TY - JOUR

T1 - Topological properties of the group of the null sequences valued in an Abelian topological group

AU - Gabriyelyan, S.

N1 - Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - For a Hausdorff Abelian topological group X, we denote by F0(X) the group of all X-valued null sequences endowed with the uniform topology. We prove that if X is an (E)-space (respectively, a strictly angelic space or a Š-space), then so is F0(X). We essentially simplify and clarify the theory of properties respected by the Bohr functor on Abelian topological groups, denoted below by X→X+. We prove that for a complete maximally almost periodic group X, the group X shares with X+ the same functionally bounded sets iff it shares the same compact sets and X+ is a μ-space. We show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X is totally disconnected, 2) F0(X) is a Schwartz group, 3) F0(X) respects compactness, 4) F0(X) has the Schur property. So, if a LCA group X is not totally disconnected, the group F0(X) is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X the group F0(X) is monothetic and every real-valued uniformly continuous function on F0(X) is bounded.

AB - For a Hausdorff Abelian topological group X, we denote by F0(X) the group of all X-valued null sequences endowed with the uniform topology. We prove that if X is an (E)-space (respectively, a strictly angelic space or a Š-space), then so is F0(X). We essentially simplify and clarify the theory of properties respected by the Bohr functor on Abelian topological groups, denoted below by X→X+. We prove that for a complete maximally almost periodic group X, the group X shares with X+ the same functionally bounded sets iff it shares the same compact sets and X+ is a μ-space. We show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X is totally disconnected, 2) F0(X) is a Schwartz group, 3) F0(X) respects compactness, 4) F0(X) has the Schur property. So, if a LCA group X is not totally disconnected, the group F0(X) is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X the group F0(X) is monothetic and every real-valued uniformly continuous function on F0(X) is bounded.

KW - (E)-space

KW - Glicksberg property

KW - Group of null sequences

KW - Locally compact group

KW - Monothetic group

KW - Schur property

KW - Strictly angelic space

KW - Š-space

UR - http://www.scopus.com/inward/record.url?scp=84965054181&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2016.04.013

DO - 10.1016/j.topol.2016.04.013

M3 - Article

AN - SCOPUS:84965054181

VL - 207

SP - 136

EP - 155

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -