A note on locally compact subsemigroups of compact groups

Semigroup Forum - 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...     

Embedding compact t-semigroups into compact uniquely divisible semigroups

Communicated by J. D. Lawson     

A compact group which is not Valdivia compact

A compact space is Valdivia compact if it can be embedded in a Tikhonov cube in such a way that the intersection is dense in , where is the sigma-product ( the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

     

A normal countably compact not absolutely countably compact space

We construct an example of a normal countably compact not absolutely countably compact space. We also prove that every hereditarily normal countably compact space is absolutely countably compact and suggest a method for construction of hereditarily normal spaces without property .

     

Hilbert compacts,compact ellipsoids,and compact extrema

We consider a system of so-called Hilbert compacts K(H) in a Hilbert space H; those Hilbert compacts admit a two-sided estimate by compact ellipsoids in H. For functionals in H, we introduce the notion of a compact extremum achieved at a certain base with respect to the imbedding in K(H). An example of the K-extremum of a variational functional in the Sobolev space W ₂¹ is considered.     

Torsion-complete and algebraically compact groups with compact endomorphism rings

We give a complete characterization of torsion-complete and algebraically compact abelian groups whose endomorphism rings admit a compact ring topology.     

Isometric coactions of compact quantum groups on compact quantum metric spaces

We propose a notion of isometric coaction of a compact quantum group on a compact quantum metric space in the framework of Rieffel, where the metric structure is given by a Lipnorm. Within this setting we study the problem of the existence of a quantum isometry group.     

Homomorphic actions of ℍ on a compact groupon a compact group

The way the additive semigroup [0, ∞) carrying the usual topology can act as a semigroup of homomorphisms on a compact group is studied in this paper. It is shown that such actions of [0, ∞) on a compact group G can be completely described in terms of continuous homomorphisms from [0, ∞) into G. Furthermore, it is proven that each such action of [0, ∞) on a compact group G has a unique extension to an action of the additive group of all real numbers endowed with the usual topology on G so that a similar characterization is also valid for actions of the real numbers on a compact group.     

Isometries of spaces of compact or compact convex subsets of metric manifolds

The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.     

Both compact and sequentially compact sets in abelian topological group

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,

(1)

if G is a Hausdorff abelian topological group and μ:N²→G is countably additive, then the range μ(N²)={μ(A):A⊆N} is compact metrizable;

(2)

if X is a Hausdorff locally convex space and {xj}⊂X, then F={_∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X^′〉-polar topology having the same subseries convergent series as the weak topology.