The Tubby Torus as a Quotient Group

Abstract

Let E be any metrizable nuclear locally convex space and $\widehat{E}$ the Pontryagin dual group of E. Then the topological group $\widehat{E}$ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

1. Introduction and Preliminaries

The Separable Quotient problem for Banach spaces has its roots in the 1930s and is due to Stefan Banach and Stanisław Mazur. While a positive answer is known for various classes of Banach spaces [1], such as reflexive Banach spaces, weakly compactly generated Banach spaces, and more generally Banach-like spaces [2], the general problem remains unsolved.

Problem 1.

(Separable quotient problem for Banach spaces) Does every infinite-dimensional Banach space have a quotient Banach space which is separable and infinite-dimensional?

The following problem stated in [3] is also unsolved, but a negative answer to it would give a negative answer to Problem 1.

Problem 2.

Does every infinite-dimensional Banach space have a quotient topological group which is homeomorphic to the countably infinite product, ${\mathbb{R}}^{\omega }$, of copies of $\mathbb{R}$?

This suggests another question which we have not seen mentioned in the literature. We state the problem and answer it.

Question 1.

Does every infinite-dimensional Banach space have a quotient topological space which is homeomorphic to ${\mathbb{R}}^{\omega }$?

Question 1 has a positive answer, although it uses very powerful machinery due to Toruńczyk. It is known [4] that every infinite-dimensional Fréchet space F (that is, a complete metrizable locally convex space) is homeomorphic to an infinite-dimensional Hilbert space H. So an infinite-dimensional Banach space B (indeed an infinite-dimensional Fréchet space) is homeomorphic to an infinite-dimensional Hilbert space H, which obviously has the infinite-dimensional separable Hilbert space ${\ell }_2$ as a quotient. Further, by the separable case of Toruńczyk’s theorem which is known as the Kadec–Anderson theorem, the separable Fréchet space ${\mathbb{R}}^{\omega }$ is homeomorphic to ${\ell }_2$, from which the positive answer to Question 1 follows.

Noting that Problem 2 remains open, it is natural to ask if every infinite-dimensional Banach space has a quotient topological group which is a separable metrizable topological group which is infinite-dimensional as a topological space. This was answered in the positive by the following theorem.

Theorem 1.

[5] Every locally convex space E, which has a subspace which is an infinite-dimensional Fréchet space, has the tubby torus, ${\mathbb{T}}^{\omega }$, as a quotient group, where $\mathbb{T}$ is the compact circle group. In particular, this is the case if E is an infinite-dimensional Banach space.

We should mention the following result.

Theorem 2.

[6] If E is any infinite-dimensional Fréchet space which is not a Banach space, then E has the locally convex space ${\mathbb{R}}^{\omega }$ as a quotient vector space.

Corollary 1.

If E is any infinite-dimensional Fréchet space which is not a Banach space, then E has the tubby torus ${\mathbb{T}}^{\omega }$ as a quotient group.

One might suspect that every infinite-dimensional locally convex space has the tubby torus as a quotient group. This is shown to be false in [5] for the free locally convex space $\phi$ on a countably infinite discrete space. Indeed in [7] it is shown that if X is a countably infinite $k_{\omega }$-space, then the free topological vector space on X, which is a connected infinite-dimensional (in the topological sense) topological group, does not have the tubby torus as a quotient group or even any infinite-dimensional (in the topological sense) metrizable quotient group.

It was recently proved that free topological groups on infinite connected compact spaces also have the tubby torus as a quotient group.

Theorem 3.

[7] Let $F_G\left(X\right)$ and $A_G\left(X\right)$ be the Graev free topological group and the Graev free abelian topological group, respectively, on an infinite connected compact Hausdorff space. Then the connected topological groups $F_G\left(X\right)$ and $A_G\left(X\right)$ have the tubby torus ${\mathbb{T}}^{\omega }$ as a quotient group.

It follows from Theorem 2.5 of [3] that every non-metrizable connected locally compact abelian group has the tubby torus as a quotient group. But as a connected locally compact abelian group G is isomorphic as a topological group to the product ${\mathbb{R}}^n\times K$, for some non-negative integer n and compact abelian group K, and ${\mathbb{R}}^n$ and all compact metrizable groups are separable, we see that if G is non-separable then it is non-metrizable. So we obtain the following result as a consequence.

Theorem 4.

Every non-separable connected locally compact abelian group has the tubby torus as a quotient group.

As mentioned earlier, Problem 1 has been aswered for dual-like groups. In particular there is the following powerful and beautiful theorem.

Theorem 5.

[8] If B is the Banach space dual of any infinite-dimensional Banach space, then B has a separable infinite-dimensional quotient Banach space.

Corollary 2.

If B is the Banach space dual of any infinite-dimensional Banach space, then B has the tubby torus as a quotient group.

Recall that if G is a (Hausdorff) abelian topological group, then we denote by $\widehat{G}$ the group of all continuous homomorphisms of G into the circle group $\mathbb{T}$, where $\widehat{G}$ has the compact-open topology. There is a natural homomorphism $\alpha :G\to \widehat{\widehat{G}}$. The Pontryagin–van Kampen duality theorem is stated below and a discussion and proof appear in [9,10].

Theorem 6.

[9,10] If G is any locally compact abelian group then the map α is an isomorphism of topological groups of G onto $\widehat{\widehat{G}}$. Also, if H is a closed subgroup of the locally compact abelian group G, then $\widehat{H}$ is a quotient group of $\widehat{G}$, and if A is a quotient group of G, then $\widehat{A}$ is isomorphic as a topological group to a closed subgroup of $\widehat{G}$. Further, the map α restricted to H is an isomorphism of topological groups of H onto the subgroup $\alpha \left(H\right)$ of $\widehat{\widehat{G}}$.

The following is less well-known.

Let E be a locally convex space. As E is a topological group, the topological group $\widehat{E}$ consisting of all continuous group homomorphisms of E into $\mathbb{T}$ with the compact-open topology is a topological group, as is $\widehat{\widehat{E}}$. As mentioned above, there is a natural homomorphism of E into $\widehat{\widehat{E}}$.

Theorem 7.

[11] Proposition 15.2. Let E be a complete metrizable locally convex space (that is a Fréchet space). Then α is an isomorphism of topological groups of E onto $\widehat{\widehat{E}}$.

We note that Theorem 7 does not tell us whether, for example $\alpha$ restricted to a closed subgroup H of E is an isomorphism of topological groups of H onto the subgroup $\alpha \left(H\right)$ of $\widehat{\widehat{E}}$. In fact this is not always true. §11 of [12] gives an example of a closed subgroup H of a Fréchet space E such that $\alpha$ restricted to H is not an isomorphism of topological groups of H onto its image in $\widehat{\widehat{E}}$. To see how badly things can go “wrong”, we note Theorem 6.1 of [11]: Let E be a metrizable locally convex space. If E is not a nuclear space, then it has a discrete subgroup H such that there are no non-trivial continuous homomorphisms from span$\left(H\right)/H$ into $\mathbb{T}$, where span$\left(H\right)$ denotes the linear span in E of H.

Theorem 5 leads us then to the natural question:

Problem 3.

If E is any infinite-dimensional Fréchet space which does not have the weak topology and $\widehat{E}$ is its dual topological group, does $\widehat{E}$ have the tubby torus as a quotient group? In particular, is this the case for E a Banach space or a Schwartz space?

This question is open, however a positive answer is given for nuclear spaces in the next section.

2. The Main Result

Definition 1.

A topological group G is said to be reflexive if the natural mapping α from G to $\widehat{\widehat{G}}$ is an isomorphism of topological groups. The topological group G is said to be strongly reflexive if every closed subgroup and every Hausdorff quotient group of G is reflexive.

Theorem 8.

[12] (Theorem 20.35) Every complete metrizable nuclear locally convex space is strongly reflexive.

Proposition 1.

[11] (Proposition 17.1(c)) Let H be a closed subgroup of a strongly reflexive topological group G. Then $\widehat{H}$ is isomorphic as a topological group to a quotient group of $\widehat{G}$.

Theorem 9.

Let E be a metrizable nuclear locally convex space. Then $\widehat{E}$ has the tubby torus ${\mathbb{T}}^{\omega }$ as a quotient group if and only if E does not have the weak topology.

Proof. 

By Theorem 2 of [13], if H is a dense subgroup of the metrizable topogical group G, then $\widehat{G}$ is isomorphic as a topological group to $\widehat{H}$. So the dual group $\widehat{E}$ of E is isomorphic as a topological group to the dual group of the completion of E. So there is no loss of generality in assuming that E is complete. Further, the completion of a metrizable nuclear locally convex space is a metrizable nuclear locally convex space by Theorems 20.34 and 20.20 of [12].

The theorem in [14] says that a locally convex space E has the weak topology if and only if every discrete subgroup of E is finitely generated. However, its proof there gives rather more. Namely, the locally convex space E does not have the weak topology if and only if E contains a discrete free abelian subgroup S which is not finitely generated.

So if the metrizable nuclear locally convex space E does not have the weak topology, then it has a subgroup S isomorphic as a topological group to a restricted direct product of ${\mathbb{Z}}_i,i=1,2,\cdots ,n,\cdots$, where each ${\mathbb{Z}}_i$ is isomorphic as a topological group to the discrete $\mathbb{Z}$ of integers. Noting §3 of [15], we see that the dual group of this restricted direct product of ${\mathbb{Z}}_i$ is the tubby torus ${\mathbb{T}}^{\omega }$, and it then follows from Theorem 8 and Proposition 1 that $\widehat{E}$ has the tubby torus as a quotient group, as required.

On the other hand if the complete metrizable locally convex space E has the weak topology, then it is isomorphic as a locally convex space to ${\mathbb{R}}^{\omega }$. So its dual group $\widehat{E}$ is isomorphic as a topological group to the locally convex space $\phi$. However, as mentioned earlier, it is proved in [5] (and generalized in [7]), that $\phi$ does not have the tubby torus as a quotient group, which completes the proof. □