The Variety of Topological Groups Generated by the Class of All Banach Spaces

2019 ◽  
pp. 319-326 ◽  
Author(s):  
Sidney A. Morris ◽  
Carolyn E. McPhail
Keyword(s):  
Banach Spaces ◽  
Topological Groups

scholarly journals EQUIVARIANT GEOMETRY OF BANACH SPACES AND TOPOLOGICAL GROUPS

2017 ◽  
Vol 5 ◽  
Author(s):  
CHRISTIAN ROSENDAL
Keyword(s):  
Banach Spaces ◽  
Topological Groups ◽  
Isometric Actions

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, that is, continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

scholarly journals Separability of Topological Groups: A Survey with Open Problems

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 3 ◽  
Author(s):  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Quotient Space ◽  
Topological Groups ◽  
Open Problems ◽  
Infinite Products ◽  
Matrix Groups ◽  
Compact Spaces ◽  
Finite Dimensional ◽  
Separable Quotient Problem

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

scholarly journals Stochastic processes on totally disconnected topological groups

2003 ◽  
Vol 2003 (48) ◽  
pp. 3067-3089 ◽  
Author(s):  
S. V. Ludkovsky
Keyword(s):  
Stochastic Processes ◽  
Banach Spaces ◽  
Lie Groups ◽  
Local Fields ◽  
Loop Groups ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Loop Spaces ◽  
Totally Disconnected ◽  
Continuous Representations

Stochastic processes on totally disconnected topological groups are investigated. In particular, they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudodifferentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including the irreducible ones of these groups. In addition, stochastic processes on general Banach-Lie groups, loop monoids, loop spaces, and path spaces of manifolds on Banach spaces over non-Archimedean local fields are also investigated.

The generalized Yosida–Hewitt theorem

1994 ◽  
Vol 116 (3) ◽  
pp. 527-533 ◽  
Author(s):  
A. Basile ◽  
A. V. Bukhvalov ◽  
M. Ya. Yakubson
Keyword(s):  
Banach Spaces ◽  
Topological Groups ◽  
Unified Approach ◽  
Vector Lattices ◽  
Vector Valued Functions ◽  
Vector Valued

The Yosida–Hewitt (YH, for short) theorem [YH] has many versions and generalizations in diverse settings, e.g. functionals on vector lattices and spaces of vector-valued functions, measures with values in Banach spaces, topological groups and vector lattices, etc. In this paper we derive a very general form of the YH theorem dealing with the much more general case of operators acting in vector lattices (VLs, for short) and Banach spaces (BSs, for short). A unified approach to all settings mentioned above may be founded on decompositions for operators in VLs.

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