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Article

A Note on the Topological Group c0

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
Axioms 2018, 7(4), 77; https://doi.org/10.3390/axioms7040077
Submission received: 28 September 2018 / Revised: 22 October 2018 / Accepted: 24 October 2018 / Published: 29 October 2018
(This article belongs to the Collection Topological Groups)

Abstract

:
A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f ∈ Tame ( c 0 ) . As an application we show that the Gromov’s compactification of c 0 is not a semigroup compactification. We discuss some questions.

1. Introduction

Recall that for every Hausdorff topological group G the algebra WAP ( G ) of all weakly almost periodic functions on G determines the universal semitopological semigroup compactification u w : G G w of G. This map is a topological embedding for many groups including the locally compact case. For some basic material about WAP ( G ) we refer to [1,2].
The question if u w always is a topological embedding (i.e., if WAP ( G ) determines the topology of G) was raised by Ruppert [2]. This question was negatively answered in [1] by showing that the Polish topological group G : = H + [ 0 , 1 ] of orientation preserving homeomorphisms of the closed unit interval has only constant WAP functions and that every continuous representation h : G I s ( V ) (by linear isometries) on a reflexive Banach space V is trivial. The WAP triviality of H + [ 0 , 1 ] was conjectured by Pestov.
Recall also that for G : = H + [ 0 , 1 ] every Asplund (hence also every WAP) function is constant and every continuous representation G Iso ( V ) on an Asplund (hence also reflexive) space V must be trivial [3]. In contrast one may show (see [4,5]) that H + [ 0 , 1 ] is representable on a (separable) Rosenthal space (a Banach space is Rosenthal if it does not contain a subspace topologically isomorphic to l 1 ).
We have the inclusions of topological G-algebras
WAP ( G ) Asp ( G ) Tame ( G ) RUC ( G ) .
For details about Tame ( G ) and definition of Asp ( G ) see [5,6,7]. We only remark that f Tame ( G ) if and only if f is a matrix coefficient of a Rosenthal representation. That is, there exist: a Rosenthal Banach space V; a continuous homomorphism h : G I s ( V ) into the topological group of all linear isometries V V with strong operator topology; two vectors v V ; ψ V * (the dual of V) such that f ( g ) = ψ ( h ( g ) v ) for every g G .
Similarly, it can be characterized f Asp ( G ) replacing Rosenthal spaces by the larger class of Asplund spaces. A Banach space is Asplund if the dual of every separable subspace is separable. Every reflexive space is Asplund and every Asplund is Rosenthal. A standard example of an Asplund but nonreflexive space is just c 0 .
Recall that c 0 , as an additive abelian topological group, is not representable on a reflexive Banach space by a well-known result of Ferri and Galindo [8]. In fact, WAP ( c 0 ) separates the points but not points and closed subsets. The group c 0 admits an injective continuous homomorphism h : c 0 I s ( V ) with some reflexive V but such h cannot be a topological embedding.
Presently it is an open question if every topological group (abelian, or not) G is Rosenthal representable and if Tame ( G ) determines the topology of G. Note that the algebra Tame ( G ) appears as an important modern tool in some new research lines in topological dynamics motivating its detailed study [5,7].
One of the good reasons to study Tame ( G ) is a special role of tameness in the dynamical Berglund-Fremlin-Talagrand dichotomy [5]; as well as direct links to Rosenthal’s l 1 -dychotomy. In a sense Tame ( G ) is a set of all functions which are not dynamically massive.
By these reasons and since H + [ 0 , 1 ] is Rosenthal representable, it seems to be an attractive concrete question if c 0 is Rosenthal representable and it is worth studying how large is Tame ( c 0 ) . In the present work we show that Tame ( c 0 ) is quite small (even for the discrete copy of c 0 , see Theorem 3).
Theorem 1.
Tame ( c 0 ) does not separate the unit sphere S and 2 S .
So, the closures of S and 2 S intersect in the universal tame compactification of c 0 (a fortiori, the same is true for the universal Asplund (HNS) semigroup compactification).
Another interesting question is if c 0 admits an embedding into a metrizable semigroup compactification. Note that any metrizable semigroup compactification of H + [ 0 , 1 ] is trivial.
In Section 3 we show that the Gromov’s compactification γ : c 0 P , which is metrizable (and γ is a G-embedding), is not a semigroup compactification.
Theorem 2.
Let γ : c 0 P be the Gromov’s compactification of the metric space ( c 0 , d 1 + d ) , where d ( x , y ) : = | | x y | | . Then γ is not a semigroup compactification.
This gives an example of a naturally defined separable unital (original topology determining) G-subalgebra of RUC ( G ) (for G = c 0 ) which is not left m-introverted in the sense of [9].

2. Tame Functions on c0

Recall that a sequence f n of real-valued functions on a set X is said to be independent if there exist real numbers a < b such that
n P f n 1 ( , a ) n M f n 1 ( b , )
for all finite disjoint subsets P , M of N . Every bounded independent sequence is an l 1 -sequence [10].
As in [6,7] we say that a bounded family F of real-valued (not necessarily continuous) functions on a set X is a tame family if F does not contain an independent sequence.
Let G be a topological group, f : G R be a real-valued function. For every g G define f g : G R as ( f g ) ( x ) = f ( g x ) (for multiplicative G). Denote by RUC ( G ) the algebra of all bounded right uniformly continuous functions on G. So, f RUC ( G ) means that f is bounded and for every ϵ > 0 there exists a neighborhood U of the identity e (of the multiplicative group G) such that | f ( u x ) f ( x ) | < ϵ for every x G and u U . This algebra RUC ( G ) corresponds to the greatest G-compactification G β G G of G (with respect to the left action), greatest ambit of G.
We say that f RUC ( G ) is a tame function if the orbit f G : = { f g } g G is a tame family. That is, f G does not contain an independent sequence; notation f Tame ( G ) .

2.1. Proof of Theorem 1

We have to show that Tame ( c 0 ) does not separate the spheres S and 2 S (where S : = { x c 0 : | | x | | = 1 } ). In fact we show the following stronger result.
Theorem 3.
Let G = c 0 be the additive group of the classical Banach space c 0 . Assume that f : c 0 R be any (not necessarily continuous) bounded function such that
f ( x ) a | | x | | = 1 b f ( x ) | | x | | = 2
for some pair a < b of real numbers. Then f is not a tame function on the discrete copy of the group c 0 .
Proof. 
For every n N consider the function
f n : c 0 R , x f ( e n + x ) ,
where e n is a vector of c 0 having 1 as its n-th coordinate and all other coordinates are 0. Clearly, f n = f g n where g n = e n c 0 . We have to check that f G is an untame family. It is enough to show that the sequence { f n } n N in f G is an independent family of functions on c 0 . We have to show that for every finite nonempty disjoint subsets I , J in N the intersection
n I f n 1 ( , a ] n J f n 1 [ b , )
is nonempty.
Define v = ( v k ) k N c 0 as follows: v j = 1 for every j J and v k = 0 for every k J . Then
(1)
v c 0 and | | v | | = 1 .
(2)
| | e i + v | | = 1 , f i ( v ) = f ( e i + v ) a for every i I .
(3)
| | e j + v | | = 2 , f j ( v ) = f ( e j + v ) b for every j J .
So we found v such that
v n I f n 1 ( , a ] n J f n 1 [ b , ) .
 □
Corollary 1.
The bounded RUC function
f : c 0 [ 1 , 1 ] , x | | x | | 1 + | | x | |
is not tame on c 0 (even on the discrete copy of the group c 0 ).
Proof. 
Observe that f ( S ) = 1 2 , f ( 2 S ) = 2 3 and apply Theorem 3. □
Theorem 3 remains true for the spheres r S and 2 r S for every r > 0 . In the case of Polish c 0 it is unclear if the same is true for any pair of different spheres around the zero. If, yes then this will imply that Tame ( c 0 ) does not separate the zero and closed subsets. The following question remains open even for any topological group [5,7].
Question 1.
Is it true that Tame ( c 0 ) separates the points and closed subsets ? Is it true that Polish group c 0 is Rosenthal representable ?

3. Gromov’s Compactification Need Not Be a Semigroup Compactification

Studying topological groups G and their dynamics we need to deal with various natural closed unital G-subalgebras A of the algebra RUC ( G ) . Such subalgebras lead to G-compactifications of G (so-called G-ambits,11]). That is we have compact G-spaces K with a dense orbit G z K such that the Gelfand algebra which corresponds to the compactification G K , g g z is just A . Frequently but not always such compactifications are the so-called semigroup compactifications, which are very useful in topological dynamics and analysis. Compactifications of topological groups already is a fruitful research line. See among others [12,13,14] and references there. In our opinion semigroup compactifications deserve even much more attention and systematic study in the context of general topological group theory.
A semigroup compactification of G is a pair ( α , K ) such that K is a compact right topological semigroup (all right translations are continuous), and α is a continuous semigroup homomorphism from G into K, where α ( G ) is dense in K and the left translation K K , x α ( g ) x is continuous for every g G .
One of the most useful references about semigroup compactifications is a book of Berglund, Junghenn and Milnes [9]. For some new directions (regarding topological groups) see also [3,4,15,16].
Question 2.
Which natural compactifications of topological groups G are semigroup compactifications? Equivalently which Banach unital G-subalgebras of RUC ( G ) are left m-introverted (in the sense of [9])?
Recall that left m-introversion of a subalgebra A of RUC ( G ) means that for every v A and every ψ M M ( A ) the matrix coefficient m ( v , ψ ) belongs to A , where
m ( v , ψ ) : G R , g ψ ( g 1 v )
and M M ( A ) A * denotes the spectrum (Gelfand space) of A .
It is not always easy to verify left m-introversion directly. Many natural G-compactifications of G are semigroup compactifications. For example, it is true for the compactifications defined by the algebras RUC ( G ) , Tame ( G ) , Asp ( G ) , WAP ( G ) . Of course, the 1-point compactification is a semitopological semigroup compactification for any locally compact group G.
As to the counterexamples. As it was proved in [3], the subalgebra UC ( G ) : = RUC ( G ) LUC ( G ) of all uniformly continuous functions is not left m-introverted for G : = H ( C ) , the Polish group of homeomorphisms of the Cantor set.
In this section we show that the Gromov’s compactification of a metrizable topological group G need not be a semigroup compactification.
Let ρ be a bounded metric on a set X. Then the Gromov’s compactification of the metric space ( X , ρ ) is a compactification γ : X P induced by the algebra A which is generated by the bounded set of functions
{ ρ z : X R , ρ z ( x ) = ρ ( z , x ) } z X .
Then γ always is a topological embedding. If X is separable then P is metrizable. Moreover, if ( X , ρ ) admits a continuous ρ -invariant action of a topological group G then γ is a G-compactification of X; see [17].
Here we examine the following particular case. Let G be a metrizable topological group. Choose any left invariant metric d on G. Denote by γ : G P the Gromov’s compactification of the bounded metric space ( G , ρ ) , where ρ = d 1 + d .
Consider the following natural bounded RUC function
f : G R , x | | x | | 1 + | | x | |
where | | x | | : = d ( e , x ) . By A f we denote the smallest closed unital G-subalgebra of RUC ( G ) which contains f G = { f g : g G } . Then A f is the algebra which corresponds to the compactification γ . Indeed, ρ g 1 ( x ) = ρ ( g 1 , x ) = ( f g ) ( x ) for every g , x G .

Proof of Theorem 2

We have to prove Theorem 2.
Proof. 
By the discussion above, the unital G-subalgebra A f of RUC ( G ) associated with γ is generated by the orbit f G , where f : G R , f ( x ) = | | x | | 1 + | | x | | . Since c 0 is separable the algebra A f is separable. Hence, P is metrizable. If we assume that γ is a semigroup compactification then the separability of A f guarantees by [4] ( Prop. 6.13) that A f Asp ( G ) . On the other hand, since Asp ( G ) Tame ( G ) , and f A f we have f Tame ( G ) . Now observe that f separates the spheres S and 2 S and we get a contradiction to Corollary 1. □
Question 3.
Is it true that the Polish group c 0 admits a semigroup compactification α : c 0 P such that P is metrizable and α is an embedding? What if P is first countable?
This question is closely related to the setting of this work. Indeed, by [4] (Prop. 6.13) (resp., by [4] (Cor. 6.20)) the metrizability (first countability) of P guarantees that the corresponding algebra is a subset of Asp ( G ) (resp. of Tame ( G ) ).

Funding

This research received no external funding.

Conflicts of Interest

The author declares no conflicts of interest.

References

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Megrelishvili, M. A Note on the Topological Group c0. Axioms 2018, 7, 77. https://doi.org/10.3390/axioms7040077

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Megrelishvili M. A Note on the Topological Group c0. Axioms. 2018; 7(4):77. https://doi.org/10.3390/axioms7040077

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Megrelishvili, Michael. 2018. "A Note on the Topological Group c0" Axioms 7, no. 4: 77. https://doi.org/10.3390/axioms7040077

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