The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups

Abstract

Let $\Gamma$ be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group $G$. In this paper we study translation-invariant (TI) subspaces of $L^2(G)$ by elements of $\Gamma$. We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that $\Gamma$ be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of $G=\mathbb {R}^n$. We also extend the notion of the spectral function in $\mathbb {R}^n$ to the LCA setting. It is shown that spectral functions, initially defined in terms of $\Gamma$, do not depend on $\Gamma$. Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of $G$ with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.

Appendix

Here we give an alternative and simpler proof of [20, Theorem (9.12)]; all number references in this appendix are to [20].

Theorem

(9.12) Let $\tau$ be a topological isomorphism of $\mathbb {R}^a\times \mathbb {Z}^b\times F$ into $\mathbb {R}^c\times \mathbb {Z}^d\times E$, where $a, b, c, d$ are nonnegative integers and $F$ and $E$ are compact groups (not necessarily abelian). Then $a\le c$ and $a+b\le c+d$.

First a lemma. An element in a topological group is compact if it belongs to a compact subgroup of the group.

Lemma

Let $\tau$ be a topological isomorphism of $H$ into $G\times K$, where $H$ and $G$ are locally compact and $K$ is a compact group. If $H$ has no compact elements other than the identity, then $H$ is topologically isomorphic to a closed subgroup of $G$.

Proof

Let $\pi$ be the projection of $G\times K$ onto $G$. Since $\tau (H)$ is closed in $G\times K$ by (5.11), the image $\pi (\tau (H))$ is closed in $G$ by (5.18). Also, $\pi$ is one-to-one on $\tau (H)$. [If $(x,k_1)$ and $(x,k_2)$ in $\tau (H)$ have the same image $x$ in $G$, then $(e,k_1k_2^{-1})$ is in $\tau (H)$. Since $K$ is compact, $(e,k_1k_2^{-1})$ is a compact element in $\tau (H)$. Since only the identity of $\tau (H)$ is a compact element, $k_1=k_2$.]

Since $\pi$ is a closed mapping by (5.18), it is also a closed mapping of the closed subgroup $\tau (H)$ onto $\pi (\tau (H))$. Since $\pi$ is one-to-one on this closed subgroup, it is also an open mapping, so that $\pi$ is a topological isomorphism of $\tau (H)$ onto $\pi (\tau (H))$. Thus $\pi \circ \tau$ is a topological isomorphism of $H$ into $G$. $\square$

Proof of (9.12)

First, we show $a+b\le c+d$. Restricting $\tau$ to $\mathbb {R}^a\times \mathbb {Z}^b\times \{e\}$ gives a topological isomorphism of $\mathbb {R}^a\times \mathbb {Z}^b$ into $\mathbb {R}^c\times \mathbb {Z}^d\times E\subset \mathbb {R}^{c+d}\times E$. Applying the Lemma with $H=\mathbb {R}^a\times \mathbb {Z}^b$, $G=\mathbb {R}^{c+d}$ and $K=E$, we see that $H$ is topologically isomorphic to a closed subgroup of $\mathbb {R}^{c+d}$. Hence $a+b\le c+d$ by (9.11).

To prove $a\le c$, note that restricting $\tau$ to $\mathbb {R}^a\times \{\mathbf {0}\}\times \{e\}$ gives a topological isomorphism of $\mathbb {R}^a$ into $\mathbb {R}^c\times \mathbb {Z}^d\times E$. Since $\mathbb {R}^a$ is connected, $\tau$ maps $\mathbb {R}^a$ into $\mathbb {R}^c\times \{\mathbf {0}\}\times E$. Applying the Lemma with $H=\mathbb {R}^a$, $G=\mathbb {R}^c$ and $K=E$, we see that $\mathbb {R}^a$ is topologically isomorphic to a closed subgroup of $\mathbb {R}^c$. Hence $a\le c$ by (9.11). $\square$