Spaces of Bounded Measurable Functions Invariant under a Group Action

Abstract

In this paper, we characterize spaces of $L^{\infty }$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work is analogous to established results concerning invariant spaces of continuous and measurable functions on a compact Hausdorff space. The case for $L^{\infty }$-functions cannot be proved in the same way when endowed with the norm-topology, but a similar argument can be used when the space of $L^{\infty }$-functions is given the weak*-topology, as we show in this paper.

1. Introduction

In this paper, we show that for a compact Hausdorff space X, acted upon continuously and transitively by a compact group G, when a certain collection $\mathcal{G}$ of spaces of continuous functions exists, then each weak*-closed space of $L^{\infty }$-functions invariant under the action can be constructed by closing the direct sum of some subcollection of $\mathcal{G}$. The $L^{\infty }$-functions in this case are defined by a measure $\mu$ that is closely related to the group action.

The motivation for this paper is to provide an analogue to the results of [1], in which it is shown that when the collection $\mathcal{G}$ exists, every uniformly closed space of continuous functions on X that is invariant under the group action can be decomposed into the closure of the direct sum of some subcollection of $\mathcal{G}$. The same is also true of the $L^p$-functions, with measure $\mu$, for $1\le p<\infty$, where closure is taken in the usual norm-topology on $L^p\left(\mu \right)$.

The case of $L^{\infty }$-functions with the norm-topology is not established in [1] because the map $\alpha \mapsto f\circ {\phi }_{\alpha }$, where ${\phi }_{\alpha }$ denotes the action of $\alpha \in G$ on X, from G into the $L^{\infty }$-functions need not be continuous under the norm-topology. However, when considering the space of $L^{\infty }$-functions with the weak*-topology, an analogous result can be established (Theorem 3), which states that when the same collection $\mathcal{G}$ of continuous functions on X from [1] exists (Definition 4), all weak*-closed invariant spaces of $L^{\infty }$-functions can be constructed by closing the direct sum of some subcollection of $\mathcal{G}$ in the weak*-topology.

This paper also serves as a generalization of a result in [2], in which the author establishes the particular case for the $L^{\infty }$-functions defined on the unit sphere of ${\mathbb{C}}^n$ acted on by the unitary group. This in turn was motivated by the work of Nagel and Rudin in [3], in which it is shown that there exists a collection $\mathcal{C}$ of (minimal and invariant) spaces of continuous functions on the unit sphere of ${\mathbb{C}}^n$ such that each closed unitarily invariant space of continuous or $L^p$-functions on the sphere decomposes into the closure of the direct sum of some subcollection of $\mathcal{C}$.

2. Preliminaries

Let X be a compact Hausdorff space and $C\left(X\right)$ the space of continuous complex functions with domain X and uniform norm $\left|\right|·\left|\right|$. Let G be a compact group (with Haar measure m) that acts continuously and transitively on X. When we wish to be explicit, the map ${\phi }_{\alpha }:X\to X$ shall denote the action of $\alpha$ on X for each $\alpha \in G$; otherwise, $\alpha x$ denotes the action of $\alpha \in G$ on $x\in X$.

Let $\mu$ denote the unique regular Borel probability measure on X that is invariant under the action of G. Specifically,

$${\int }_{X}fd\mu ={\int }_{X}f\circ {\phi }_{\alpha }d\mu ,$$

(1)

for all $f\in C\left(X\right)$ and $\alpha \in G$. The existence of such a measure is a result of André Weil from [4], and a construction of $\mu$ can be found in [5] (Theorem 6.2). Throughout the paper, $\mu$ shall refer to this measure.

The notation $L^p\left(\mu \right)$ denotes the usual Lebesgue spaces, for $1\le p\le \infty$. Recall the norm $\left|\right|·{\left|\right|}_p$ on $L^p\left(\mu \right)$ for $1\le p<\infty$ is given by

$${\left|\right|f\left|\right|}_p={\left({\int }_X{\left|f\right|}^{p}d\mu \right)}^{1/p},$$

and the norm ${\left|\right|f\left|\right|}_{\infty }$ of $f\in L^{\infty }\left(\mu \right)$ is given by the essential supremum of $\left|f\right|$.

For $Y\subset C\left(X\right)$, the uniform closure of Y is denoted $\overline{Y}$, and for $Y\subset L^p\left(\mu \right)$, the norm-closure of Y in $L^p\left(\mu \right)$ is denoted ${\overline{Y}}^p$. When $L^{\infty }\left(\mu \right)$ is equipped with the weak*-topology, ${\overline{Y}}^{\ast }$ denotes the weak*-closure of $Y\subset L^{\infty }\left(\mu \right)$.

Remark 1.

For $Y\subset L^{\infty }\left(\mu \right)$ convex and $1\le p<\infty$, we have

$${\overline{Y}}^{\ast }\subset {\overline{Y}}^p\cap L^{\infty }\left(\mu \right).$$

This follows from the local convexity of $L^p\left(\mu \right)$, and from the fact that the weak*-topology on $L^{\infty }\left(\mu \right)$ is stronger than the topology which $L^{\infty }\left(\mu \right)$ inherits from each $L^p\left(\mu \right)$ endowed with the weak topology.

The following definition has appeared in several sources, such as [1,6,7], but no attribution is given. The last citation is the specific case of the unitary group acting on the unit sphere in ${\mathbb{C}}^n$.

Definition 1.

A space of complex functions Y defined on X is invariant under G (G-invariant) if $f\circ {\phi }_{\alpha }\in Y$ for every $f\in Y$ and every $\alpha \in G$.

Remark 2.

Since the action is continuous, $C\left(X\right)$ is G-invariant. Conversely, if $C\left(X\right)$ is G-invariant, then each action ${\phi }_{\alpha }$ must be continuous.

Remark 3.

The invariance property (1) means $\mu \left(\alpha E\right)=\mu \left(E\right)$ for every Borel set $E\subset X$ and every $\alpha \in G$. Consequently, (1) holds for every $L^p$-function, and $L^p\left(\mu \right)$ is G-invariant for all $1\le p\le \infty$.

The following is consequence of (1) and the G-invariance of the spaces $L^p\left(\mu \right)$:

Remark 4.

Let $1\le p<\infty$ and let $p^{\prime }$ be its conjugate exponent. Then

$${\int }_X(f\circ {\phi }_{\alpha })·gd\mu ={\int }_{X}f·(g\circ {\phi }_{{\alpha }^{-1}})d\mu ,$$

for $f\in L^p\left(\mu \right)$, $g\in L^{p^{\prime }}\left(\mu \right)$, and $\alpha \in G$.

Definition 2

(Definition 2.6 [1]). A space $Y\subset C\left(X\right)$ is G-minimal if it is G-invariant and contains no nontrivial G-invariant spaces.

Definition 3

(Definition 3.1 [1]). For each $x\in X$, the space $H\left(x\right)$ is the set of all continuous functions that are unchanged by the action of any element of G which stabilizes x. That is,

$$H\left(x\right)=\{f\in C\left(X\right):f=f\circ {\phi }_{\alpha },\mathrm{for}\mathrm{all}\alpha \in G\mathrm{such}\mathrm{that}\alpha x=x\}.$$

Definition 4

(Definition 3.2 [1]). Let $\mathcal{G}$ be a collection of spaces in $C\left(X\right)$ with these properties:

(1)

Each $H\in \mathcal{G}$ is a closed G-minimal space.

(2)

Each pair $H_1$ and $H_2$ in $\mathcal{G}$ is orthogonal (in $L^2\left(\mu \right)$): If $f_1\in H_1$ and $f_2\in H_2$, then

$${\int }_X{f}_1\overline{f_2}d\mu =0.$$

(3)

$L^2\left(\mu \right)$ is the direct sum of the spaces in $\mathcal{G}$.

We say $\mathcal{G}$ is a G-collection if it also possesses the following property:

(*)

$dim(H\cap H(x\left)\right)=1$ for each $x\in X$ and each $H\in \mathcal{G}$.

Throughout the paper, $\mathcal{G}$ shall denote a G-collection of $C\left(X\right)$, indexed by I, whose elements are denoted $H_i$, for $i\in I$, and further, we assume that a G-collection exists for X.

Remark 5.

It should be stressed that we are not implying that a G-collection always exists for any X and G. However, a collection of spaces in $C\left(X\right)$ lacking at most only property $(\ast )$ of Definition 4 always exists, as a consequence of the Peter-Weyl theorem from [8]. This collection is necessarily unique.

Definition 5

(Definition 3.7 [1]). We define ${\pi }_i$ to be the projection of $L^2\left(\mu \right)$ onto $H_i$.

Remark 6.

Theorem 3.5 of [1] shows that each ${\pi }_i$ commutes with G, in the sense that

$${\pi }_i(f\circ {\phi }_{\alpha })=\left({\pi }_{i}f\right)\circ {\phi }_{\alpha }$$

for every $f\in L^2\left(\mu \right)$ and every $\alpha \in G$, and further, to each $x\in X$ there exists a unique $K_x\in H_i$ such that

$${\pi }_{i}f={\int }_{X}f\left(x\right)K_{x}d\mu \left(x\right),$$

for all $f\in L^2\left(\mu \right)$. The domain of ${\pi }_i$ can then be extended to $L^1\left(\mu \right)$ by defining ${\pi }_{i}f$ to be the above integral for all $f\in L^1\left(\mu \right)$.

Definition 6

(Definition 3.8 [1]). For $\Omega \subset I$, $E_{\Omega }$ denotes the direct sum of the spaces $H_i$ for $i\in \Omega$.

Remark 7.

The G-invariance of each $E_{\Omega }$ is a natural consequence of the definition.

Remark 8.

Definition 4 yields that each $f\in L^2\left(\mu \right)$ has a unique expansion $f=\sum f_i$, with each $f_i\in H_i$, which converges unconditionally to f in the $L^2$-norm. Since ${\pi }_i$ is the identity map on $H_i$ and the spaces $H_i$ are pairwise orthogonal, we have $f_i={\pi }_{i}f$ for $i\in I$. Thus,

$$f=\sum {\pi }_{i}f.$$

Each ${\pi }_i$ is continuous as the orthogonal projection of $L^2\left(\mu \right)$ onto the closed subspace $H_i$. Thus, ${\pi }_i$ annihilates a subset of $L^2\left(\mu \right)$ if and only if it annihilates its closure. The following is a consequence of this and Remark 8:

Remark 9.

For each set $\Omega \subset I$, we have

$${\overline{E}}_{\Omega }^2=\{f\in L^2\left(\mu \right):{\pi }_{i}f=0\mathrm{when}i\notin \Omega \}.$$

Finally, the classical results used in this paper can be found in many texts, with the reference given in each instance being just one such place.

3. Closures of G-Invariant Sets

In this section, we verify that G-invariance is preserved by closures in the spaces $C\left(X\right)$ and $L^p\left(\mu \right)$ for $1\le p\le \infty$ (Corollaries 1 and 2) by showing that G induces classes of isometries on $L^p\left(\mu \right)$ and on $C\left(X\right)$ (Theorem 1), as well as a class of weak*-homeomorphisms on $L^{\infty }\left(\mu \right)$ (Theorem 2). Consequently, each space $C\left(X\right)$ or $L^p\left(\mu \right)$ for $1\le p\le \infty$ contains at least as many closed G-invariant spaces as there are subsets of I.

Theorem 1.

Suppose $\mathcal{X}$ is any of the spaces $C\left(X\right)$ or $L^p\left(\mu \right)$ for $1\le p\le \infty$ and $\alpha \in G$. If $L_{\alpha }:\mathcal{X}\to \mathcal{X}$ is the map given by $L_{\alpha }f=f\circ {\phi }_{\alpha }$, then $L_{\alpha }$ is a bijective linear isometry.

Proof. 

The bijectivity of each $L_{\alpha }$ is clear because each has an inverse map $L_{{\alpha }^{-1}}$. The linearity of each $L_{\alpha }$ is also clear. Further, the invariance property (1) of $\mu$ yields that each $L_{\alpha }$ is an isometry on $L^p\left(\mu \right)$ (the case for $L^{\infty }\left(\mu \right)$ follows from Remark 3).

To show the same on $C\left(X\right)$, we observe that

$$|\left(L_{\alpha }f\right)\left(x\right)|=|f\left(\alpha x\right)|\le |\left|f\right|\left|\mathrm{and}\right|f\left(x\right)|=|\left(L_{\alpha }f\right)\left({\alpha }^{-1}x\right)|\le ||L_{\alpha }f\left|\right|$$

for all $x\in X$. These inequalities yield that $\left|\right|L_{\alpha }f\left|\right|=\left|\right|f\left|\right|$. □

Corollary 1.

Suppose $\mathcal{X}$ is any of the spaces $C\left(X\right)$ or $L^p\left(\mu \right)$ for $1\le p\le \infty$. If $Y\subset \mathcal{X}$ is G-invariant, then the closure of Y in $\mathcal{X}$ is G-invariant.

Theorem 2.

Let $\alpha \in G$. If $L_{\alpha }:L^{\infty }\left(\mu \right)\to L^{\infty }\left(\mu \right)$ is the map given by $L_{\alpha }\left(f\right)=f\circ {\phi }_{\alpha }$, then $L_{\alpha }$ is a weak*-homeomorphism.

Proof. 

Recall the weak*-topology on $L^{\infty }\left(\mu \right)$ is a weak topology induced by the maps on $L^{\infty }\left(\mu \right)$ of the form

$${\Lambda }_{g}f={\int }_{X}fgd\mu ,$$

for some $g\in L^1\left(\mu \right)$. Thus, $L_{\alpha }$ is continuous with respect to the weak*-topology if and only if ${\Lambda }_g\circ L_{\alpha }$ is continuous for all maps ${\Lambda }_g$.

Fix $g\in L^1\left(\mu \right)$. We observe that

$$({\Lambda }_g\circ L_{\alpha })\left(f\right)={\Lambda }_g(f\circ {\phi }_{\alpha })={\int }_X(f\circ {\phi }_{\alpha })·gd\mu ={\int }_{X}f·(g\circ {\phi }_{{\alpha }^{-1}})d\mu ={\Lambda }_{g\circ {\phi }_{{\alpha }^{-1}}}f,$$

for every $f\in L^{\infty }\left(\mu \right)$, by Remark 4. We conclude $L_{\alpha }$ is continuous on $L^{\infty }\left(\mu \right)$ with respect to the weak*-topology.

Finally, the map $L_{{\alpha }^{-1}}:L^{\infty }\left(\mu \right)\to L^{\infty }\left(\mu \right)$ given by $L_{{\alpha }^{-1}}\left(f\right)=f\circ {\phi }_{{\alpha }^{-1}}$ is the inverse of $L_{\alpha }$. By a similar argument, $L_{{\alpha }^{-1}}$ is continuous with respect to the weak*-topology, and thus $L_{\alpha }$ is a weak*-homeomorphism. □

Corollary 2.

If $Y\subset L^{\infty }\left(\mu \right)$ is G-invariant, then ${\overline{Y}}^{\ast }$ is G-invariant.

Remark 10.

From Remark 7 and Corollaries 1 and 2, the closure of each $E_{\Omega }$ in any space $C\left(X\right)$ or $L^p\left(\mu \right)$ for $1\le p\le \infty$ is G-invariant.

4. Characterization of Weak*-Closed G-Invariant Subspaces of $L^{\infty }\left(\mu \right)$

In this section, we state and prove our main result (Theorem 3), which shows that the spaces ${\overline{E}}_{\Omega }^{\ast }$ are the only weak*-closed G-invariant subspaces of $L^{\infty }\left(\mu \right)$.

Theorem 3.

If Y is a weak*-closed G-invariant subspace of $L^{\infty }\left(\mu \right)$, then $Y={\overline{E}}_{\Omega }^{\ast }$ for some $\Omega \subset I$.

This result is an analogue to Theorem 4.1 of [1], which is used in its proof:

Theorem 4

(Theorem 4.1 [1]). Let $\mathcal{X}$ be any of the spaces $C\left(X\right)$ or $L^p\left(\mu \right)$ for $1\le p<\infty$. If Y is a closed G-invariant subspace of $\mathcal{X}$, then Y is the closure of $E_{\Omega }$ for some $\Omega \subset I$.

The set $\Omega$ from Theorems 3 and 4 is the set $\{i\in I:{\pi }_{i}Y\ne 0\}$. The proof of Theorem 3 further requires Lemma 1, which we prove in Section 5.

Lemma 1.

Let $Y\subset L^{\infty }\left(\mu \right)$ be a G-invariant space. Then for $g\in L^{\infty }\left(\mu \right)$, we have that $g\notin {\overline{Y}}^2$ whenever $g\notin {\overline{Y}}^{\ast }$.

Remark 11.

From Remark 1 and Lemma 1, for any G-invariant space $Y\subset L^{\infty }\left(\mu \right)$,

$${\overline{Y}}^{\ast }={\overline{Y}}^2\cap L^{\infty }\left(\mu \right).$$

Remark 12.

Remarks 11 and 9 give a description of the sets ${\overline{E}}_{\Omega }^{\ast }$:

$${\overline{E}}_{\Omega }^{\ast }={\overline{E}}_{\Omega }^2\cap L^{\infty }\left(\mu \right)=\{f\in L^{\infty }\left(\mu \right):{\pi }_{i}f=0\mathrm{when}i\notin \Omega \}.$$

Proof of Theorem 3.

Let $Y\subset L^{\infty }\left(\mu \right)$ be a weak*-closed G-invariant space. Then

$$Y={\overline{Y}}^{\ast }={\overline{Y}}^2\cap L^{\infty }\left(\mu \right)$$

from Remark 11. Since Y is G-invariant, so is ${\overline{Y}}^2$ from Corollary 1. By Theorem 4,

$${\overline{Y}}^2={\overline{E}}_{{\Omega }^{\prime }}^2,$$

where ${\Omega }^{\prime }=\{i\in I:{\pi }_i{\overline{Y}}^2\ne 0\}$.

We define $\Omega =\{i\in I:{\pi }_{i}Y\ne 0\}$. Then, Remark 12 yields

$${\overline{E}}_{\Omega }^2\cap L^{\infty }\left(\mu \right)={\overline{E}}_{\Omega }^{\ast }.$$

We have $\Omega ={\Omega }^{\prime }$ by the continuity of each ${\pi }_i$, and thus ${\overline{E}}_{\Omega }^2={\overline{E}}_{{\Omega }^{\prime }}^2$. □

5. Proof of Lemma 1

In this section, we prove Lemma 1, which we note is an analogue to Lemma 4.4 of [1] and a generalization of Lemma 4.2 from [2]. The proof requires the following lemmas:

Lemma 2.

If $f\in C\left(X\right)$, then the map $\alpha \mapsto f\circ {\phi }_{\alpha }$ is a continuous map of G into $C\left(X\right)$.

Proof. 

For $\alpha \in G$, we define the map $\varphi :G\to C(X,X)$ by $\varphi \left(\alpha \right)={\phi }_{\alpha }$, where $C(X,X)$ denotes the continuous maps from X into X. Then $\varphi$ is continuous when $C(X,X)$ is given the compact-open topology (Theorem 46.11 of [9]). We note that the continuity of the group action is used here.

We define the map $T_f:C(X,X)\to C\left(X\right)$ for $f\in C\left(X\right)$ by $T_f\left(\phi \right)=f\circ \phi$, and we endow both spaces with the compact-open topology. Let $f\circ \phi \in C\left(X\right)$ for $\phi \in C(X,X)$ and suppose $f\circ \phi \in V$, where $V=V(K,U)$ is a subbasis element in $C\left(X\right)$. Explicitly,

$$K\subset {(f\circ \phi )}^{-1}\left(U\right).\mathrm{That}\mathrm{is}\mathrm{to}\mathrm{say},K\subset {\phi }^{-1}\left(f^{-1}\left(U\right)\right).$$

Then $V^{\prime }=V^{\prime }(K,f^{-1}\left(U\right))$ is a subbasis element in $C(X,X)$ and $\phi \in V^{\prime }$. Further, $V^{\prime }\subset T_f^{-1}\left(V\right)$, so that $T_f$ is continuous when $C(X,X)$ and $C\left(X\right)$ are endowed with the respective compact-open topologies. We finally observe that since X is compact, the norm topology and the compact-open topology on $C\left(X\right)$ coincide. □

Lemma 3.

If $f\in L^p\left(\mu \right)$ for $1\le p<\infty$, then the map $\alpha \mapsto f\circ {\phi }_{\alpha }$ of G into $L^p\left(\mu \right)$ is continuous.

Proof. 

We verify the map $\alpha \mapsto f\circ {\phi }_{\alpha }$ is continuous at the identity in G, which suffices to conclude it is continuous on G. Let $ϵ>0$. Then there exists $g\in C\left(X\right)$ such that

$$\left|\right|f-{g\left|\right|}_p<ϵ/3.$$

Since the map $\alpha \mapsto g\circ {\phi }_{\alpha }$ is continuous (Lemma 2), and in particular continuous at the identity in G, there exists a neighborhood N of the identity in G such that

$$\left|\right|g-g\circ {\phi }_{\alpha }\left|\right|<ϵ/3$$

for all $\alpha \in N$. Since

$$\left|\right|f-f\circ {\phi \left|\right|}_p\le \left|\right|f-{g\left|\right|}_p+\left|\right|g-g\circ {\phi }_{\alpha }{\left|\right|}_p+\left|\right|(g-f)\circ {\phi }_{\alpha }{\left|\right|}_p,$$

the G-invariance of $L^p\left(\mu \right)$ yields that $\left|\right|f-f\circ {\phi }_{\alpha }{\left|\right|}_p<ϵ$ for all $\alpha \in N$. □

Lemma 4.

Let $g\in L^{\infty }\left(\mu \right)$. Then the map $\varphi :G\to L^{\infty }\left(\mu \right)$ given by $\varphi \left(\alpha \right)=g\circ {\phi }_{\alpha }$ is weak*-continuous.

Proof. 

To show that $\varphi$ is weak*-continuous, we verify each ${\Lambda }_h\circ \varphi$ is continuous, where ${\Lambda }_h$ is the map $L^{\infty }\left(\mu \right)\to \mathbb{C}$ given by integration against the function $h\in L^1\left(\mu \right)$.

Observe the map ${\Lambda }_h\circ \varphi$ is given by

$$({\Lambda }_h\circ \varphi )\left(\alpha \right)={\int }_X(g\circ {\phi }_{\alpha })·hd\mu .$$

From Lemma 3, the map $\alpha \mapsto h\circ {\phi }_{\alpha }$ is continuous from G into $L^1\left(\mu \right)$. Thus, the map $\alpha \mapsto g·(h\circ {\phi }_{{\alpha }^{-1}})$ is continuous. We apply Remark 4 to get that

$$\alpha \mapsto {\int }_{X}g·(h\circ {\phi }_{{\alpha }^{-1}})d\mu ={\int }_X(g\circ {\phi }_{\alpha })·hd\mu$$

is continuous, as desired. □

Proof of Lemma 1.

Suppose $g\in L^{\infty }\left(\mu \right)$ and $g\notin {\overline{Y}}^{\ast }$. Then there exists a weak*-continuous linear functional $\mathsf{\Gamma }$ on $L^{\infty }\left(\mu \right)$ such that $\mathsf{\Gamma }f=0$ for $f\in Y$, and $\mathsf{\Gamma }g=1$, due to the Hahn–Banach theorem (Theorem 3.5 [10]). Since each weak*-continuous linear functional on $L^{\infty }\left(\mu \right)$ is induced by an element of $L^1\left(\mu \right)$, there exists $h\in L^1\left(\mu \right)$ such that $\mathsf{\Gamma }F={\int }_{X}Fhd\mu$ for $F\in L^{\infty }\left(\mu \right)$.

From Lemma 4, there exists a neighborhood N of the identity in G such that

$$\mathrm{Re}{\int }_X(g\circ {\phi }_{\alpha })·hd\mu >\frac{1}{2}$$

for $\alpha \in N$. We choose a continuous map $\psi :G\to [0,\infty )$ such that $\int \psi dm=1$ and the support of $\psi$ is contained in N (recall m denotes the Haar measure on G).

We now define a map $\Lambda$ on $L^{\infty }\left(\mu \right)$ by

$$\Lambda F={\int }_{X}h\left(x\right){\int }_G\psi \left(\alpha \right)·F\left(\alpha x\right)dm\left(\alpha \right)d\mu \left(x\right),\mathrm{for}F\in L^{\infty }\left(\mu \right).$$

We fix $F\in L^{\infty }\left(\mu \right)$ and $x\in X$ and define the map ${\mathcal{F}}_x:G\to \mathbb{C}$ by $\alpha \mapsto F\left(\alpha x\right)$. Since

$${\int }_G|{\mathcal{F}}_x{|}^{2}dm={\int }_G{\left|F\left(\alpha x\right)\right|}^{2}dm\left(\alpha \right)={\int }_X{\left|F\right|}^{2}d\mu ={\left|\right|F\left|\right|}_2^2<\infty ,$$

we get ${\mathcal{F}}_x\in L^2\left(G\right)$, and since $\psi$ is continuous, we have that $\psi \in L^2\left(G\right)$. Further,

$$\left|{\int }_G\psi {\mathcal{F}}_{x}dm\right|\le {\left({\int }_G{\left|\psi \right|}^{2}dm\right)}^{\frac{1}{2}}{\left({\int }_G{\left|{\mathcal{F}}_x\right|}^{2}dm\right)}^{\frac{1}{2}}={\left|\right|\psi \left|\right|}_2·{\left|\right|F\left|\right|}_2,$$

so that for $F\in L^{\infty }\left(\mu \right)$,

$$|\Lambda F|\le {\left|\right|\psi \left|\right|}_2·{\left|\right|F\left|\right|}_2{\int }_X\left|h\right|d\mu ={\left|\right|\psi \left|\right|}_2·{\left|\right|F\left|\right|}_2·{\left|\right|h\left|\right|}_1.$$

The linearity of $\Lambda$ on $L^{\infty }\left(\mu \right)$ is clear. Thus, $\Lambda$ defines an $L^2$-continuous linear functional on $L^{\infty }\left(\mu \right)$, and hence extends to an $L^2$-continuous linear functional ${\Lambda }_1$ on $L^2\left(\mu \right)$ by the Hahn–Banach theorem (Theorem 3.6 [10]). By interchanging the integrals in the definition of $\Lambda$, we see that ${\Lambda }_1$ annihilates Y, since Y is G-invariant. Further,

$$\mathrm{Re}{\Lambda }_{1}g={\int }_G\psi \left(\alpha \right)\left(\mathrm{Re}{\int }_{X}g\left(\alpha x\right)·h\left(x\right)d\mu \left(x\right)\right)dm\left(\alpha \right)>{\int }_N\psi \left(\alpha \right)·\frac{1}{2}dm\left(\alpha \right)=\frac{1}{2}.$$

We conclude that $g\notin {\overline{Y}}^2$. □

6. Future Questions

(1)

Does a G-collection exist for all groups G acting continuously and transitively on X? What conditions might exist on G or X that yield a collection lacking (*)?

(2)

Under what conditions can the restrictions on X, G, and the action of G on X be loosened? Can the compactness of X and G be substituted with local compactness? Can the continuity of the action be substituted with separate continuity?

(3)

Suppose H is a subgroup of G and $\mathcal{H}$ is a collection of closed H-minimal spaces satisfying the same conditions as $\mathcal{G}$. What is the relationship between $\mathcal{H}$ and $\mathcal{G}$? What if $\mathcal{H}$ and $\mathcal{G}$ lack $(\ast )$? The uniqueness of $\mu$ shows that H does not induce a new H-invariant measure on X. Further, G-invariance implies H-invariance (of a space).

We note that (3) is prompted from the study of $\mathcal{M}$-invariant and $\mathcal{U}$-invariant spaces of continuous functions on the unit sphere of ${\mathbb{C}}^n$ from [3], in which it is shown that there are infinitely many $\mathcal{U}$-invariant spaces and only six $\mathcal{M}$-invariant spaces. These six $\mathcal{M}$-invariant spaces are found by combining the $\mathcal{U}$-minimal spaces in a specific way (see Lemma 13.1.2 of [7]), and we are curious if this method can be generalized.

(4)

Under what conditions can a G-collection characterize the closed G-invariant algebras of continuous functions? We note that the case for the unitary group acting on the unit sphere of ${\mathbb{C}}^n$ is discussed in [11] and is also summarized in [7].