiomatic approach to the averaging operation (sample: [2, 3, 10]). If X denotes a connected topological space, then the existence of H-space structures places severe restrictions on the structure of X. (See, for instance, [6] or [16].) This is due to the presence of homotopy identities on both sides. If, however, one considers idempotent multiplications μ : X ×X → X, that is, multiplications which satisfy μ(x, x) = x for all x ∈ X, then no restriction follows from the presence of such multiplications, since every space X allows the two idempotent multiplications p1, p2 : X × X → X, p(x, y) = x and q(x, y) = y for all x, y ∈ X. These are the so-called trivial multiplications. On the other hand, the existence of nontrivial idempotent multiplication again forces restrictions on the space. We wish to illustrate this by discussing idempotent multiplications on suitable classes of spaces. The Received by the editors on September 1, 1988, and in revised form on October 24, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium

Two commuting real matrices A and B have commuting exponentials exp A and exp B, a fact observed for instance in linear algebra or differential equations courses. The converse implication is false. A clarification of this phenomenon is proposed that makes use of the theory of the exponential function exp: g → G of a real Lie group G and its singularities. In Section 1, a catalog of low-dimensional examples illustrates various ways that, for two elements X,Y ∈ g, the commuting of exp X and exp Y in G may fail to entail the commuting of X and Y in g. In Section 2, consequences of the relation [exp X,exp Y ]= 1 are inspected, whereby certain regularity assumptions on X and Y are made. A regular element Y of the Lie algebra g determines a Cartan subalgebra h = g 0 (Y )o fg and a certain subgroup WY of the (finite!) Weyl group of g with respect to the Cartan subalgebra h. If, additionally, the exponential function is regular at X and at Y , then the ordered pair (X,Y )i ss a i dt ob ei ngeneral position. If (X,Y) is in general position, then the relation [exp X,exp Y ]= 1 in G permits the definition of a certain element w(X,Y) ∈W Y .L e tz(g) denote the center of g. It is shown that, if exp X and exp Y commute in G for (X,Y) in general position

In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special proper- ties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given. The spectral theory of lattices serves the purpose of representing a lattice L as a lattice of open sets of a topological space X. The spectral theory of rings and algebras practically reduces to this situation in view of the fact that for the most part one considers the lattice of ring (or algebra) ideals and then develops the spectral theory of that lattice. (The occasional complications due to the fact that ideal products are not intersections have been dealt with elsewhere, e.g. (4).) The lattice of all ring (or algebra) ideals forms a particular kind of continuous lattice, namely an algebraic lattice. It should be the case, however, that more general continuous lattices arise in the study of certain objects endowed with both an algebraic and a topological structure. Indeed the first author has shown in a seminar report using the concept of Pedersen's ideal that the closed ideals of a C*-algebra always form a distributive continuous lattice with respect to intersection. How widely continuous lattices occur in such contexts is, at this point, a largely uncharted sea. We show that the spectrum of a distributive continuous lattice is a locally quasicompact sober space (see 2.6 for the definition of sobriety). This implies, e.g., that the space of closed two sided prime ideals of a C*-algebra is locally quasicompact in the hull-kernel topology. (This is usually proved for primitive ideals by different methods.) On the other hand, the question of what topological consequences follow

In recent year there has been considerable interest in local semigroups in Lie groups, partly from the view point of geometric control theory, partly from the view point of differential geometry (for the former aspect see [Br, Hi, JS,l, for the latter [Lo, O11 and 2, Pa, Ro, Vi]). Lie theory relates local subgroups of a Lie group with subalgebras of its Lie algebra; in just the same spirit it relates certain local subsemigroups with Lie semialgebras in its Lie algebra. Here a Lie semialgebra in a finite dimensional real Lie algebra is a wedge (i.e., a topologically closed subset which is stable under addition and multiplication by non-negative scalars) for which there is a neighborhood B of 0 in L such that for all X, Y~ B the CampbellHausdorff product X * Y = X + Y+ 89 Y-l+ ... is well defined and in addition stays in W for X, Ye W. If W is a Lie semialgebra in L, then W W is a Lie subalgebra [HL 2,l. If W is any wedge in L and if W contains the Lie algebra [W,W] generated by all commutators w,w" with w,w'~W, then W is a Lie semialgebra. Lie semialgebras arising in this fashion we will call trivial. We shall prove the following

We prove a Baire category theorem for continuous lattices and derive category theorems for non-Hausdorff spaces which imply a category theorem of Isbell's and have applications to the spectral theory of C*-algebras. The same lattice theoretical methods yield a proof of de Groot's category theorem for regular subcompact spaces.

LetS be an open subsemigroup of a Lie groupG with\(1 \in \bar S\). We shall show that for every congruence ℋ onS with closed congruence classes there exists an open neighborhoodU of1 inG and a foliation ofS ∩U whose leaves locally coincide with both the congruence classes of ℋ and the cosets of a normal analytic subgroup ofG which is uniquely determined by ℋ.

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

The structure of the three dimensional Lie group (2,ℝ) and its universal covering group G is surveyed in an explicit fashion with detailed computational and geometrical information on their exponential functions and one parameter groups. In particular, a new global parametrisation of the group G is given which allows a convenient description of the exponential function and its singularities.

S(X) is the semigroup of all continuous self maps of the topological space X and for any semigroup S, Cong(S) will denote the complete lattice of congruences on S. Cong(S) has a zero Z and a unit U. Specifically, Z = {(a, a):a ∈ S} and U = S × S. Evidently, Z and U are distinct if S has at least two elements. By a proper congruence on S we mean any congruence which differs from each of these. Since S(X) has more than one element when X is nondegenerate, we will assume without further mention that the spaces we discuss in this paper have more than one point. We observed in [4] that there are a number of topological spaces X such that S(X) has a largest proper congruence, that is, Cong(S(X)) has a unique dual atom which is greater than every other proper congruence on S(X). On the other hand, we also found out in [5] that it is also common for S(X) to fail to have a largest proper congruence. We will see that the situation is quite different at the other end of the spectrum in that it is rather rare for S(X) not to have a smallest proper congruence. In other words, for most spaces X, Cong(S(X)) has a unique atom which is smaller than every other proper congruence.

Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

We show that compact extensions of compactly generated (locally compact) pro-Lie groups are pro-Lie, and that compactly generated (locally com- pact) nilpotent groups are pro-Lie. As a consequence compact extensions of com- pactly generated nilpotents are pro-Lie. We give examples indicating limitations to extending our results. 0. Introduction. In work which culminated in (5), the Fourier-Stieltjes algebra B(G) of certain locally compact groups G was studied. The groups in question were all compact modulo a closed normal nilpotent subgroup, and at first results were obtained only for such groups which in addition were either Lie groups, or else were compactly generated. It then became clear that these results could be simplified if all such groups could be shown to be pro-Lie, and that is the purpose of the present paper. To fix notation, we consider locally compact groups G with a closed normal subgroup N and quotient group K = G/N. We obtain various results leading up to the main point of the paper: First, if N is pro-Lie and K is compact, then the compact normal co-Lie subgroups of N which intersect in (1) may be chosen to be /^-invariant. If, on the other hand, G is compactly generated, K is pro-Lie, and JV is a compactly generated central subgroup of G, then G is again pro-Lie. Finally, if K is compact and N is a compactly generated pro-Lie group, then G is also pro-Lie. These results are then used to establish our main result: If N is compactly generated nilpotent and K is compact, then G is pro-Lie. In order to put our results in historical perspective, we point out that the fundamental result that almost connected groups are pro-Lie fits into our setting, with N — G0 (see (6, Theorem 4.6)). Grosser and Moskowitz (3, Theorem 2.11), extending a result of S. P. Wang, showed that (SIN)B groups are B pro-Lie for compact groups B containing the inner automorphism group of G. In particular, all (Z) groups are pro-Lie. The authors gratefully acknowledge the support of the National Science Founda- tion during the course of the research contained herein. 1. Definition. Suppose G is a locally compact group and B a group of automor- phisms of G. Then G is B pro-Lie if there is a family {Ka} of compact normal

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions ( x , y ) → x ° y , x \ y , x / y : M × M → M (respectively, U × U → M for an open neighbourhood U of e in M ) such that the following conditions are satisfied: (i) x ° e = e ° x = e , (ii) x ° ( x \ y ) = y , and (iii) ( x / y )° y = x for all x , y e M (respectively, U ). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L ( M ) in e is equipped with an anticommutative bilinear operation ( X , Y ) →[ X , Y ] and a trilinear operation ( X , Y , Z ) →〈 X , Y , Z 〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L ( M ) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L ( M ) by defining X ° Y = (exp| B ) −1 ((exp X )° (exp Y )) for X and Y in a neighbourhood C of 0 in B such that (exp C ) ° (exp C ) ⊂ V . Similarly, we transport / and \.

We shall prove the following theorem concerning compact connected abelian groups: For any element x in a compact connected abelian group there is an element y such that the closure of the cyclic group generated by y is connected and contasns x. This is a generalisation of the well known fact that any compact connected group whose topology admits a basis of at most continuum cardinality is monothetic (i.e. contains a dense cyclic subgroup) [11]. Our proof leans heavily on the theory of duality for locally compact abelian groups which seems to be an appropriate procedure in view of the fact that the character of this theorem is typically abelian; it must remain undecided whether there is an approach using the fact that a compact connected abelian group can be approximated by torus groups. We shall use the theorem mentioned above to determine the structure of compact loops in which every pair of elements generates an abelian subgroup. These loops are called di-associative and commutative. In a compact group the connected component of the identity is the set of all divisible elements [10]; we have been unable to establish the corresponding result for compact di-associative and commutative loops; it is easy to see that divisibility implies connectedness; as long as the converse is not established, the concept of divisibility has to replace the notion of connectivity wherever it occurs in the theory of abelian groups to obtain the analogous results for compact diassociative commutative loops. Our main result is the following: The set D of all divisible elements in a compact di-associative commutative loop G is a closed subgroup of the center of G; no element in G/D other than the identity is divisible and G/D contains no nontrivial connected subgroup. This implies in particular that every compact di-associative commutative divisible loop is a group. Moreover, G/D is a direct product of closed characteristic subloops, one for each prime p, having the property to be divisible by all natural numbers relatively prime to p. Up to the structure of these constituents the structure of compact di-associative and commutative loops is now rather completely described. One would, however, like to know whether the connected component C of the unit in G can actually be larger than the set D of divisible elements.