TY - GEN

T1 - Maximally almost periodic groups and respecting properties

T2 - 2nd Meeting in Topology and Functional Analysis, TFA 2018

AU - Gabriyelyan, Saak

N1 - Funding Information:
Acknowledgements The author gratefully acknowledges the financial support of the Operations Research Center (CIO) and the Department of Statistics, Mathematics and Informatics of the Uni-versidad Miguel Hernández.
Publisher Copyright:
© Springer Nature Switzerland AG 2019.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A maximally almost periodic topological (MAP) group G respects p if p(G)=p(G+), where G+ is the group G endowed with the Bohr topology and p stands for the subsets of G that have the property p. For a Tychonoff space X, we denote by (formula presented) the family of topological properties p of being a convergent sequence or a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. We study relations between different respecting properties from (formula presented) and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of (formula presented). We characterize respecting properties from (formula presented) in wide classes of MAP topological groups including the class of metrizable MAP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group (formula presented) is c0 -barrelled or g-barrelled, respectively. We prove that an lqc abelian Kω-group respects all properties (formula presented). As an application of the obtained results we show that a reflexive abelian group of finite exponent is a Mackey group.

AB - A maximally almost periodic topological (MAP) group G respects p if p(G)=p(G+), where G+ is the group G endowed with the Bohr topology and p stands for the subsets of G that have the property p. For a Tychonoff space X, we denote by (formula presented) the family of topological properties p of being a convergent sequence or a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. We study relations between different respecting properties from (formula presented) and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of (formula presented). We characterize respecting properties from (formula presented) in wide classes of MAP topological groups including the class of metrizable MAP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group (formula presented) is c0 -barrelled or g-barrelled, respectively. We prove that an lqc abelian Kω-group respects all properties (formula presented). As an application of the obtained results we show that a reflexive abelian group of finite exponent is a Mackey group.

KW - Free locally convex space

KW - Glicksberg property

KW - K-group

KW - Locally quasi-convex group

KW - Schur property

UR - http://www.scopus.com/inward/record.url?scp=85067355650&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-17376-0_7

DO - 10.1007/978-3-030-17376-0_7

M3 - Conference contribution

AN - SCOPUS:85067355650

SN - 9783030173753

T3 - Springer Proceedings in Mathematics and Statistics

SP - 103

EP - 136

BT - Descriptive Topology and Functional Analysis II - In Honour of Manuel López-Pellicer Mathematical Work, 2018

A2 - Ferrando, Juan Carlos

PB - Springer New York LLC

Y2 - 7 June 2018 through 8 June 2018

ER -