Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAlgebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters
186
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Assuming the existence of $mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $mathfrak c$ that admit a countably compact group topology. We show that for each $kappa in [mathfrak c, 2^mathfrak c]$ each of these groups has a countably compact group topology of weight $kappa$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^mathfrak c$ selective ultrafilters, there are at least $2^mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.
rate research
Read More
We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).
We prove that if there are $mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $kappa$ such that $kappa^omega=kappa$, there exists a group topology on the free Abelian group of cardinality $kappa$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).
We obtain a set of generalized eigenvectors that provides a generalized spectral decomposition for a given unitary representation of a commutative, locally compact topological group. These generalized eigenvectors are functionals belonging to the dual space of a rigging on the space of square integrable functions on the character group. These riggings are obtained through suitable spectral measure spaces.
In his seminal work cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if $H$ is a compact subgroup of a locally compact group $G$ and $S$ is a small (in the sense of Palais) $H$-slice in a proper $G$-space, then the action map $Gtimes Sto G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)to G/H$ is continuos and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach-Mazur compacta.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
