# Veech’s Theorem of $G$ acting freely on $G^{\textsl{{LUC}}}$ and Structure
Theorem of a.a. flows

Xiongping Dai
Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
E-mail<EMAIL_ADDRESS>Hailan Liang
Department of Mathematics, Fuzhou University, Fuzhou 350116, P.R. China
E-mail<EMAIL_ADDRESS>Zhengyu Yin
Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
E-mail<EMAIL_ADDRESS>
(July 13, 2023)

###### Abstract

Veech’s Theorem claims that if $G$ is a locally compact (LC) Hausdorff
topological group, then it may act freely on $G^{\textsl{{LUC}}}$. We prove
Veech’s Theorem for $G$ being only locally quasi-totally bounded, not
necessarily LC. And we show that the universal a.a. flow is the maximal almost
1-1 extension of the universal minimal a.p. flow and is unique up to almost
1-1 extensions. In particular, every endomorphism of Veech’s hull flow induced
by an a.a. function is almost 1-1; for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$
acts freely on its canonical universal a.a. space. Finally, we characterize
Bochner a.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$
(due to Veech 1965 for the special case that $G$ is abelian, LC,
$\sigma$-compact, and first countable).

††2020 _Mathematics Subject Classification_ : Primary 37B05; Secondary 22A10,
54H15.†† _Key words and phrases_ : Maximal ideal space, $G^{\textsl{{LUC}}}$,
Veech’s Theorem, a.a. functions.

##  1 Introduction

The maximal ideal space, also named as the structure space, of a commutative
Banach algebra $\mathcal{A}$ is an important tool for the study of “universal
elements”, for example, the universal $\\{\textrm{minimal}\\}$
$\\{\textrm{minimal equicontinuous}\\}$ $\\{\textrm{point-transitive}\\}$
flow, in the abstract topological dynamical systems (cf. [14, 17, 12]). Now we
will present self-closely a revisit to this concept in the special case that
$\mathcal{A}$ is a Banach subalgebra of uniformly continuous functions on
groups/semigroups in $\S$2, $\S$3, and $\S$4. The new ingredient is that we
shall introduce the analysis on Hausdorff left-topological groups/semigroups
instead of Hausdorff topological groups and discrete semigroups. We will give
applications in $\S$5 and $\S$6.

Let $G^{\textsl{{LUC}}}$ be the left-uniformly continuous compactification of
a Hausdorff topological group $G$ (see Def. 2.8.4). Veech’s Theorem claims
that if $G$ is locally compact, then $G$ acts freely on $G^{\textsl{{LUC}}}$
([31, Thm. 2.2.1]). In $\S$5, we shall extend Veech’s Theorem for $G$ being
only a “locally quasi-totally bounded” group, not necessarily locally compact
(see Def. 5.4.4 and Thm 5.7). This implies that the universal
$\\{\textrm{point-transitive}\\}$$\\{\textrm{minimal}\\}$ compact flow of any
locally quasi-totally bounded group is free.

In $\S$6.1 we shall study left almost automorphic (a.a.) functions on
Hausdorff left-topological groups. Further, in $\S\S$6.2 and 6.3 we shall
simply reprove Veech’s Structure Theorem of a.a. flows and a.a. functions
(Thm. 6.2.6 and Thm. 6.3.11); and moreover, show that the universal a.a. flow
of $G$ is existent and is exactly the maximal almost 1-1 extension of the
universal minimal equicontinuous flow of $G$ unique up to almost 1-1
extensions (Thm. 6.2.7D). In particular, we will prove that every endomorphism
of Veech’s hull flow induced by an a.a. function is almost 1-1 (Thm. 6.3.12).
In addition, for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$ acts freely on its
canonical universal a.a. space (Corollary to Thm. 6.2.8).

Recall that if $f$ is a (von Neumann) almost periodic function on a non-
abelian group $G$ (Def. 6.3.14), then the left hull of $f$,
$\overline{Gf}^{\textrm{p}}=\overline{Gf}^{\|\cdot\|_{\infty}}$, need not be a
compact topological group. However, we shall show in $\S$6.3 that $f$ is (von
Neumann) almost periodic iff Veech’s bi-hull,

$\overline{G\xi_{f}}^{\textrm{p}}=\overline{G\xi_{f}}^{\|\cdot\|_{\infty}}\quad\textrm{where
}\xi_{f}(s,t)=f(st)\ \forall s,t\in G,$

is a compact topological group (see Thm. 6.3.14C). In fact, almost automorphic
functions may be characterized in terms of almost periodic functions (see Thm.
C.3 in Appendix C).

Finally, using bi-hulls of uniformly continuous functions on $G$, we shall
characterize in $\S$6.4 Bochner a.a. functions on a locally compact group $G$
in terms of Bohr a.a. function on $G$ (due to Veech 1965 for the special case
that $G$ is abelian, locally compact, $\sigma$-compact, and first countable);
see Thm. 6.4.6.

######  1.1 Standing hypotheses.

a. Unless specified otherwise, assume $T$ is a noncompact, Hausdorff, left-
topological monoid with identity $e$; that is, $T$ is a noncompact Hausdorff
space, which has a semigroup structure: $T\times T\xrightarrow{(x,y)\mapsto
xy}T$, with identity $e$, such that the left-translate $L_{y}\colon
T\xrightarrow{x\mapsto yx}T$ is continuous for all $y\in T$.

b. Given $\tau\in T$, $\mathfrak{N}_{\tau}(T)$ stands for the filter of
neighborhoods of $\tau$ in $T$; and LC is short for “locally compact”.

For example, if $G$ is a group and $\lambda\colon X\times G\rightarrow X$ is a
right-action nonequicontinuous phase mapping on a noncompact Hausdorff space
$X$, then $T=\overline{\\{X\xrightarrow{x\mapsto xg}X\,|\,g\in G\\}}$,
equipped with the topology of pointwise convergence, is a noncompact,
Hausdorff, left-topological semigroup under the composition of mappings. In
this case there need generally not exist left/right-uniformity structures on
the left-topological semigroup $T$.

######  1.2 Basic notation.

As usual, we say that $(T,X)$ is a semiflow, denoted $\mathscr{X}$ or
$T\curvearrowright X$, if there exists a continuous mapping $\pi\colon T\times
X\xrightarrow{(t,x)\mapsto tx}X$, called the phase mapping and where $X$ is a
Hausdorff space, such that $ex=x$ and $(st)x=s(tx)$ for all $x\in X$ and
$s,t\in T$. Let $\Delta_{X}=\\{(x,x)\,|\,x\in X\\}$.

a. If $X$ is compact, $\mathscr{X}$ will be referred to as a compact semiflow.
If meanwhile $X\curvearrowleft T$ is a compact right-action semiflow, with
phase mapping $(x,t)\mapsto xt$, such that $(sx)t=s(xt)$ for all $s,t\in T$
and $x\in X$, then $(T,X,T)$ will be called a compact bi-semiflow.

b. If the orbit closure $\overline{Tx}$ is compact for all $x\in X$, then
$\mathscr{X}$ will be called Lagrange stable (cf., e.g., [27]).

c. The pair $(\mathscr{X},x)$ will be called an ambit of $T$ as in [4] if
$\mathscr{X}$ is a semiflow with $\overline{Tx}=X$; and in this case, $x$ is
called a transitive point of $\mathscr{X}$. If no point
$y\in\overline{Tx}\setminus\\{x\\}$ such that
$\overline{T(x,y)}\cap\Delta_{X}\not=\emptyset$, then $x$ will be called a
distal point for $\mathscr{X}$.

d. If the phase mapping $\pi\colon(t,x)\mapsto tx$ is only separately
continuous here, then $\mathscr{X}$ or $T\curvearrowright X$ will be called a
dynamic instead of a semiflow.

e. For a Lagrange stable dynamic $\mathscr{X}$, a point $x\in X$ is almost
periodic (a.p.) iff $\overline{Tx}$ is a minimal set under $\mathscr{X}$
(i.e., $\overline{Ty}=\overline{Tx}$ for all $y\in\overline{Tx}$). It is a
well-known fact that a distal point is an a.p. point in a Lagrange stable
dynamic [14, 4].

f. Given $x\in X$ and $U\in\mathfrak{N}_{x}(X)$, set $N_{T}(x,U)=\\{t\in
T\,|\,tx\in U\\}$ as in [16].

If $T$ is a group, a semiflow will be called a flow in the later.

##  2 Definition of the maximal ideal spaces

A collection $\mathcal{F}$ of complex-valued functions on a space $W$ is
called an algebra if for any $f,g\in\mathcal{F}$ and $c\in\mathbb{C}$ we have
$cf+g,fg\in\mathcal{F}$. Next, $\mathcal{F}$ is said to separate points of $W$
if for all $w_{1}\not=w_{2}$ in $W$, we have $f(w_{1})\not=f(w_{2})$ for some
$f\in\mathcal{F}$. We say that $\mathcal{F}$ separates points and closed sets
in $W$ if for all point $w\in W$ and closed set $A\subset W$ with $w\notin A$,
there exists an $f\in\mathcal{F}$ such that $f(w)=0$, $f=1$ on $A$, and $0\leq
f\leq 1$. $\mathcal{F}$ is called self-adjoint if $\bar{f}=g-ih\in\mathcal{F}$
whenever $f=g+ih\in\mathcal{F}$ where $g$ and $h$ are real-valued functions on
$W$.

By $\mathcal{C}(W)$ we denote the set of all continuous complex-valued
functions on $W$; let $\|f\|_{\infty}=\sup_{x\in W}|f(x)|$ be the supremum
norm of $f\in\mathcal{C}(W)$ and write $\mathcal{C}_{b}(W)$ for
$\\{f\in\mathcal{C}(W)\,|\,\|f\|_{\infty}<\infty\\}$.

Using the maximal ideal spaces, we shall prove that every uniformly continuous
complex-valued bounded function on $T$ may come from a compact ambit of $T$
(see Thm. 3.3), and, construct the universal a.a. flows of groups (see Def.
6.2.7a).

######  2.1 Theorem (Stone-Weierstrass Theorem, cf. [9, Thm. V.8.1]).

Let $W$ be a compact Hausdorff space. Let $\mathcal{A}$ be a subalgebra of
$\mathcal{C}(W)$, which separates the points of $W$, contains the constants
and is self-adjoint. Then $\mathcal{A}$ is norm dense in $\mathcal{C}(W)$.

######  2.2Left-uniformly continuous function on $T$.

Let $\mathbf{1}\in\mathcal{C}_{b}(T)$ be the constant function
$\mathbf{1}(x)=1\ \forall x\in T$. For all $f\in\mathcal{C}_{b}(T)$ and $t\in
T$,

  1. 1.

$ft$ is defined by $T\xrightarrow{x\mapsto f(tx)}\mathbb{C}$, called the right
$t$-shift of $f$. Clearly, $ft\in\mathcal{C}_{b}(T)$ since $x_{n}\to x$ in $T$
implies that $tx_{n}\to tx$;

  2. 2.

$tf$ is defined by $T\xrightarrow{x\mapsto f(xt)}\mathbb{C}$, called the left
$t$-shift of $f$. Notice here that $tf$ need not belong to
$\mathcal{C}_{b}(T)$, since $T$ need generally not be a right-topological
semigroup.

  3. 3.

Although there is no any uniformity structure on $T$ here, we can introduce
left-uniformly continuous function on $T$ as follows: Let

$\verb"LUC"(T)=\\{f\in\mathcal{C}_{b}(T)\,|\,ft_{n}\to ft\textrm{
$\|\cdot\|_{\infty}$-uniformly as }t_{n}\to t\textrm{ in }T\\},$

called the set of left-uniformly continuous complex-valued bounded functions
on $T$.

    1. i.

If $T$ is a Hausdorff topological group, then $f\in\verb"LUC"(T)$ iff to any
$\varepsilon>0$ there exists some $V\in\mathfrak{N}_{e}(T)$ such that
$|f(x)-f(vx)|<\varepsilon$ for all $x\in T$ and all $v\in V$ (cf., e.g., [24,
7]).

    2. ii.

If $T$ has a “left-uniformity” $\mathscr{U}^{L}$ (i.e., a uniformity on $T$
such that $\\{R_{t}\colon T\xrightarrow{x\mapsto xt}T\,|\,t\in T\\}$ is
$\mathscr{U}^{L}$-equicontinuous on $T$; for instance, $T=\mathbb{R}_{+}$),
then $f\in\verb"LUC"(T)$ whenever $f\in\mathcal{C}_{b}(T)$ is
$\mathscr{U}^{L}$-uniformly continuous.

###### Proof.

Assume $f$ is $\mathscr{U}^{L}$-uniformly continuous. Let $\varepsilon>0$.
Then there is a $V\in\mathscr{U}^{L}$ such that $|f(x)-f(y)|<\varepsilon$
$\forall(x,y)\in V$. Pick some $U\in\mathscr{U}^{L}$ such that $(x,y)\in U$
implies $(xt,yt)\in V$ $\forall t\in T$. So, as $t_{n}\to t$ in $T$, we have
that $(t_{n},t)\in U$ eventually so that $|ft_{n}(x)-ft(x)|<\varepsilon$
$\forall x\in X$. Thus, $f\in\verb"LUC"(T)$. ∎

For example, if $T$ is a topological group, then
$\mathscr{U}^{L}=\\{V_{L}\,|\,V\in\mathfrak{N}_{e}(T)\\}$, where
$V_{L}=\\{(x,y)\,|\,yx^{-1}\in V\\}$, is the canonical left-uniformity on $T$.

  4. 4.

Clearly, $\verb"LUC"(T)$ is a norm closed subalgebra of the commutative Banach
algebra $\mathcal{C}_{b}(T)$, which is self-adjoint and contains the
constants; and moreover, $\verb"LUC"(T)$ is left and right $T$-invariant
(i.e., if $f\in\verb"LUC"(T)$, then $sft\in\verb"LUC"(T)$ for all $s,t\in T$)
(cf. [24] or [14, Lem. 3.1, p.84] for $T$ a topological group).

  5. 5.

We could similarly define $\texttt{RUC}(T)$—the set of right-uniformly
continuous complex-valued bounded functions on $T$ as follows:

$\verb"RUC"(T)=\\{f\in\mathcal{C}_{b}(T)\,|\,t_{n}f\to tf\textrm{
$\|\cdot\|_{\infty}$-uniformly as }t_{n}\to t\textrm{ in }T\\}.$

  6. 6.

Let $\verb"UC"(T)=\verb"LUC"(T)\cap\verb"RUC"(T)$. If $\hat{T}$ is the
continuous character group of $T$, then $\hat{T}\subseteq\verb"UC"(T)$.

######  Caution.

$\verb"LUC"(T)$ is sometimes written as $\texttt{RUC}(T)$ in some literature
(cf. [12]). We agree with [23, 24, 28, 7] here.

######  2.3.

If $\mathscr{X}$ is a dynamic of $T$ and $x\in X$, then
$f\in\mathcal{C}_{b}(T)$ is called coming from $(\mathscr{X},x)$, denoted
$f\in\mathbb{F}(\mathscr{X},x)$, whenever there is an $F\in\mathcal{C}(X)$
such that $f(t)=F(tx)$ for all $t\in T$. We write $F^{x}(t)=F(tx)$. Clearly,

$tF^{x}=F^{tx}\quad\textrm{and}\quad
t\mathbb{F}(\mathscr{X},x)=\mathbb{F}(\mathscr{X},tx)\quad\forall x\in X,t\in
T,F\in\mathcal{C}(X).$

######  2.4.

If $\mathscr{X}$ is a Lagrange stable semiflow of $T$, then
$\mathbb{F}(\mathscr{X},x)\subseteq\verb"LUC"(T)$ for all $x\in X$ (see [12,
Prop. IV.5.6] for $\mathscr{X}$ a compact flow with $T$ a topological group).

###### Proof.

We may assume $X$ is compact with uniformity $\mathscr{U}$ by replacing $X$ by
$\overline{Tx}$ if necessary. Let $F\in\mathcal{C}(X)$ and set $f=F^{x}$.
Given $\varepsilon>0$ and $s_{n}\to s$ in $T$, there is an open
$\alpha\in\mathscr{U}$ such that
$|F(x^{\prime})-F(x^{\prime\prime})|<\varepsilon$ whenever
$(x^{\prime},x^{\prime\prime})\in\alpha$. Further, there is a set
$V\in\mathfrak{N}_{s}(T)$ such that
$Vx^{\prime}\times\\{sx^{\prime}\\}\subset\alpha^{2}$ for all $x^{\prime}\in
X$. Indeed, by continuity of

$(T\times X)\times(T\times
X)\xrightarrow{((t_{1},x_{1}),(t_{2},x_{2}))\mapsto(t_{1}x_{1},t_{2}x_{2})}X\times
X,$

for all $x^{\prime}\in X$ there exist some
$V_{x^{\prime}}\in\mathfrak{N}_{s}(T)$ and some
$U_{x^{\prime}}\in\mathfrak{N}_{x^{\prime}}(X)$ such that
$V_{x^{\prime}}U_{x^{\prime}}\times
V_{x^{\prime}}U_{x^{\prime}}\subset\alpha$. Since $X$ is compact, there exists
a finite set $\\{x_{1},\dotsc,x_{n}\\}\subset X$ with
$X=\bigcup_{i=1}^{n}U_{x_{i}}$. Clearly, $V=\bigcap_{i=1}^{n}V_{x_{i}}$ is as
desired. This implies that $fs_{n}(t)\to fs(t)$ uniformly for $t\in T$. ∎

Let $(\mathscr{X},x)$ be a compact ambit of $T$, then we can define

$F^{\mathscr{X}}=\\{F^{y}\,|\,y\in
X\\}\quad\textrm{and}\quad\varrho^{x}\colon\mathcal{C}(X)\xrightarrow{F\mapsto
F^{x}}\verb"LUC"(T).$

Then $\mathbb{F}(\mathscr{X},x)=\varrho^{x}[\mathcal{C}(X)]$, and
$\varrho^{x}$ is an isometric algebra homomorphism from $\mathcal{C}(X)$ into
$\verb"LUC"(T)$. Thus, $\mathbb{F}(\mathscr{X},x)$ is a norm closed right-
invariant subalgebra of $\verb"LUC"(T)$. In fact,

$F^{x}t=(Ft)^{x}\in\mathbb{F}(\mathscr{X},x)\quad\forall
F\in\mathcal{C}(X),t\in T.$

Moreover, $\mathbb{F}(\mathscr{X},x)$ is self-adjoint and contains the
constants.

######  2.4.3 Lemma.

Let $\mathscr{X}$ be a compact semiflow of $T$ and $x_{0},x_{1}\in X$ with
$\overline{Tx_{0}}=\overline{Tx_{1}}=X$. Let $F\in\mathcal{C}(X)$,
$f_{0}=F^{x_{0}}$, and $f_{1}=F^{x_{1}}$. Then

$\overline{Tf_{0}}^{\textrm{p}}=F^{\mathscr{X}}=\overline{Tf_{1}}^{\textrm{p}}\subseteq\texttt{LUC}(T)$

where $\overline{B}^{\textrm{p}}$ denotes the pointwise closure of a set $B$
in $\mathbb{C}^{T}$.

###### Proof.

To show the desired equalities, it is sufficient to prove
$F^{\mathscr{X}}={\overline{Tf_{0}}}^{\textrm{p}}$. Let
$t_{n}f_{0}\to_{\textrm{p}}\phi$ in $\mathbb{C}^{T}$. Then for $t\in T$,
$t_{n}f_{0}(t)\to\phi(t)$ and $f_{0}(tt_{n})\to\phi(t)$. Thus
$F(tt_{n}x_{0})\to\phi(t)$. We can assume (a subnet of) $t_{n}x_{0}\to x\in
X$, for $X$ is compact. Then $F^{x}(t)=\phi(t)$. This shows
${\overline{Tf_{0}}}^{\textrm{p}}\subseteq F^{\mathscr{X}}$. On the other
hand, let $x\in X$. There is a net $t_{n}\in T$ such that $t_{n}x_{0}\to x$,
for $Tx_{0}$ is dense in $X$. Clearly, $t_{n}f_{0}\to_{\textrm{p}}F^{x}$ and
${\overline{Tf_{0}}}^{\textrm{p}}\supseteq F^{\mathscr{X}}$. Thus,
${\overline{Tf_{0}}}^{\textrm{p}}=F^{\mathscr{X}}$. The proof is completed. ∎

Even for a minimal compact flow $\mathscr{X}$ with $T$ abelian, there exists
no the equality $\mathbb{F}(\mathscr{X},x_{0})=\mathbb{F}(\mathscr{X},x_{1})$
for $x_{0}\not=x_{1}$ in general, according to Theorem 3.1 below. If $T$ is
abelian and $f\in\mathbb{F}(\mathscr{X},x_{0})$, then it is easy to verify
$Tf=fT\subset\mathbb{F}(\mathscr{X},x_{0})$. However,
$\mathbb{F}(\mathscr{X},x_{0})$ need not be pointwise or compact-open closed,
so
$\overline{Tf}^{\textrm{p}}\nsubseteq\mathbb{F}(\mathscr{X},x_{0})\varsubsetneq\bigcup_{F\in\mathcal{C}(X)}F^{\mathscr{X}}$
generally.

Note here that if $\mathscr{X}=X\curvearrowleft T$ is defined by $T$ acting
from right-hand side on $X$, then we have that
$\mathbb{F}(\mathscr{X},x)\subseteq\texttt{RUC}(T)$. In this case, we have for
$F\in\mathcal{C}(X)$ and $x\in X$ that $F_{x}(t)=F(xt)$.

######  Caution.

The pointwise topology is weaker than the weak topology on
$\mathbb{F}(\mathscr{X},x)$. Indeed, it holds that
$\mathbb{F}(\mathscr{X},x)\subsetneq\overline{\mathbb{F}(\mathscr{X},x)}^{\textrm{p}}$.

######  2.5.

It is easy to verify that

  1. $\lambda\colon\verb"LUC"(T)\times T\rightarrow\verb"LUC"(T),\ (f,t)\mapsto ft$

is a jointly continuous phase mapping; that is, if
$f_{n}\to_{\|\cdot\|_{\infty}}f$ in $\verb"LUC"(T)$ and $t_{n}\to t$ in $T$,
then $f_{n}t_{n}\to_{\|\cdot\|_{\infty}}ft$ in $\verb"LUC"(T)$.

######  Caution.

If we replace $\verb"LUC"(T)$ by $\mathcal{C}_{b}(T)$, then $\lambda$ need not
be jointly continuous unless $T$ is a LC Hausdorff topological group.

######  2.6.

Let $\verb"LUC"(T)^{*}$ be the dual space of $\verb"LUC"(T)$—the space of
bounded linear functionals on the Banach space
$(\verb"LUC"(T),\|\cdot\|_{\infty})$, endowed with the weak-$*$ topology. Then

  1. $\verb"LUC"(T)^{*}\times\verb"LUC"(T)\rightarrow\mathbb{C},\ (\mu,f)\mapsto\langle\mu,f\rangle=\mu(f)$

is jointly continuous.

###### Proof.

If $\mu_{n}\to_{\textrm{wk}^{*}}\mu$ in $\verb"LUC"(T)^{*}$ and
$f_{n}\to_{\|\cdot\|_{\infty}}f$ in $\verb"LUC"(T)$, then by the uniform
boundedness principle (cf. [9, Thm. III.14.1]) it follows that

$|\mu_{n}(f_{n})-\mu(f)|\leq\|\mu_{n}\|\cdot\|f_{n}-f\|_{\infty}+|\mu_{n}(f)-\mu(f)|\to
0.$

The proof is complete. ∎

######  2.7.

Given $\mu\in\verb"LUC"(T)^{*}$ and $t\in T$ we define $t\mu$ by

  1. $\langle t\mu,f\rangle=\langle\mu,ft\rangle\ \forall f\in\verb"LUC"(T)$.

Then

  1. $T\times\verb"LUC"(T)^{*}\rightarrow\verb"LUC"(T)^{*},\ (t,\mu)\mapsto t\mu$

is a jointly continuous phase mapping.

###### Proof.

Let $t_{n}\to t$ in $T$ and $\mu_{n}\to_{\textrm{wk}^{*}}\mu$ in
$\verb"LUC"(T)^{*}$ then for all $f\in\verb"LUC"(T)$ we have $|\langle
t_{n}\mu_{n},f\rangle-\langle t\mu,f\rangle|=|\mu_{n}(ft_{n})-\mu(ft)|\to 0$
by 2.6, for $ft_{n}\to_{\|\cdot\|_{\infty}}ft$ by 2.5. Thus,
$t_{n}\mu_{n}\to_{\textrm{wk}^{*}}t\mu$. Clearly, $(st)\mu=s(t\mu)$ for all
$s,t\in T$ and $\mu\in\verb"LUC"(T)^{*}$. ∎

######  2.8.

Let $\mathbb{V}$ be a norm closed subalgebra of $\verb"LUC"(T)$ such that
$\mathbb{V}$ is right $T$-invariant (i.e., $ft\in\mathbb{V}$ whenever $t\in T$
and $f\in\mathbb{V}$), and that $\mathbb{V}$ is self-adjoint and contains the
constants.

Given $t\in T$, define $t^{*}\in\mathbb{V}^{*}$ by
$t^{*}\colon\mathbb{V}\xrightarrow{f\mapsto f(t)}\mathbb{C}$. Clearly,
$\|t^{*}\|=1$ because of $\mathbf{1}\in\mathbb{V}$. Write

  1. $|\mathbb{V}|=\overline{\\{t^{*}\,|\,t\in T\\}}^{\textrm{wk}^{*}}$,

which is a compact Hausdorff space by Alaoglu’s theorem (cf. [9, Thm. V.3.1]).
Since $st^{*}(f)=f(st)=(st)^{*}(f)$ for all $s,t\in T$ and $f\in\mathbb{V}$,
then by 2.7 it follows that $st^{*}\in|\mathbb{V}|$ and that

  1. $\rho\colon T\times|\mathbb{V}|\rightarrow|\mathbb{V}|,\ (t,\mu)\mapsto t\mu$

is a jointly continuous phase mapping (cf. [24] or [14, Lem. 3.2, p.85] for
$T$ a topological group) so that

$(\mathscr{V},e^{*}):=(T,|\mathbb{V}|,e^{*})$ is a compact ambit. In fact, if
$\tau\in T$ such that $\overline{T\tau}=T$, then $(\mathscr{V},\tau^{*})$ is a
compact ambit of $T$.

Now for every $f\in\mathbb{V}$, define the Fourier transformation of $f$ as
follows:

  1. $\hat{f}\colon|\mathbb{V}|\rightarrow\mathbb{C},\ \mu\mapsto\langle\mu,f\rangle$.

Clearly,

  1. $\hat{{}}\,\colon\mathbb{V}\rightarrow\mathcal{C}(|\mathbb{V}|),\ f\mapsto\hat{f}$

is a linear mapping such that:

  1. a.

$\|\hat{f}\|=\|f\|_{\infty}\ \forall f\in\mathbb{V}$ (by 2.6 and 2.8.1).

  2. b.

$\widehat{fg}=\hat{f}\hat{g}\ \forall f,g\in\mathbb{V}$; i.e., $\hat{{}}$ is
multiplicative.

Indeed, we have for every $t\in T$ and all $f,g\in\mathbb{V}$ that

$\widehat{fg}(t^{*})=t^{*}(fg)=f(t)g(t)=\hat{f}(t^{*})\hat{g}(t^{*});$

thus, $\widehat{fg}=\hat{f}\hat{g}$.

  3. c.

If $f\equiv c$ then $\hat{f}\equiv c$; and $\hat{\bar{f}}=\bar{\hat{f}}$. If
$\alpha\not=\mu$ in $|\mathbb{V}|\,(\subseteq\mathbb{V}^{*})$, then
$\alpha(f)\not=\mu(f)$ for some $f\in\mathbb{V}$ so that
$\hat{f}(\alpha)\not=\hat{f}(\mu)$.

  4. d.

$\widehat{ft}=\hat{f}t\ \forall f\in\mathbb{V}$ and $t\in T$.

This shows that $\mathcal{A}=\\{\hat{f}\,|\,f\in\mathbb{V}\\}$ is a norm
closed subalgebra of $\mathcal{C}(|\mathbb{V}|)$ such that $\mathcal{A}$
separates points of $|\mathbb{V}|$, is self-adjoint and contains the
constants.

In particular, write

  1. $T^{\textsl{{LUC}}}=|\verb"LUC"(T)|$ and $\mathscr{T}^{\textsl{{LUC}}}=(T,T^{\textsl{{LUC}}})$.

Let $\iota\colon T\rightarrow T^{\textsl{{LUC}}}$ be defined by $t\mapsto
t^{*}\ \forall t\in T$. Clearly, $\iota$ is continuous. Indeed, if $t_{n}\to
t$ in $T$, then $f(t_{n})\to f(t)$ for all $f\in\mathbb{V}$ so that
$t_{n}^{*}\to_{\textrm{wk}^{*}}t^{*}$. As usual, $(\iota,T^{\textsl{{LUC}}})$
is called the LUC-compactification of $T$, such that

  1. i.

$\iota[T]$ is dense in $T^{\textsl{{LUC}}}$;

  2. ii.

if $f\colon T\rightarrow Y$ is a left-uniformly continuous function on $T$ to
a compact Hausdorff space $Y$ (i.e., $ft_{n}\to ft$ uniformly as $t_{n}\to t$
in $T$), then $f$ may be extended uniquely to a continuous function
$f^{\textsl{{LUC}}}$ on $T^{\textsl{{LUC}}}$ to $Y$.

(See Knapp 1966 [24] and Flor 1967 [15] for $T$ a topological group.)

If $\verb"LUC"(T)$ separates points of $T$, then $\iota\colon T\rightarrow
T^{\textsl{{LUC}}}$ is obviously 1-1 (i.e, $\iota$ is an imbedding).

######  2.8.6 Embedding Lemma.

If $\verb"LUC"(T)$ separates points and closed sets in $T$ (e.g., $T$ is a
topological group; cf. Cor. 5.5.2), then $\iota\colon
T\rightarrow\iota[T]\,(\subseteq T^{\textsl{{LUC}}})$ is an open 1-1 onto map.
Moreover,

  1. i.

$T$ is LC iff $\iota[T]$ is open in $T^{\textsl{{LUC}}}$; and

  2. ii.

if $T$ is discrete and $A,B\subset T$, then $\overline{A\cap
B}=\overline{A}\cap\overline{B}$ relative to $T^{\textsl{{LUC}}}$.

Consequently, if $T$ is a LC topological group, then $\mu\in
T^{\textsl{{LUC}}}$ is a transitive point for $\mathscr{T}^{\textsl{{LUC}}}$
iff $\mu=t^{*}$ for some $t\in T$.

###### Proof.

Clearly, $\iota$ is a bijection. Now, let $\tau\in T$ and let
$U\in\mathfrak{N}_{\tau}(T)$. To prove openness of $\iota\colon
T\rightarrow\iota[T]$, it is sufficient to show that $\iota[U]$ contains the
intersection of $\iota[T]$ and a neighborhood of $\tau^{*}$ in
$T^{\textsl{{LUC}}}$. By the separating hypothesis, we can choose
$f\in\verb"LUC"(T)$ such that $f(\tau)=0$, $f=1$ on $T\setminus U$, and $0\leq
f\leq 1$. By shrinking $U$ if necessary, we may assume $0\leq f<1$ on $U$. Let
$V=\\{\mu\in T^{\textsl{{LUC}}}\,|\,|\mu(f)-\tau^{*}(f)|<1\\}$. Since $V$ is
an open neighborhood of $\tau^{*}$ in $T^{\textsl{{LUC}}}$ such that
$V\cap\iota[T]\subseteq\iota[U]$, $\iota$ is an open map of $T$ onto
$\iota[T]$.

i). If $T$ is additionally LC, then $\iota[T]$ is open in $T^{\textsl{{LUC}}}$
by [23, Problem 5.G].111Lemma ([23, 5.G]). If $X$ is a Hausdorff space and $Y$
is a dense LC subspace of $X$, then $Y$ is open in $X$.

###### Proof.

Suppose to the contrary that there is a point $y\in Y$ and a net $x_{n}\in
X\setminus Y$ with $x_{n}\to y$. Let $U\in\mathfrak{N}_{y}(Y)$ be compact.
Then $\textrm{cls}_{X}U=U$ and there exists an open $V\in\mathfrak{N}_{y}(X)$
such that $V\cap Y\subseteq U$. Clearly, $x_{n}\in V\setminus U$ eventually.
Since $V\setminus U$ is open in $X$ and $Y$ is dense in $X$, we may pick
$x_{n}^{\prime}\in Y$ with $x_{n}^{\prime}\in V\setminus U$ such that
$x_{n}^{\prime}\to y$. However, since $U\in\mathfrak{N}_{y}(Y)$,
$x_{n}^{\prime}\in U$ eventually. So we reach a contradiction. ∎

ii). Assume $T$ is discrete and $A,B\subset T$. First $\overline{A\cap
B}\subseteq\overline{A}\cap\overline{B}$. To show the opposite inclusion, let
$\mu\in\overline{A}\cap\overline{B}$. Then there are nets $\\{a_{n}\\}$ in $A$
and $\\{b_{n}\\}$ in $B$ with $a_{n}^{*}\to\mu$ and $b_{n}^{*}\to\mu$. If
there is a subnet $\\{a_{i}\\}$ from $\\{a_{n}\\}$ with $\\{a_{i}\\}\subset
B$, then $\mu\in\overline{A\cap B}$. However, if there is no such a subnet,
then we can find some $n_{0}$ such that $a_{n}\not\in B$ for all $n\geq
n_{0}$. Now define $f\in\verb"LUC"(T)$ such that $f=0$ on $B$ and $f(a_{n})=1$
for all $n\geq n_{0}$. Then $1=\lim_{n}f(a_{n})=\mu(f)=\lim_{n}f(b_{n})=0$, a
contradiction. The proof is completed. ∎

######  2.9.

Under the context of 2.8, it follows from Theorem 2.1 that
$\mathcal{C}(|\mathbb{V}|)$ is isometric onto $\mathbb{V}$. Now $|\mathbb{V}|$
is called the maximal ideal space or structure space of $\mathbb{V}$. It is
clear that

$None$ $\mathbb{F}(\mathscr{V},e^{*})=\mathbb{V}$

(see [12, Thm. IV.5.18] for $T$ a Hausdorff topological group). Indeed, for
all $f\in\mathbb{V}$ or $\hat{f}\in\mathcal{C}(|\mathbb{V}|)$, we have that
$\hat{f}^{e^{*}}(t)=\hat{f}(te^{*})=\hat{f}(t^{*})=f(t)$ for all $t\in T$. In
fact, if $T$ is a group, we can expect more as follows:

######  2.9B Lemma.

If $T$ is a left-topological group, then
$\mathbb{F}(\mathscr{T}^{\textsl{{LUC}}},\tau^{*})=\verb"LUC"(T)$ for all
$\tau\in T$.

###### Proof.

Let $\tau\in T$. Clearly
$\mathbb{F}(\mathscr{T}^{\textsl{{LUC}}},\tau^{*})\subseteq\verb"LUC"(T)$ by
2.4.1 and 2.8.2. Now let $f\in\verb"LUC"(T)$. Then
$\tau^{-1}f\in\verb"LUC"(T)$ (by Lem. 4.2 below) so that

$\widehat{\tau^{-1}f}^{\tau^{*}}(t)=\tau^{-1}f(t\tau)=f(t)\quad\forall t\in
T.$

Thus, $\widehat{\tau^{-1}f}^{\tau^{*}}=f$ and
$\texttt{LUC}(T)\mathbb{F}\subseteq(\mathscr{T}^{\textsl{{LUC}}},\tau^{*})$.
The proof is completed. ∎

Therefore, every $f\in\verb"LUC"(T)$ may come from a compact ambit of $T$. See
[25, $\S\S$20 $\thicksim$ 26] or [20, Appendix C] for the general theory of
the maximal ideal spaces of commutative Banach algebras.

##  3 The universal compact ambit of $T$

A compact ambit of $T$ is called universal iff any other compact ambit of $T$
must be one of its factors (cf. [14, Def. 7.4] or [4, p.117]). We shall prove
that the universal compact ambit of $T$ may be realized by $T$ acting on
$T^{\textsl{{LUC}}}$ (Thm. 3.3).

######  3.1 Theorem (see [12, Prop. IV.5.12] for compact flows).

Let $(\mathscr{Y},y_{0})$ and $(\mathscr{Z},z_{0})$ be two point-transitive
compact dynamics of $T$. Then there exists an extension
$\phi\colon(\mathscr{Y},y_{0})\rightarrow(\mathscr{Z},z_{0})$ iff
$\mathbb{F}(\mathscr{Y},y_{0})\supseteq\mathbb{F}(\mathscr{Z},z_{0})$. Hence
$(\mathscr{Y},y_{0})$ is isomorphic with $(\mathscr{Z},z_{0})$ iff
$\mathbb{F}(\mathscr{Y},y_{0})=\mathbb{F}(\mathscr{Z},z_{0})$.

###### Proof.

Necessity is obvious. Now conversely, suppose
$\mathbb{F}(\mathscr{Y},y_{0})\supseteq\mathbb{F}(\mathscr{Z},z_{0})$. To
construct an extension
$\phi\colon(\mathscr{Y},y_{0})\rightarrow(\mathscr{Z},z_{0})$, we first define
a mapping $\phi\colon Ty_{0}\rightarrow Tz_{0}$ by $ty_{0}\mapsto tz_{0}$. To
see that $\phi$ is a well-defined mapping, let $t_{1},t_{2}\in T$ with
$t_{1}y_{0}=t_{2}y_{0}$. If $t_{1}z_{0}\not=t_{2}z_{0}$, we can select
$F\in\mathcal{C}(Z)$ such that $F(t_{1}z_{0})\not=F(t_{2}z_{0})$ and set
$F^{z_{0}}(t)=F(tz_{0})$ for all $t\in T$. So
$F^{z_{0}}(t_{1})\not=F^{z_{0}}(t_{2})$ and
$F^{z_{0}}\in\mathbb{F}(\mathscr{Z},z_{0})$. By
$\mathbb{F}(\mathscr{Y},y_{0})\supseteq\mathbb{F}(\mathscr{Z},z_{0})$, there
exists a function $\eta\in\mathcal{C}(Y)$ such that
$\eta^{y_{0}}(t)=\eta(ty_{0})=F^{z_{0}}(t)$ for every $t\in T$. Thus,
$\eta^{y_{0}}(t_{1})\not=\eta^{y_{0}}(t_{2})$. However,
$\eta^{y_{0}}(t_{1})=\eta(t_{1}y_{0})=\eta(t_{2}y_{0})=\eta^{y_{0}}(t_{2})$, a
contradiction. Thus, $\phi\colon Ty_{0}\rightarrow Tz_{0}$ is a well-defined
mapping.

Next, we shall extend $\phi$ from $Ty_{0}$ to $Y$ continuously. For this, let
$y\in Y$. Select two nets $s_{n},t_{n}\in T$ with $s_{n}y_{0}\to y$ and
$t_{n}y_{0}\to y$. Then $s_{n}z_{0}\to z$ and $t_{n}z_{0}\to z$, for some
$z\in Z$, and set $\phi(y)=z$. Indeed, assume
$s_{i}^{\prime},t_{j}^{\prime\prime}\in T$ are two subnets of $\\{s_{n}\\}$
and $\\{t_{n}\\}$, respectively, such that $s_{i}^{\prime}z_{0}\to z^{\prime}$
and $t_{j}^{\prime\prime}z_{0}\to z^{\prime\prime}$ in $Z$ with
$z^{\prime}\not=z^{\prime\prime}$. We can pick a function $F\in\mathcal{C}(Z)$
such that $F(z^{\prime})\not=F(z^{\prime\prime})$. Since
$F^{z_{0}}\in\mathbb{F}(\mathscr{Z},z_{0})\subseteq\mathbb{F}(\mathscr{Y},y_{0})$,
there exists $\eta\in\mathcal{C}(Y)$ with $\eta^{y_{0}}=F^{z_{0}}$. So

$F^{z_{0}}(s_{i}^{\prime})=\eta^{y_{0}}(s_{i}^{\prime})=\eta(s_{i}^{\prime}y_{0})\to\eta(y)$

and

$F^{z_{0}}(t_{j}^{\prime\prime})=\eta^{y_{0}}(t_{j}^{\prime\prime})=\eta(t_{j}^{\prime\prime}y_{0})\to\eta(y).$

Thus, $F(z^{\prime})=\eta(y)=F(z^{\prime\prime})$, a contradiction. So
$\phi\colon Y\rightarrow Z$ is well defined and continuous.

Since $\phi(ty_{0})=tz_{0}=t\phi(y_{0})$ and $\overline{Ty_{0}}=Y$, hence
$\phi$ is $T$-equivariant with $z_{0}=\phi(y_{0})$. The proof is completed. ∎

Therefore, if $\mathscr{X}$ is a minimal non-distal compact semiflow such that
$(x_{0},x_{1})$ is not a.p. under $\mathscr{X}\times\mathscr{X}$, then we have
that $\mathbb{F}(\mathscr{X},x_{0})\not=\mathbb{F}(\mathscr{X},x_{1})$.
Otherwise, there exists
$\phi\colon(\mathscr{X},x_{0})\rightarrow(\mathscr{X},x_{1})$ so that
$(x_{0},x_{1})$ is an a.p. point under $\mathscr{X}\times\mathscr{X}$.

######  3.2Regular and coalescent dynamics.

A dynamic $\mathscr{X}$ is called regular as in [3] if whenever $(x,y)\in
X\times X$ is a.p. under $\mathscr{X}\times\mathscr{X}$, then there exists
$\rho\in\texttt{Aut}(\mathscr{X})$ such that $y=\rho x$. If every endomorphism
of $\mathscr{X}$ is an automorphism of $\mathscr{X}$, then $\mathscr{X}$ is
said to be coalescent (cf. [4, p.81]).

For a non-LC group, the regularity/coalescence of the universal minimal
compact flow is proved in the literature [4, p.116] and [19] by using the
maximal a.p. set and transfinite induction, respectively. Here Ellis’
algebraic proof is still valid for left-topological monoids; see Theorem 3.3
below.

According to 2.8, $(\mathscr{T}^{\textsl{{LUC}}},e^{*})$ is a compact ambit of
$T$. It turns out that $\mathscr{T}^{\textsl{{LUC}}}$ is universal in the
category of compact ambits of $T$.

######  3.3 Theorem (cf. [24], [14, $\S\S$7,9], [17, p.35], [4, p.122] for $T$
a topological group).

Let $T$ be a Hausdorff left-topological monoid. Then:

  1. 1.

$(\mathscr{T}^{\textsl{{LUC}}},e^{*})$ is the unique (up to isomorphisms)
universal compact ambit of $T$. Moreover, it holds that
$\mathbb{F}(\mathscr{T}^{\textsl{{LUC}}},e^{*})=\verb"LUC"(T)$.

  2. 2.

$T^{\textsl{{LUC}}}$ is a compact Hausdorff right-topological semigroup with
identity $e^{*}$, where the multiplication is the extension of
$(s^{*},t^{*})\mapsto(st)^{*}$ for all $s,t\in T$. (Hence, if
$\phi\colon\mathscr{T}^{\textsl{{LUC}}}\rightarrow\mathscr{T}^{\textsl{{LUC}}}$
is a homomorphism, then there exists $q\in T^{\textsl{{LUC}}}$ with
$\phi=R_{q}$; cf. [14, Prop. 7.9].)

  3. 3.

If $T$ is a LC topological group, then $\mathscr{T}^{\textsl{{LUC}}}$ is
coalescent. (Thus, if $\mathscr{Z}$ is point transitive and
$\varphi\colon\mathscr{Z}\rightarrow\mathscr{T}^{\textsl{{LUC}}}$ is an
extension, then $\varphi$ is an isomorphism; cf. [14, Cor. 7.10].)

Hence the universal minimal compact semiflow of $T$ is:

  1. a)

regular (so unique up to isomorphisms);

  2. b)

a factor of $\mathscr{T}^{\textsl{{LUC}}}$; and

  3. c)

a subsystem of $\mathscr{T}^{\textsl{{LUC}}}$.

###### Proof.

(1): First, $(\mathscr{T}^{\textsl{{LUC}}},e^{*})$ is a compact ambit of $T$
by 2.8.1 and 2.8.4. By 2.9, it follows that
$\mathbb{F}(\mathscr{T}^{\textsl{{LUC}}},e^{*})=\verb"LUC"(T)$. Suppose
$(\mathscr{Y},y_{0})$ is any compact ambit of $T$. Since
$\mathbb{F}(\mathscr{Y},y_{0})\subseteq\verb"LUC"(T)$ by 2.4.1, hence
$(\mathscr{Y},y_{0})$ is a factor of $(\mathscr{T}^{\textsl{{LUC}}},e^{*})$
from Theorem 3.1. If $(\mathscr{Y},y_{0})$ is another universal compact ambit
of $T$, then there exists a homomorphism
$(\mathscr{Y},y_{0})\rightarrow(\mathscr{T}^{\textsl{{LUC}}},e^{*})$. Whence
$\verb"LUC"(T)\supseteq\mathbb{F}(\mathscr{Y},y_{0})\supseteq\mathbb{F}(\mathscr{T}^{\textsl{{LUC}}},e^{*})=\verb"LUC"(T)$.
Thus, it follows by Theorem 3.1 that
$(\mathscr{Y},y_{0})\cong(\mathscr{T}^{\textsl{{LUC}}},e^{*})$.

(2): It follows from 1. that $(\mathscr{T}^{\textsl{{LUC}}},e^{*})$ is
isomorphic to its Ellis enveloping pointed semiflow
$(T,E(\mathscr{T}^{\textsl{{LUC}}}),\textrm{id}_{T^{\textsl{{LUC}}}})$ such
that the homeomorphism $E(\mathscr{T}^{\textsl{{LUC}}})\xrightarrow{p\mapsto
pe^{*}}T^{\textsl{{LUC}}}$ induces the desired multiplication on
$T^{\textsl{{LUC}}}$.

(3): By Lemma 2.9B and Lemma 2.8.6.

Finally, let $(T,M)$ be a minimal left ideal in $T^{\textsl{{LUC}}}$. By 1.,
$(T,M)$ is a universal minimal compact semiflow of $T$, which is a factor and
a subsystem of $\mathscr{T}^{\textsl{{LUC}}}$. Moreover, $M=\bigcup_{v}vM$ and
$vM$ are groups with identity $v$ for all idempotents $v\in M$. So, for every
$q\in M$, the right translation $R_{q}\colon M\rightarrow M$ is an
automorphism of $(T,M)$. Then $(T,M)$ is regular and it is exactly the unique
(up to isomorphisms) universal minimal compact semiflow of $T$.

The proof is thus completed. ∎

The simple observation Theorem 3.3.2 has been useful in proving the
regularity/unicity of the universal minimal compact system of $T$.

######  3.4 Remark.

$\iota\colon T\xrightarrow{t\mapsto t^{*}}T^{\textsl{{LUC}}}$ is left-
uniformly continuous in the sense that as $t_{n}\to t$ in $T$, $\iota
t_{n}=\iota\circ L_{t_{n}}\to\iota t=\iota\circ L_{t}$ uniformly; that is,
given any $f_{1},\dotsc,f_{k}\in\verb"LUC"(T)$ and $\varepsilon>0$,
$\exists\,n_{0}$ s.t. if $n\geq n_{0}$ then

$|f_{i}(t_{n}x)-f_{i}(tx)|<\varepsilon\quad\forall x\in T$

for $i=1,\dotsc,k$.

######  3.5 Remark.

If $T$ is Tychonoff (completely regular $T_{1}$), then the Stone-Čech
compactification $(\mathfrak{e},\beta T)$ is well defined ([23, Thm. 5.24]).
So every continuous function from $T$ to a compact Hausdorff space may be
uniquely continuously extended on $\beta T$. However,
$(\iota,T^{\textsl{{LUC}}})$ can only extend left-uniformly continuous
functions on $T$ (see 2.8.4-ii). This shows that $T^{\textsl{{LUC}}}$ is
“smaller” than $\beta T$ in general; that is, there is a continuous map
$\psi\colon(\beta T,e)\rightarrow(T^{\textsl{{LUC}}},e^{*})$ with
$\iota=\psi\circ\mathfrak{e}$.

######  3.6 Remark.

If $T$ is a LC topological group or if $T$ is a discrete monoid, then $(\beta
T,e)\cong(T^{\textsl{{LUC}}},e^{*})$.

###### Proof.

It follows by Ellis’s joint continuity theorem that the canonical
multiplication $T\times\beta T\rightarrow\beta T$ is jointly continuous so
that $(T,\beta T,e)$ is a universal compact ambit. Then by Theorem 3.3,
$(T,\beta T,e)\cong(T,T^{\textsl{{LUC}}},e^{*})$. The proof is complete. ∎

######  3.7 Remark.

By Theorem 3.3 or 2.8.1, $\rho\colon T\times T^{\textsl{{LUC}}}\rightarrow
T^{\textsl{{LUC}}}$ is jointly continuous. However, $\lambda\colon
T^{\textsl{{LUC}}}\times T\rightarrow T^{\textsl{{LUC}}}$ need not be jointly
continuous here. To see this, let’s borrow an example from [29] (also see [7,
Prop. 1]) as follows: Let $G$ be a LC Hausdorff topological group, which is
not distal in the sense of Rosenblatt. So there is some $g\in G$ with
$g\not=e$ and a net $g_{n}\in G$ such that $g_{n}^{-1}gg_{n}\to e$ in $G$.
Passing to a subnet if necessary, we can assume $g_{n}\to x$ in
$G^{\textsl{{LUC}}}$. If $\lambda$ is jointly continuous, then

$gg_{n}\to gx\quad\textrm{and}\quad gg_{n}=g_{n}(g_{n}^{-1}gg_{n})\to x,$

contrary to $gx\not=x$ by Veech’s Theorem (see Thm. 5.7 below).

######  3.8Left-invariant means on $\texttt{LUC}(T)$ and invariant measures on
$T^{\textsl{{LUC}}}$.

A linear functional $\mathfrak{m}\colon\verb"LUC"(T)\rightarrow\mathbb{C}$ is
called a left-invariant mean if

$\displaystyle\overline{\mathfrak{m}(f)}=\mathfrak{m}(\bar{f})\quad\forall
f\in\texttt{LUC}(T),$ $\displaystyle\mathfrak{m}(\mathbf{1})=1,$
$\displaystyle\mathfrak{m}(f)\geq 0\quad\textrm{if
}f\in\texttt{LUC}(T)\textrm{ s.t. }f\geq 0,$ $\displaystyle
t\mathfrak{m}=\mathfrak{m}\ (\textrm{or
equivalently},\mathfrak{m}(ft)=\mathfrak{m}(f)\ \forall
f\in\texttt{LUC}(T))\quad\forall t\in T.$

$T$ is called left-amenable iff there exists a left-invariant mean on
$\verb"LUC"(T)$.

Then:

######  Theorem (see [17, Thm. III.3.1, (1)$\Leftrightarrow$(2)] for $T$ a
topological group).

$\mathscr{T}^{\textsl{{LUC}}}$ admits a regular Borel probability measure iff
$\verb"LUC"(T)$ have a left-invariant mean. In other words,
$\mathscr{T}^{\textsl{{LUC}}}$ admits regular Borel probability measures iff
$T$ is left-amenable.

###### Proof.

First from 2.8.3d the Fourier transformation
$\hat{}\colon\verb"LUC"(T)\rightarrow\mathcal{C}(T^{\textsl{{LUC}}})$ is a
topological isomorphism under $\|\cdot\|_{\infty}$ such that

$\widehat{ft}=\hat{f}t\quad\forall f\in\verb"LUC"(T)\textrm{ and }t\in T.$

If $\mathfrak{m}$ is a left-invariant mean on $\verb"LUC"(T)$, then define a
linear functional as follows:

$\mu\colon\mathcal{C}(T^{\textsl{{LUC}}})\rightarrow\mathbb{C}\quad\textrm{by}\
\hat{f}\mapsto\mathfrak{m}(f).$

By the Riesz representation theorem, $\mu$ may be a regular Borel probability
measure on $T^{\textsl{{LUC}}}$ such that $\mu(\varphi)=\mu(\varphi t)$ for
all $\varphi\in\mathcal{C}(T^{\textsl{{LUC}}})$ and $t\in T$. Thus, $\mu$ is
$T$-invariant on $T^{\textsl{{LUC}}}$ and $\mathscr{T}^{\textsl{{LUC}}}$
admits $\mu$.

Conversely, suppose $\mu$ is an invariant regular Borel probability measure on
$T^{\textsl{{LUC}}}$. Define
$\mathfrak{m}\colon\verb"LUC"(T)\rightarrow\mathbb{C}$ by
$f\mapsto\mathfrak{m}(f)=\mu(\hat{f})$. Clearly, $\mathfrak{m}$ is a left-
invariant mean on $\verb"LUC"(T)$. The proof is complete. ∎

##  4 Hulls of left-uniformly continuous functions

Given $f\in\verb"LUC"(T)$, we have seen that $f$ may come from a compact ambit
of $T$ (Thm. 3.3). In fact, $(\mathscr{T}^{\textsl{{LUC}}},e^{*})$ is the
maximal such ambit of $T$. Now we will find the “smallest” one (see Thm. 4.4).

######  4.1Left compact-open/pointwise hulls of functions.

Given $f\in\mathcal{C}_{b}(T)$, define

  1. (1)

$\verb"H"_{\textrm{co}}^{L}[f]={\overline{Tf}}^{\textrm{co}}$ and

  2. (2)

$\verb"H"_{\textrm{p}}^{L}[f]={\overline{Tf}}^{\textrm{p}}$.

Here ${}_{\textrm{co}}$ and ${}_{\textrm{p}}$ stand for the compact-open and
pointwise topologies on $\verb"LUC"(T)$, respectively.

  1. (3)

Let $\mathcal{F}\subseteq\mathcal{C}_{b}(T)$. We say $\mathcal{F}$ is left-
equicontinuous on $T$ if given $\varepsilon>0$ and $s_{n}\to s$ in $T$, there
exists $n_{0}$ such that as $n\geq n_{0}$,
$\|fs_{n}-fs\|_{\infty}<\varepsilon$ for all $f\in\mathcal{F}$. In this case,
$\mathcal{F}\subseteq\verb"LUC"(T)$ by Def. 2.2.3.

Then it is easy to verify that

  1. (4)

If $\mathcal{F}\subseteq\mathcal{C}_{b}(T)$ is left-equicontinuous on $T$,
then $\mathcal{F}$ is equicontinuous on $T$ (i.e., $\forall x\in T$ and
$\forall\varepsilon>0$, $\exists U\in\mathfrak{N}_{x}(T)$ s.t.
$|f(y)-f(x)|<\varepsilon$ $\forall y\in U$ and $\forall f\in\mathcal{F}$).

Indeed, let $x\in T$. If $\mathcal{F}$ is not equicontinuous at $x$, then
there is some $\varepsilon>0$ such that for every $U\in\mathfrak{N}_{x}(T)$ we
can pick some $f\in\mathcal{F}$ and $u\in U$ with
$|f(u)-f(x)|\geq\varepsilon$. Further, there is a net $u_{n}\to x$ in $T$ and
$f_{n}\in\mathcal{F}$ such that $|f_{n}u_{n}(e)-f_{n}x(e)|\geq\varepsilon$,
contrary to (3).

######  4.2 Lemma (cf. [12, Lem. IV.5.2, Cor. IV.5.3] for $T$ a topological
group).

Let $f\in\mathcal{C}_{b}(T)$; then the following conditions are pairwise
equivalent:

  1. (1)

$f\in\verb"LUC"(T)$;

  2. (2)

$Tf$ is left-equicontinuous on $T$;

  3. (3)

$\verb"H"_{\textrm{co}}^{L}[f]$ is left-equicontinuous on $T$;

  4. (4)

$\verb"H"_{\textrm{co}}^{L}[f]\subseteq\verb"LUC"(T)$.

If one of (1) $\sim$ (4) holds, then
$\verb"H"_{\textrm{co}}^{L}[f]=\verb"H"_{\textrm{p}}^{L}[f]$ is a compact
left-invariant subset of $\verb"LUC"(T)$ and

  1. $\rho\colon T\times\verb"H"_{\textrm{co}}^{L}[f]\rightarrow\verb"H"_{\textrm{co}}^{L}[f],\quad(t,\phi)\mapsto t\phi$

is a jointly continuous phase mapping, under the compact-open topology on
$\verb"H"_{\textrm{co}}^{L}[f]$. Consequently,
$(\mathscr{H}_{\textrm{co}}^{L}[f],f):=(T,\verb"H"_{\textrm{co}}^{L}[f],f)$ is
a compact ambit and $f$ comes from $(\mathscr{H}_{\textrm{co}}^{L}[f],f)$, for
all $f\in\verb"LUC"(T)$.

###### Proof.

(1)$\Rightarrow$(2): Let $\varepsilon>0$ and $s_{n}\to s$ in $T$ be arbitrary.
By

$|tfs_{n}(x)-tfs(x)|=|fs_{n}(xt)-fs(xt)|<\varepsilon\quad\forall x\in
T\textrm{ and }t\in T$

as $n$ sufficiently big, it follows that $Tf$ is left-equicontinuous on $T$.

(2)$\Rightarrow$(3): Let $\varepsilon>0$ and $s_{n}\to s$ in $T$. By condition
(2) and 4.1-(3), there is $n_{0}$ such that as $n\geq n_{0}$,

$|tfs_{n}(x)-tfs(x)|=|fs_{n}(xt)-fs(xt)|<\varepsilon\quad\forall x\in
T\textrm{ and }t\in T.$

Now let $\phi\in\verb"H"_{\textrm{co}}^{L}[f]$ and select a net $t_{i}\in T$
with $t_{i}f\to_{\textrm{co}}\phi$. Then, as $n\geq n_{0}$,

$|\phi s_{n}(x)-\phi
s(x)|={\lim}_{i}|t_{i}fs_{n}(x)-t_{i}fs(x)|\leq\varepsilon\quad\forall x\in
X.$

Thus, $\verb"H"_{\textrm{co}}^{L}[f]$ is left-equicontinuous on $T$.

(3)$\Rightarrow$(4): By 4.1-(3).

(4)$\Rightarrow$(1): Obvious.

Now suppose one of (1) $\sim$ (4) is fulfilled. By 2.2.4 and condition (3),
$\verb"H"_{\textrm{co}}^{L}[f]$ is a left-invariant subset of $\verb"LUC"(T)$.
Moreover, it follows from 4.1-(4) and Arzelà-Ascoli theorem (cf. [12, A.2.5])
that $\verb"H"_{\textrm{co}}^{L}[f]$ is compact under the compact-open
topology on $\verb"LUC"(T)$. Thus,
$\verb"H"_{\textrm{co}}^{L}[f]=\verb"H"_{\textrm{p}}^{L}[f]$ and the compact-
open topology coincides with the pointwise topology on it (cf. [25, 3A]).

It remains to show that $\rho\colon(t,\phi)\rightarrow t\phi$ is continuous
from $T\times\verb"H"_{\textrm{co}}^{L}[f]$ to
$\verb"H"_{\textrm{co}}^{L}[f]$. For this, let $t_{n}\to t$ in $T$ and
$f_{n}\to\phi$ in $\verb"H"_{\textrm{co}}^{L}[f]$; we need prove that
$t_{n}f_{n}\to_{\textrm{p}}t\phi$. For this, let $x\in T$ then by $xt_{n}\to
xt$ in $T$ (noting $T$ is a left-topological semigroup), condition (3) and

$\begin{split}|t_{n}f_{n}(x)-t\phi(x)|&=|f_{n}(xt_{n})-\phi(xt)|\\\
&\leq|f_{n}(xt_{n})-f_{n}(xt)|+|f_{n}(xt)-\phi(xt)|\\\
&=|f_{n}xt_{n}(e)-f_{n}xt(e)|+|f_{n}(xt)-\phi(xt)|,\end{split}$

it follows that $\lim_{n}|t_{n}f_{n}(x)-t\phi(x)|=0$. Thus, $\rho$ is
continuous.

Let $\mathfrak{e}\colon\verb"H"_{\textrm{co}}^{L}[f]\rightarrow\mathbb{C}$ be
defined by $\phi\mapsto\phi(e)$. Then
$\mathfrak{e}\in\mathcal{C}(\verb"H"_{\textrm{co}}^{L}[f])$ and
$f(t)=\mathfrak{e}_{f}(t)$ for all $t\in T$. So
$f\in\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$ for all
$f\in\verb"LUC"(T)$. The proof is completed. ∎

Next we proceed to show that $(\mathscr{H}_{\textrm{co}}^{L}[f],f)$ is
actually the smallest compact ambit generated by $f\in\verb"LUC"(T)$.

######  4.3 Theorem (cf. [14, Prop. 9.8] and [12, Thm. IV.5.8] for $T$ a
group).

If $(\mathscr{X},x_{0})$ is a compact ambit of $T$, then
$(\mathscr{X},x_{0})\cong(T,|\mathbb{F}(\mathscr{X},x_{0})|,e^{*})$.

###### Proof.

By 2.4.2, 2.9 and Theorem 3.1. ∎

We note here that our proof line of Theorem 4.3 is different with that
available in literature (e.g. [12]). Here Theorem 4.3 follows from Theorem
3.1. In fact, Theorem 3.1 may in turn follow from Theorem 4.3 as to be dealt
with in [12, p.348]

######  4.4 Theorem.

Let $f\in\mathcal{C}_{b}(T)$ then:

  1. (1)

$f\in\verb"LUC"(T)$ iff $f$ comes from a compact ambit of $T$.

  2. (2)

If $f\in\verb"LUC"(T)$, then $\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$
is the smallest norm closed right-invariant self-adjoint subalgebra of
$\verb"LUC"(T)$ containing $f$ and $\mathbf{1}$.

###### Proof.

(1): By Lemma 4.2 and 2.4.1.

(2): By Lemma 4.2 and 2.4.2, $(\mathscr{H}_{\textrm{co}}^{L}[f],f)$ is a
compact ambit such that $\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$ is a
norm closed right-invariant self-adjoint subalgebra of $\verb"LUC"(T)$
containing $f$ and $\mathbf{1}$. Let $\mathbb{V}$ be the smallest closed
right-invariant self-adjoint subalgebra of $\verb"LUC"(T)$ containing $f$ and
$\mathbf{1}$. Then
$\mathbb{V}\subseteq\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$. By 2.9,
$(\mathscr{V},e^{*})$ is a compact ambit such that
$\mathbb{F}(\mathscr{V},e^{*})=\mathbb{V}$ and
$\mathcal{C}(|\mathbb{V}|)=\\{\hat{g}\,|\,g\in\mathbb{V}\\}$. Following 2.4.2,
we define

$\varrho_{\hat{f}}\colon|\mathbb{V}|\rightarrow\verb"LUC"(T),\quad\mu\mapsto\hat{f}^{\mu}\quad\textrm{s.t.
}te^{*}\mapsto tf\ \forall t\in T,$

where $\hat{f}^{\mu}(t)=\hat{f}(t\mu)=\langle\mu,ft\rangle\ \forall t\in T$,
for all $\mu\in|\mathbb{V}|$. Since $e^{*}$ is a transitive point for
$\mathscr{V}$, it follows by Lemma 4.2 that $\varrho_{\hat{f}}(e^{*})=f$ and
$\varrho_{\hat{f}}[|\mathbb{V}|]=\verb"H"_{\textrm{co}}^{L}[f]$. Moreover, it
is easy to show that
$\varrho_{\hat{f}}\colon(|\mathbb{V}|,e^{*})\rightarrow(\verb"H"_{\textrm{co}}^{L}[f],f)$
is a homomorphism. So by Theorem 3.1,
$\mathbb{V}\supseteq\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$ and
$\mathbb{V}=\mathbb{F}(\mathscr{H}_{\textrm{co}}^{L}[f],f)$. The proof is
completed. ∎

######  4.5 Theorem (cf. [12, Cor. IV.5.20] for $T$ a topological group).

Let $T$ be a Hausdorff left-topological monoid. Then

$T\curvearrowright(T^{\texttt{LUC}},e^{*})\cong T\curvearrowright(X,x_{0}),$

where

$X=\overline{Tx_{0}}\quad\textrm{and}\quad
x_{0}=(f)_{f\in\texttt{LUC}(T)}\in{\prod}_{f\in\texttt{LUC}(T)}\verb"H"_{p}^{L}[f].$

###### Proof.

Using Theorems 3.1 and 3.3 and the fact
$\mathbb{F}(\mathscr{X},x_{0})=\texttt{LUC}(T)$. ∎

##  5 On locally quasi-totally bounded groups and Veech’s Theorem

If $T$ is a LC Hausdorff topological group, then
$\mathscr{T}^{\textsl{{LUC}}}$ is free by Veech [31, Thm. 2.2.1] (the discrete
case of which is due to Ellis [13], also see [14, Prop. 8.14]). In fact, we
shall extend Veech’s theorem to locally quasi-totally bounded topological
groups (Def. 5.4.4 and Thm. 5.7).

######  5.1Effective and strongly effective.

A dynamic $\mathscr{X}=(T,X)$ is said to be effective if we have for all $t\in
T$ that $tx=x\ \forall x\in X$ implies $t=e$. If, for all $t\in T$, condition
$tx=x$ for some $x\in X$ implies $t=e$, then $\mathscr{X}$ is called strongly
effective or free.

######  5.2.

Let $t\in T$ with $t\not=e$ and let $\alpha\in T^{\textsl{{LUC}}}$. If
$t_{n}\in T$ with $t_{n}^{*}\to\alpha$ in $T^{\textsl{{LUC}}}$, then by 2.8.3c
and 2.9, $t\alpha=\alpha$ iff
$\lim_{n}f(tt_{n})=\hat{f}(t\alpha)=\hat{f}(\alpha)=\lim_{n}f(t_{n})$ for all
$f\in\verb"LUC"(T)$.

######  5.3 Theorem.

If $\verb"LUC"(T)$ separates the points and closed sets in $T$ and $T$ is
right cancelable (for instance, $T$ is a Hausdorff topological group), then
$\mathscr{T}^{\textsl{{LUC}}}$ is effective and $\iota\colon T\rightarrow
T^{\textsl{{LUC}}}$, $t\mapsto t^{*}$ is an embedding.

###### Proof.

By 5.2 and 2.8.6. ∎

######  5.4Locally quasi-totally bounded group.

Now we will consider a class of topological groups—locally quasi-totally
bounded topological groups that are more general than LC topological groups.

######  5.4.1 Standing hypothesis.

In the sequel of this section, let $G$ be a Hausdorff topological group with
identity $e$.

######  5.4.2.

$G$ is called locally totally bounded if there exists a set
$U\in\mathfrak{N}_{e}(G)$ which is totally bounded in the sense that given any
$V\in\mathfrak{N}_{e}(G)$ one can find a finite subset $F$ of $U$ such that
$VF\supseteq U$; that is, $U$ can be covered by a finite number of right
translates of $V$.

######  5.4.3 Examples.

A LC Hausdorff topological group is obviously locally totally bounded;
however, not vice versa.

  1. a.

$(\mathbb{Q},+)$, as a subspace of the Euclidean space $\mathbb{R}$, is
locally totally bounded but not LC.

  2. b.

$\texttt{SL}(2,\mathbb{Q})$, with the topology inherited from the Euclidean
space $\mathbb{R}^{2\times 2}$, is locally totally bounded non-LC.

######  5.4.4.

$G$ is called locally quasi-totally bounded if for every $t\in G$ with
$t\not=e$ there exists a symmetric set $V\in\mathfrak{N}_{e}(G)$ such that
$tV^{3}\cap V^{3}=\emptyset$ and that $V^{3}tV^{3}t^{-1}\cup
V^{3}t^{-1}V^{3}t$ can be covered by a finite number of right translates of
$V$.

A locally totally bounded group must be locally quasi-totally bounded; but not
vice versa as shown by the following examples.

######  5.4.5 Examples.

  1. 1.

First we consider the group $(\mathbb{Z}^{\mathbb{Z}},+)$. Given any integers
$n\geq 1$ and $n$ elements $x_{1},\dotsc,x_{n}\in\mathbb{Z}$, let
$V[x_{1},\dotsc,x_{n}]$ be the subgroup of $\mathbb{Z}^{\mathbb{Z}}$
consisting of the elements $f\in\mathbb{Z}^{\mathbb{Z}}$ such that
$f(x_{i})=0$ for all $1\leq i\leq n$. It is easy to see that there exists a
Hausdorff topology $\mathfrak{T}$ on $\mathbb{Z}^{\mathbb{Z}}$ for which
$(\mathbb{Z}^{\mathbb{Z}},\mathfrak{T})$ is a topological group with a
neighborhood base

$\displaystyle\\{V[x_{1},\dotsc,x_{n}]\,|\,x_{1},\dotsc,x_{n}\in\mathbb{Z}\textrm{
and }n\geq 1\\}$

of the zero element $\textbf{o}=(\dotsc,0,0,0,\dotsc)$ in
$\mathbb{Z}^{\mathbb{Z}}$.

    1. $\bullet$

$(\mathbb{Z}^{\mathbb{Z}},\mathfrak{T})$ is an abelian locally quasi-totally
bounded, but not locally totally bounded (further non-LC), Hausdorff
topological group.

###### Proof.

It is obvious that $(\mathbb{Z}^{\mathbb{Z}},\mathfrak{T})$ is locally quasi-
totally bounded for $V[x_{1},\dotsc,x_{n}]$ is a subgroup of
$\mathbb{Z}^{\mathbb{Z}}$. Now we show that it is not locally totally bounded;
otherwise, there exists a totally bounded neighborhood $V[x_{1},\dotsc,x_{n}]$
of o. Let $y$ be an integer with $y\not\in\\{x_{1},\dotsc,x_{n}\\}$. Then
there exist elements $f_{1},\dotsc,f_{r}\in\mathbb{Z}^{\mathbb{Z}}$ such that

$V[x_{1},\dotsc,x_{n}]\subseteq(f_{1}+V[y])\cup\dotsm\cup(f_{r}+V[y]).$

So, for every $\varphi\in V[x_{1},\dotsc,x_{n}]$ we have
$\varphi(y)\in\\{f_{1}(y),\dotsc,f_{r}(y)\\}$. Thus,
$\mathbb{Z}\subseteq\\{f_{1}(y),\dotsc,f_{r}(y)\\}$. This is a contradiction.
∎

  2. 2.

Next we will construct a non-abelian locally quasi-totally bounded Hausdorff
topological group, which is not locally totally bounded as follows. Let
$\\{H_{n}\\}_{n\in\mathbb{N}}$ be a family of Hilbert spaces and $H$ the
direct sum of them. We consider the non-abelian multiplicative group

$G=\left\\{L\in B(H)\,|\,L\textrm{ is invertible s.t. }LH_{n}=H_{n}\ \forall
n\in\mathbb{N}\right\\}$

where $B(H)$ be the set of bounded operators of $H$ into itself. For every
finite subset $\gamma$ of $\mathbb{N}$, define a normal subgroup $N_{\gamma}$
of $G$ by $L\in N_{\gamma}$ iff $L|_{H_{n}}=\textrm{id}_{H_{n}}\ \forall
n\in\gamma$. Let $\Gamma$ be the collection of all finite nonempty subsets of
$\mathbb{N}$.

Note that $N_{A}\cap N_{B}=N_{A\cup B}$ for $A,B\in\Gamma$ and
$\bigcap_{\gamma\in\Gamma}N_{\gamma}=\\{\textrm{id}_{H}\\}$. It is easy to see
that there exists a Hausdorff topology $\mathfrak{T}$ such that
$(G,\mathfrak{T})$ becomes a topological group with the neighborhood base
$\\{N_{\gamma}\,|\,\gamma\in\Gamma\\}$ of $e$ where $e=\textrm{id}_{H}$.

    1. $\bullet$

$(G,\mathfrak{T})$ is locally quasi-totally bounded, but not locally totally
bounded, non-abelian Hausdorff topological group.

###### Proof.

It is obvious that $G$ is locally quasi-totally bounded. Now we show that $G$
is not locally totally bounded. Otherwise, there exists a totally bounded
neighborhood $N_{\gamma}$ of $e$. Let $i$ be an integer with $i\not\in\gamma$.
Then there exist $L_{1},\dotsc,L_{r}\in G$ such that $N_{\gamma}\subseteq
L_{1}N_{\\{i\\}}\cup\dotsm\cup L_{r}N_{\\{i\\}}$. So for all $L\in N_{\gamma}$
and $x\in H_{i}$ we have $L(x)\in\\{L_{1}(x),\dotsc,L_{r}(x)\\}$. It is a
contradiction (if we pick carefully $H_{i}$, e.g., $H_{i}=\mathbb{R}$). ∎

######  5.5 Lemma (Urysohn-wise lemma; cf. Katětov 1959 [21]).

Let $(X,\mathscr{U})$ be any uniform space and $A,B\subset X$ nonempty. If
there is an $\alpha\in\mathscr{U}$ such that $A\cap\alpha[B]=\emptyset$, then
there is a uniformly continuous function $f\colon X\rightarrow[0,1]$ such that
$f=1$ on $A$ and $f=0$ on $B$.

###### Proof.

First we can take some $\gamma\in\mathscr{U}$ such that
$\overline{A}\cap\gamma[\overline{B}]=\emptyset$. Further there is some
$\delta\in\mathscr{U}$ such that
$\delta[\overline{A}]\cap\delta[\overline{B}]=\emptyset$. Now by [23, Thm
6.15], it follows that there is a pseudo-metric $\rho$, which is uniformly
continuous on $X\times X$, such that $\\{(x,y)\in X\times
X\,|\,\rho(x,y)<r\\}\subseteq\delta$ for some $r>0$. Thus, relative to $\rho$,
the ${r}/{3}$-neighborhoods $\mathrm{B}_{r/3}(\overline{A})$ and
$\mathrm{B}_{r/3}(\overline{B})$ are such that
$\mathrm{B}_{r/3}(\overline{A})\cap\mathrm{B}_{r/3}(\overline{B})=\emptyset$.
Since $f_{B}\colon x\mapsto 3r^{-1}\rho(\overline{B},x)$ from $X$ to
$[0,\infty)$ is uniformly continuous by the triangle inequality, hence
$f\colon X\rightarrow[0,1]$ defined by $x\mapsto\min\\{1,f_{B}(x)\\}$ is
uniformly continuous. Since $f|B\equiv 0$ and $f|A\equiv 1$, the proof is
completed. ∎

######  5.5.1 Corollary (cf. [25, Thm. 3E] using Urysohn’s lemma and one-point
compactification).

If $X$ is a LC Hausdorff space and if $C$ and $U$ are respectively compact and
open sets such that $C\subset U$, then there exists a real-valued continuous
function $f$ on $X$ such that $f=0$ on $C$, $f=1$ on $X\setminus U$ and $0\leq
f\leq 1$.

######  5.5.2 Corollary.

Let $T$ have a left-uniformity (cf. 2.2.3-ii). Then $\verb"LUC"(T)$ separates
points and closed sets in $T$, and in particular, if $t\in T$ and $t\not=e$
there exists a function $f\in\verb"LUC"(T)$ such that $tf\not=f$.

###### Proof.

Let $x\in T$ and $A\subset T$ be a closed set with $x\notin A$. Then there
exists an index $\alpha\in\mathscr{U}^{L}$ such that $\alpha[x]\cap
A=\emptyset$. By Lemma 5.5, there is a $\mathscr{U}^{L}$-uniformly continuous
function $f\colon X\rightarrow[0,1]$ with $f(x)=0$ and $f=1$ on $A$. Further,
$f\in\verb"LUC"(T)$ by 2.2.3-ii.

Now for $t\in T$ with $t\not=e$, there exists some $f\in\verb"LUC"(T)$ such
that $f(e)\not=f(t)=tf(e)$. So $tf\not=f$. The proof is complete. ∎

######  5.5.3 Theorem.

The universal point-transitive compact flow of $G$ is effective.

###### Proof.

By Theorem 5.3 or by Theorem 4.5 and Corollary 5.5.2. ∎

######  5.6 Lemma (uniformly separated finite partition lemma).

Let $t\in G$ with $t\not=e$ such that there exists a symmetric set
$V\in\mathfrak{N}_{e}(G)$ with the properties:

  1. $(\mathrm{a})$

$tV^{3}\cap V^{3}=\emptyset$;

  2. $(\mathrm{b})$

$V^{3}tV^{3}t^{-1}\cup V^{3}t^{-1}V^{3}t$ can be covered by a finite number of
right translates of $V$.

Then there exists a finite partition $G=P_{1}\cup\dotsm\cup P_{n}$ such that
$tP_{k}\cap VP_{k}=\emptyset$ for all $k=1,\dotsc,n$.

###### Proof.

By Zorn’s Lemma, there is a maximal subset $Y$, with respect the inclusion, of
$G$ such that $Vy\cap Vy^{\prime}=\emptyset$ for all $y\not=y^{\prime}$ in
$Y$. Thus

$None$ $|Vx\cap Y|\leq 1\quad\forall x\in G,\quad\textrm{where }|N|\textrm{ is
the cardinality of }N.$

Let $U=V^{3}tV^{3}t^{-1}\cup V^{3}t^{-1}V^{3}t$. By hypothesis (b) of the
lemma, there are a finite number of elements $x_{1},\dotsc,x_{m}$ of $G$ such
that $U\subseteq Vx_{1}\cup\dotsm\cup Vx_{m}$ so that $Ux\subseteq
Vx_{1}x\cup\dotsm\cup Vx_{m}x$ for all $x\in G$. Thus, by (5.6.1), there
exists a positive integer $\mathfrak{n}$ such that

$None$ $|Ux\cap Y|\leq\mathfrak{n}\quad\forall x\in G.$

Then put $K=\\{k\in\mathbb{N}\,|\,1\leq k\leq 2\mathfrak{n}+1\\}$ and we can
define the disjoint subsets $Y_{k}$, $k\in K$, of $Y$ inductively as follows:

Step 1. Let $Y_{1}$ be the maximal subset of $Y$ such that $tV^{3}Y_{1}\cap
V^{3}Y_{1}=\emptyset$. (Since $tV^{3}x\cap V^{3}x=\emptyset$ for all $x\in G$
by the choice of $V$, we can always pick such $Y_{1}$.)

Step 2. If the sets $Y_{1},\dotsc,Y_{k}$, for some $k\in K$ with
$k<2\mathfrak{n}+1$, have been chosen, then let $Y_{k+1}$ be the maximal
subset of $Y\setminus(\bigcup_{j=1}^{k}Y_{j})$ such that $tV^{3}Y_{k+1}\cap
V^{3}Y_{k+1}=\emptyset$; note that we might have $Y_{k}=\emptyset$ for some
$k\in K$ if and only if $Y=\bigcup_{j=1}^{k-1}Y_{j}$.

We now can assert that $Y=\bigcup_{k\in K}Y_{k}$. Suppose to the contrary that
there exists an element $y\in Y\setminus(\bigcup_{k\in K}Y_{k})$. Then
$Y_{k}\not=\emptyset$ for all $k\in K$. Let $k\in K$ be arbitrary. Since $y\in
Y\setminus Y_{k}$, it follows that

$tV^{3}(Y_{k}\cup\\{y\\})\cap V^{3}(Y_{k}\cup\\{y\\})\not=\emptyset.$

Then the choice of the sets $V$ and $Y_{k}$ implies that $tV^{3}Y_{k}\cap
V^{3}y\not=\emptyset$ or $tV^{3}y\cap V^{3}Y_{k}\not=\emptyset$ or both. Put

$K_{1}=\left\\{k\in K\,|\,tV^{3}Y_{k}\cap V^{3}y\not=\emptyset\right\\}\
\textrm{ and }\ K_{2}=\left\\{k\in K\,|\,tV^{3}y\cap
V^{3}Y_{k}\not=\emptyset\right\\}.$

Then $K=K_{1}\cup K_{2}$, and so either $|K_{1}|\geq\mathfrak{n}+1$ or
$|K_{2}|\geq\mathfrak{n}+1$. Assume first $|K_{1}|\geq\mathfrak{n}+1$. If
$k\in K_{1}$, then $Y_{k}\cap V^{3}t^{-1}V^{3}y\not=\emptyset$ so that
$Y_{k}\cap Ut^{-1}y\not=\emptyset$. Since the sets $Y_{k}$ are disjoint, the
set $Ut^{-1}y$ contains at least $\mathfrak{n}+1$ members of $Y$, a
contradiction to (5.6.2). Similarly, $|K_{2}|\geq\mathfrak{n}+1$ will lead to
a contradiction to (5.6.2). Thus have concluded our assertion.

$G=V^{2}Y$ because of the maximality of $Y$. Therefore $G=\bigcup_{k\in
K}V^{2}Y_{k}$ such that for all $k\in K$, $tV^{2}Y_{k}\cap
V(V^{2}Y_{k})\subseteq tV^{3}Y_{k}\cap V^{3}Y_{k}=\emptyset$. Now let
$n=2\mathfrak{n}+1$ and $P_{k}=V^{2}Y_{k}$. Then $G=P_{1}\cup\dotsm\cup P_{n}$
with $tP_{k}\cap VP_{k}=\emptyset$ for $1\leq k\leq n$. If
$P_{1},\dotsc,P_{n}$ are not pairwise disjoint, then by a standard surgery we
can obtain the disjointness. The proof is completed. ∎

######  5.6a Corollary.

Let $G$ be a discrete group and $t\in G$ with $t\not=e$. Then there exists a
finite partition $G=P_{1}\cup\dotsm\cup P_{n}$ such that $tP_{k}\cap
P_{k}\not=\emptyset$ for all $k=1,\dotsc,n$.

This corollary is comparable with the following well-known classical result:

######  5.6b Theorem (“three-sets” lemma [6, 22, 28]).

Let $G$ be a discrete group and $t\in G$ with $t\not=e$. Then there exists a
partition $G=P_{1}\cup P_{2}\cup P_{3}$ such that $tP_{k}\cap
P_{k}\not=\emptyset$ for all $k=1,2,3$.

######  5.7 Theorem (cf. Veech [31, Thm. 2.2.1] and [8, 28, 2] for $G$ to be
LC).

Let $t\in G$ with $t\not=e$. If there exists a symmetric set
$V\in\mathfrak{N}_{e}(G)$ such that

  1. 1.

$tV^{3}\cap V^{3}=\emptyset$, and

  2. 2.

$V^{3}tV^{3}t^{-1}\cup V^{3}t^{-1}V^{3}t$ can be covered by a finite number of
right translates of $V$.

Then the canonical action $\rho\colon G\times G^{\textsl{{LUC}}}\rightarrow
G^{\textsl{{LUC}}}$ is $t$-free; that is, $tx\not=x$ for all $x\in
G^{\textsl{{LUC}}}$.

Consequently, if $G$ is locally quasi-totally bounded, then $G$ acts freely on
$G^{\textsl{{LUC}}}$, and, the universal compact
$\\{\textrm{ambit}\\}\\{\textrm{minimal flow}\\}$ of $G$ is free ([14,
Question, p.64]).

###### Proof.

In view of Theorem 5.3 and Corollary 5.5.2, without loss of generality, we can
identity $G$ with $\iota[G]$; i.e., $G$ is a dense subset of
$G^{\textsl{{LUC}}}$. Let $t\in G$ with $t\not=e$ and $x\in
G^{\textsl{{LUC}}}$ be any given. Then by Lemma 5.6, there are
$V\in\mathfrak{N}_{e}(G)$ and a partition $G=P_{1}\cup\dotsm\cup P_{n}$ such
that $tP_{k}\cap VP_{k}=\emptyset$ for all $k=1,\dotsc,n$.

By
$G^{\textsl{{LUC}}}=\bigcup_{k=1}^{n}\textrm{cls}_{G^{\textsl{{LUC}}}}{P_{k}}$,
we can pick some $k$ such that $x\in\textrm{cls}_{G^{\textsl{{LUC}}}}{P_{k}}$.
Since $tP_{k}\cap VP_{k}=\emptyset$, by Lemma 5.5 there exists an
$f\in\texttt{LUC}(G)$ such that $f_{|P_{k}}\equiv 0$ and $f_{|tP_{k}}\equiv
1$. Then by 2.8.4-ii, $f$ can be uniquely extended to
$f^{\textsl{{LUC}}}=\hat{f}$ on $G^{\textsl{{LUC}}}$ to $[0,1]$. Moreover,
because $x\in\textrm{cls}_{G^{\textsl{{LUC}}}}{P_{k}}$, $\hat{f}(x)=0$ and
$tx\in
t\textrm{cls}_{G^{\textsl{{LUC}}}}{P_{k}}=\textrm{cls}_{G^{\textsl{{LUC}}}}{tP_{k}}$
so that $\hat{f}(tx)=1$. Therefore, $x\not=tx$ and
$\mathscr{T}^{\textsl{{LUC}}}$ is free. Then by Theorem 3.3, the universal
minimal compact flow of $G$ is also free. The proof is completed. ∎

######  5.7.1 Corollary (Veech’s theorem in locally totally bounded groups).

If $G$ is locally totally bounded, then the canonical action $\rho\colon
G\times G^{\textsl{{LUC}}}\rightarrow G^{\textsl{{LUC}}}$ is free.

######  5.7.2 Remark.

Since $G^{\textsl{{LUC}}}$ is a compact Hausdorff right-topological semigroup
and we may identify $G$ with a dense subgroup of $G^{\textsl{{LUC}}}$ (by Thm.
5.3), Theorem 5.7 says that if $G$ is locally quasi-totally bounded, then
$gx\not=x$ for all $g\in G\setminus\\{e\\}$ and all $x\in G^{\textsl{{LUC}}}$.

######  5.7.3 Remark.

However, even for $G$ to be LC, it is possible that $xg=x$ for some $g\in
G\setminus\\{e\\}$ and some $x\in G^{\textsl{{LUC}}}$. For instance, let
$g_{n}^{-1}\to y\in G^{\textsl{{LUC}}}$ in Remark 3.7. Then

$g_{n}^{-1}g\to yg\quad\textrm{and}\quad
g_{n}^{-1}g=(g_{n}^{-1}gg_{n})g_{n}^{-1}\to y$

so that $yg=g$, as desired.

######  5.8Another application of Theorem 3.1.

Let $(\mathscr{X},x_{0})$ be a compact ambit of $T$ with phase mapping
$(t,x)\mapsto tx$, where $T$ is a topological group. Given any $a\in T$ we can
define a flow

$\mathscr{X}^{a}\colon T\times X\rightarrow X,\quad(t,x)\mapsto
t\cdot_{a}x:=ata^{-1}x.$

Clearly $\mathscr{X}^{a}$ is also point-transitive with
$\overline{T\cdot_{a}x_{0}}=X$. Since $t\mapsto ata^{-1}$ is an inner-
automorphism of $T$, then $\mathscr{X}^{a}$ is minimal (resp. proximal,
distal) if so is $\mathscr{X}$. Given any $F\in\mathcal{C}(X)$, $x\in X$, and
$a\in T$, write $F^{x}(t|\mathscr{X})=F(tx)$ and
$F^{x}(t|\mathscr{X}^{a})=F(t\cdot_{a}x)$ for $t\in T$.

######  5.8.1.

Given $F\in\mathcal{C}(X)$ and $a\in T$, we have that

$\displaystyle F^{ax_{0}}(t|\mathscr{X}^{a})$
$\displaystyle=F(t\cdot_{a}ax_{0})$
$\displaystyle=F(ata^{-1}ax_{0})=F(atx_{0})$
$\displaystyle=F^{x_{0}}(at)=F^{x_{0}}a(t)=(Fa)^{x_{0}}(t|\mathscr{X})$

and

$\displaystyle F^{x_{0}}(t|\mathscr{X})$ $\displaystyle=F(tx_{0})$
$\displaystyle=(Fa^{-1})(atx_{0})=(Fa^{-1})(ata^{-1}ax_{0})$
$\displaystyle=(Fa^{-1})(t\cdot_{a}ax_{0})$
$\displaystyle=(Fa^{-1})^{ax_{0}}(t|\mathscr{X}^{a})$

Noting $Fa,Fa^{-1}\in\mathcal{C}(X)$, we can conclude that

(5.8.2) $\mathbb{F}(\mathscr{X},x_{0})=\mathbb{F}(\mathscr{X}^{a},ax_{0}).$

Then by Theorem 3.1 and (5.8.2), we could obtain the following:

######  5.8.3 Proposition (cf. [17, 11] for $\mathscr{X}$ the universal
minimal (proximal) flows of $T$).

Given $a\in T$, there exists an isomorphism
$\lambda_{a}\colon\mathscr{X}\rightarrow\mathscr{X}^{a}$ with
$\lambda_{a}(x_{0})=ax_{0}$. Hence $\lambda_{a}(x)=ax$ for all $x\in X$.

##  6 Almost automorphic functions and a.a. flows

In this section we will present an alternative complete simple proofs for
Veech’s Structure Theorems of a.a. flows and a.a. functions (Thm. 6.2.6 and
Thm. 6.3.11). Moreover, we shall prove that every endomorphism of Veech’s flow
of a.a. function is almost 1-1 (Thm. 6.3.12). Theorem 6.4.6 will give us an
equivalent description of Bochner a.a. functions on a LC group.

######  6.0 Standing hypotheses.

Unless otherwise specified:

  1. 1.

Let $G$ be a Hausdorff left-topological group and by $G_{\\!d}$ we denote the
discretization of $G$.

  2. 2.

Let $l^{\infty}(G)$ be the set of bounded complex-valued functions, continuous
or not, on $G$.

  3. 3.

If $G$ is LC, $\beta G$ stands for the Stone-Čech compactification of $G$ on
which there exists a compact Hausdorff right-topological semigroup structure
so that $G$ is an open dense subsemigroup of $\beta G$.

###  6.1 Almost automorphic functions

We shall introduce the fundamental theory of a.a. functions on the Hausdorff
left-topological group $G$ in $\S$6.1.

######  6.1.1a.a. functions on $G$.

A complex-valued function $f$ on $G$ is said to be left almost automorphic
(l.a.a.), denoted $f\in\verb"A"_{\textrm{c,L}}(G)$, if every net $\\{t_{n}\\}$
in $G$ has a subnet $\\{t_{n^{\prime}}\\}$ and some $\phi\in\mathcal{C}(G)$
such that both

$t_{n^{\prime}}f\to_{\textrm{p}}\phi\quad\textrm{and}\quad
t_{n^{\prime}}^{-1}\phi\to_{\textrm{p}}f.$

We could define r.a.a. and $\verb"A"_{\textrm{c,R}}(G)$ analogously.

Clearly,
$\verb"A"_{\textrm{c,L}}(G)\cup\verb"A"_{\textrm{c,R}}(G)\subseteq\mathcal{C}(G)$
by considering a special net $\\{t_{n}\\}=\\{e\\}$ so that
$f=\phi\in\mathcal{C}(G)$.

Write
$\texttt{A}_{\textrm{c}}(G)=\texttt{A}_{\textrm{c,L}}(G)\cap\texttt{A}_{\textrm{c,R}}(G)$.
If $f\in\verb"A"_{\textrm{c}}(G)$, then $f$ is called an (Bochner) a.a.
function on $G$.

######  Note.

An a.a. function in our sense above was called a “continuous” a.a. function by
Veech 1965 [30, Def. 4.1.1]. This kind of a.a. functions was studied in Veech
[30] only for $G$ to be an abelian, LC, $\sigma$-compact, and first countable
Hausdorff topological group.

######  6.1.2cf. [30, Thm. 1.2.1-iv].

If $f\in\verb"A"_{\textrm{c,L}}(G)$, then $f\in\mathcal{C}_{b}(G)$. For
otherwise, we can find a net $\\{t_{n}\\}$ in $G$ with $|f(t_{n})|\to\infty$.
So there is no subnet $\\{t_{n^{\prime}}\\}$ of $\\{t_{n}\\}$ such that
$t_{n^{\prime}}f\to_{\textrm{p}}\phi$ for some $\phi\in\mathcal{C}(G)$,
contrary to Def. 6.1.1a.

######  6.1.3.

Let $\\{t_{n}\\}$ be any net in $G$. We can select a subnet
$\\{t_{n^{\prime}}\\}$ of $\\{t_{n}\\}$ such that $t_{n^{\prime}}\to p$ and
$t_{n^{\prime}}^{-1}\to q$ in $\beta G_{\\!d}$ or in $\beta G$ if $G$ is a LC
topological group. So $st_{n^{\prime}}\to sp$ and $st_{n^{\prime}}^{-1}\to sq$
for all $s\in G$. Now for $f\in\mathcal{C}_{b}(G)$, let $f^{\beta}\colon\beta
G_{\\!d}\rightarrow\mathbb{C}$ or $f^{\beta}\colon\beta
G\rightarrow\mathbb{C}$ be the canonical extension of $f$.

Thus, we have for each $f\in\verb"A"_{\textrm{c,L}}(G)$ and each $s\in G$ that

$t_{n^{\prime}}f(s)\to\phi(s)=f^{\beta}(sp)\ \textrm{ and }\
t_{n^{\prime}}^{-1}(pf^{\beta})(s)=f^{\beta}(st_{n^{\prime}}^{-1}p)\to
f(s)=f^{\beta}(sqp).$

That is, $qpf^{\beta}(s)=f(s)$ for all $s\in G$.

If $G$ is a LC topological group and $t_{n^{\prime}}f\to_{\textrm{p}}\phi$,
then $\phi\in\mathcal{C}_{b}(G)$ since $\phi(t)=f^{\beta}(tp)$ for all $t\in
G$.

######  Note.

Given $f\in\mathcal{C}_{b}(G)$, $f^{\beta}$ is itself independent of the
right-topological semigroup structure $\cdot_{\\!R}$ on $\beta G$, since so is
$pt$ and $tp$, for $p\in\beta G$ and $t\in G$. However, $f^{\beta}(spqt)$
depends generally upon this structure for $p,q\in\beta G\setminus G$ and
$s,t\in G$.

######  6.1.4 Theorem (cf. [30, Thm. 1.2.1, Thm. 1.3.1] for $G$ a discrete
group).

The following properties are enjoyed by $\verb"A"_{\textrm{c,L}}(G)$:

  1. 1.

If $f,f_{1},f_{2}\in\verb"A"_{\textrm{c,L}}(G)$ and $\lambda\in\mathbb{C}$,
then $\lambda f,\bar{f},f_{1}+f_{2}$,
$f_{1}f_{2},\mathbf{1}\in\verb"A"_{\textrm{c,L}}(G)$.

  2. 2.

If $f\in\verb"A"_{\textrm{c,L}}(G)$, then
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\mathcal{C}_{b}(G)$ and
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a compact dynamic.

  3. 3.

$\verb"A"_{\textrm{c,L}}(G)$ is (left and right) invariant; i.e., if
$f\in\verb"A"_{\textrm{c,L}}(G)$ and $s,t\in G$, then
$sft\in\verb"A"_{\textrm{c,L}}(G)$.

  4. 4.

$\verb"A"_{\textrm{c,L}}(G)$ is norm closed such that
$\verb"A"_{\textrm{c,L}}(G)\subseteq\verb"A"_{\textrm{c,L}}(G_{\\!d})=\verb"A"_{\textrm{c,R}}(G_{\\!d})$.

  5. 5.

If $f\in\verb"LUC"(G)$, then $f\in\verb"A"_{\textrm{c,L}}(G)$ iff
$f\in\verb"A"_{\textrm{c,L}}(G_{\\!d})$ iff
$f\in\verb"A"_{\textrm{c,R}}(G_{\\!d})$. (Thus, if $G$ is a topological group
and $f\in\verb"UC"(G)$, then $f\in\verb"A"_{\textrm{c,L}}(G)$ iff
$f\in\verb"A"_{\textrm{c,R}}(G)$.)

  6. 6.

If $G$ is a LC topological group, then

$\verb"A"_{\textrm{c}}(G_{\\!d})\cap\mathcal{C}(G)=\verb"A"_{\textrm{c,L}}(G)=\verb"A"_{\textrm{c,R}}(G)=\verb"A"_{\textrm{c}}(G)\subseteq\verb"UC"(G)$

(cf. [30, Lem. 4.1.1] for $G$ to be abelian, LC, $\sigma$-compact, and first
countable).

Hence $\verb"A"_{\textrm{c,L}}(G)$ is a norm closed, right $G$-invariant, and
self-adjoint subalgebra of $\mathcal{C}_{b}(G)$ containing the constants.

###### Proof.

1.: Obvious.

2.: First, $f\in\mathcal{C}_{b}(G)$ by 6.1.2. Now let
$\phi\in\verb"H"_{\textrm{p}}^{L}[f]$. Then there is a net $t_{n}\in G$ such
that $t_{n}f\to_{\textrm{p}}\phi$. By Def. 6.1.1, there is a subnet
$\\{t_{n^{\prime}}\\}$ of $\\{t_{n}\\}$ such that
$t_{n^{\prime}}f\to_{\textrm{p}}\phi\in\mathcal{C}_{b}(G)$. Thus,
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\mathcal{C}_{b}(G)$. On the other hand,
since $t_{n}f(xt)\to\phi(xt)=t\phi(x)$ for all $x,t\in G$, hence
$t\phi\in\verb"H"_{\textrm{p}}^{L}[f]$. Thus, $\rho\colon
G\times\verb"H"_{\textrm{p}}^{L}[f]\rightarrow\verb"H"_{\textrm{p}}^{L}[f]$,
$(t,\phi)\mapsto t\phi$, is a well-defined separately continuous phase
mapping. That is, $G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a compact
dynamic.

If $G$ is a LC topological group, then by Ellis’s Joint Continuity Theorem,
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a compact flow. So
$f\in\texttt{LUC}(G)$ by 2.4.1, since $f$ may come from the compact ambit
$(\mathscr{H}_{\textrm{p}}^{L}[f],f)$ of $G$.

3.: Obvious.

4.: Let $f_{i}\in\verb"A"_{\textrm{c,L}}(G)$ be a sequence with
$f_{i}\to_{\|\cdot\|_{\infty}}f\in\mathcal{C}_{b}(G)$ as $i\to\infty$. Let
$\\{t_{n}\\}$ be any net in $G$ with a subnet $\\{t_{n^{\prime}}\\}$ as in
6.1.3. Then, as $i\to\infty$,

$|qpf_{i}^{\beta}(x)-qpf^{\beta}(x)|=|(f_{i}-f)^{\beta}(xqp)|\leq\|f_{i}-f\|_{\infty}\to
0$

uniformly for $x\in G$. Thus, by the triangle inequality and
$qpf_{i}^{\beta}=f_{i}$ on $G$, we have as $i\to\infty$ that

$|qpf^{\beta}(x)-f(x)|\leq|qpf^{\beta}(x)-qpf_{i}^{\beta}(x)|+|f_{i}(x)-f(x)|\to
0$

uniformly for $x\in G$. Moreover, since

$|f_{i}^{\beta}(xp)-f^{\beta}(xp)|=|(f_{i}-f)^{\beta}(xp)|\leq\|f_{i}-f\|_{\infty}\to
0$

uniformly for $x\in G$, we have that $pf^{\beta}$ is continuous restricted to
$G$. Thus, $f\in\verb"A"_{\textrm{c,L}}(G)$ by Def. 6.1.1a.

$\verb"A"_{\textrm{c,L}}(G)\subseteq\verb"A"_{\textrm{c,L}}(G_{\\!d})$ is
obvious.

$\verb"A"_{\textrm{c,L}}(G_{\\!d})=\verb"A"_{\textrm{c,R}}(G_{\\!d})$ is
exactly [30, Thm. 1.3.1]. We will, however, present an alternative simple
proof here. Let $f\in\verb"A"_{\textrm{c,L}}(G_{\\!d})$ then it is enough to
prove that $f\in\verb"A"_{\textrm{c,R}}(G_{\\!d})$. Suppose there exists a net
$\\{t_{n}\\}$ in $G$ and $\phi,\psi\in l^{\infty}(G)$ such that
$ft_{n}\to_{\textrm{p}}\phi$ and $\phi t_{n}^{-1}\to_{\textrm{p}}\psi$. To
prove $f=\psi$, we may assume $t_{n}\to p$ and $t_{n}^{-1}\to q$ in $\beta
G_{\\!d}$. Then, for all $x\in G$,

$\psi(x)={\lim}_{n}\phi
t_{n}^{-1}(x)={\lim}_{n}(t_{n}^{-1}x)\phi(e)=\phi^{\beta}(qx)$

and

$\phi(x)={\lim}_{n}ft_{n}(x)={\lim}_{n}(t_{n}x)f(e)=p(xf)(e)=f^{\beta}p(x).$

So, $\phi^{\beta}=f^{\beta}p$ and then $\psi(x)=pqxf(e)$. Since
$xf\in\verb"A"_{\textrm{c,L}}(G)$, we have that $pqxf=xf$ by 6.1.3 and
$\psi=f$.

5.: By 4. and Lemma 4.2.

6.: By 4. and 6.1.3b, it follows that
$\verb"A"_{\textrm{c,L}}(G)=\verb"A"_{\textrm{c}}(G_{\\!d})\cap\mathcal{C}(G)=\verb"A"_{\textrm{c,R}}(G)$.
Finally by 2. and Ellis’s Joint Continuity Theorem,
$G\curvearrowright\texttt{H}_{\textrm{p}}^{L}[f]$ is a compact flow so that
$f\in\texttt{LUC}(G)$ by 2.4.1. Thus,
$\verb"A"_{\textrm{c,R}}(G)=\verb"A"_{\textrm{c}}(G)\subseteq\verb"UC"(G)$.

The proof of Theorem 6.1.4 is therefore completed. ∎

######  Corollary.

Let $f=g+ih\in\mathcal{C}(G)$, where $g,h$ are continuous real functions on
$G$. Then $f\in\verb"A"_{\textrm{c,L}}(G)$ iff
$g,h\in\verb"A"_{\textrm{c,L}}(G)$.

######  6.1.5 Lemma.

If $f\in l^{\infty}(G)$ is a.a. on $G_{\\!d}$, then for every finite set
$\\{t_{1},\dotsc,t_{k}\\}$ in $G$, $(t_{1}f,\dotsc,t_{k}f)$ is an a.p. point
for $G_{\\!d}\curvearrowright X$, where

$X=\stackrel{{\scriptstyle
k\textrm{-times}}}{{\overbrace{l^{\infty}(G)\times\dotsm\times
l^{\infty}(G)}}}$

is under the topology of pointwise convergence. Thus, for all
$f\in\verb"A"_{\textrm{c,L}}(G)$,
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a minimal compact dynamic.

###### Proof.

By Theorem 6.1.4, $\overline{G(t_{1}f,\dotsc,t_{k}f)}^{\textrm{p}}$ is a
compact set in $X$ so that by Zorn’s Lemma we can select an a.p. point, say
$(\phi_{1},\dotsc,\phi_{k})$, in it for $G_{\\!d}\curvearrowright X$. Then
there is a net $\\{t_{n}\\}$ in $G$ with
$t_{n}(t_{1}f,\dotsc,t_{k}f)\to_{\textrm{p}}(\phi_{1},\dotsc,\phi_{k})$. Since
$t_{1}f,\dotsc,t_{k}f$ all are l.a.a. on $G_{\\!d}$ by Theorem 6.1.4, there is
a subnet $\\{t_{n^{\prime}}\\}$ of $\\{t_{n}\\}$ such that
$t_{n^{\prime}}^{-1}(\phi_{1},\dotsc,\phi_{k})\to_{\textrm{p}}(t_{1}f,\dotsc,t_{k}f)$.
Thus, $(t_{1}f,\dotsc,t_{k}f)$ is also a.p. under $G_{\\!d}\curvearrowright
X$. The proof is completed. ∎

In fact, it is known that a l.a.a. function is a “point-distal function” as
follows. See [30] by Structure Theorem of a.a. flows and [16, 10] by
$\Delta^{\\!*}$-recurrence. Here we will present a simple short proof.

######  6.1.6 Lemma.

Let $f\in\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$. Then
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a minimal compact flow and
$(f,\phi)$ is a.p. under
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]\times\verb"H"_{\textrm{p}}^{L}[f]$
for all $\phi\in\verb"H"_{\textrm{p}}^{L}[f]$.

###### Proof.

First by Lemma 4.2, $G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a
compact flow. Let $\phi\in\verb"H"_{\textrm{p}}^{L}[f]$. By Lemmas 6.1.5,
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a minimal compact flow. Let
$\mathbb{I}$ be a minimal left ideal in $\beta G_{\\!d}$. Then
$\verb"H"_{\textrm{p}}^{L}[f]=\mathbb{I}f$. Let $q\in\mathbb{I}$ with
$\phi=qf$; then we can choose a net $t_{n}\in G$ with $t_{n}^{-1}\to q$ and
$t_{n}\to p$ in $\beta G_{\\!d}$. So

$(f,qf)=(qpf,qf)=q(pf,f)\in\verb"H"_{\textrm{p}}^{L}[f]\times\verb"H"_{\textrm{p}}^{L}[f]$

is a.p. for
$G_{\\!d}\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]\times\verb"H"_{\textrm{p}}^{L}[f]$
and then for
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]\times\verb"H"_{\textrm{p}}^{L}[f]$.
The proof is completed. ∎

######  6.1.7.

Recall that a set $A$ in $G$ is (left-)syndetic [18] if $G=K^{-1}A$, or
equivalently, $Kt\cap A\not=\emptyset$ $\forall t\in G$, for some compact set
$K$ in $G$. If there is a finite set $L\subset G$ with $G=LA=AL$, then $A$ is
called relatively dense in $G$ (cf. [30]). Then $A$ is relatively dense in $G$
iff it is left-syndetic and right-syndetic in $G_{\\!d}$.

Given any $f\in l^{\infty}(G)$, a finite nonempty subset $N$ of $G$ and a real
number $\epsilon>0$, put

$None$ $U(f;N,\epsilon)=\\{\phi\in l^{\infty}(G)\,|\,{\max}_{t\in
N}|\phi(t)-f(t)|<\epsilon\\}$

that is an open neighborhood of $f$ in $l^{\infty}(G)$ under the pointwise
topology; and

$None$ $C(f;N,\epsilon)=\\{\tau\in G\,|\,{\max}_{s,t\in N}|f(s\tau
t)-f(st)|<\epsilon\\}.$

Then,

$None$ $C(f;N,\epsilon)=\\{\tau\in G\,|\,fs\tau\in U(fs;N,\epsilon)\ \forall
s\in N\\}$

and

$None$ $C(f;N,\epsilon)=\\{\tau\in G\,|\,\tau tf\in U(tf;N,\epsilon)\ \forall
t\in N\\}.$

######  6.1.8 Lemma.

Let $f\in\verb"A"_{\textrm{c,L}}(G)$. Given a finite set $N\subset G$ and a
real number $\epsilon>0$, there exists a finite superset $M$ of $N$ and a
positive real number $\delta<\epsilon$ such that

$N_{G}(f,U(f;M,\delta))^{-1}\cup N_{G}(f,U(f;M,\delta))\subseteq
N_{G}(f,U(f;N,\epsilon))$

where $N_{G}(f,U)$ is as in Def. 1.2f under $G\curvearrowright l^{\infty}(G)$.

###### Proof.

Otherwise, we can select a net $\\{t_{n}\\}$ in $G$ such that
$t_{n}f\to_{\textrm{p}}f$ but $t_{n}^{-1}f\notin U(f;N,\epsilon)$. Thus,
$t_{n}f\to_{\textrm{p}}f$ and $t_{n}^{-1}f\not\to_{\textrm{p}}f$, contrary to
$f\in\verb"A"_{\textrm{c,L}}(G_{\\!d})$. The proof is complete. ∎

######  6.1.9 Lemma.

Let $f\in l^{\infty}(G)$ be a.a. on $G_{\\!d}$. If $N$ is a finite subset of
$G$ and $\epsilon>0$, then $C(f;N,\epsilon)$ contains a symmetric syndetic set
in $G_{\\!d}$.

###### Proof.

By Lemma 6.1.5, Lemma 6.1.8, and (6.1.7d). ∎

######  Corollary (cf. [30, Lem. 2.1.1] by a different proof).

Let $f\in l^{\infty}(G)$ be a.a. on $G_{\\!d}$. If $N$ is a finite subset of
$G$ and $\epsilon>0$, then $C(f;N,\epsilon)$ is relatively dense in $G$.

###### Proof.

By Lemma 6.1.9 and the fact that a syndetic symmetric set in $G_{\\!d}$ is
relatively dense. ∎

######  6.1.10 Lemma (cf. [30, Lem. 2.1.2] or [10, Thm. 1.9]).

Let $f\in l^{\infty}(G)$ be a.a. on $G_{\\!d}$. Given $\epsilon>0$ and a
finite set $N\subset G$ there exist $\delta>0$ and a finite superset $M$ of
$N$ such that $\sigma^{-1}\tau\in C(f;N,\epsilon)$ for all $\sigma,\tau\in
C(f;M,\delta)$.

###### Proof.

See Appendix A. ∎

In fact, we shall show in $\S$6.4 that the converse of Lemma 6.1.10 is also
true; see Theorem 6.4.6 in terms of the so-called Bohr a.a. functions.

###  6.2 Almost automorphic flows

######  6.2.1.

Let $\mathscr{X}=(G,X)$ be a compact flow and by $\mathscr{U}_{X}$ we denote
the uniformity structure of $X$.

$\mathscr{X}$ is called equicontinuous if given
$\varepsilon\in\mathscr{U}_{X}$ there exists $\delta\in\mathscr{U}_{X}$ such
that $G\delta\subseteq\varepsilon$. Later $\mathscr{X}_{\\!eq}$ will stand for
the maximal a.p. factor of $\mathscr{X}$ (see [14, 17, 4, 12] for the
existence of $\mathscr{X}_{\\!eq}$).

Let

$\begin{split}&\texttt{P}(\mathscr{X})=\\{(x,y)\,|\,\overline{G(x,y)}\cap\Delta_{X}\not=\emptyset\\},\\\
&\texttt{RP}(\mathscr{X})=\\{(x,y)\,|\,\overline{G(\varepsilon[x]\times\varepsilon[y])}\cap\Delta_{X}\not=\emptyset\
\forall\varepsilon\in\mathscr{U}_{X}\\}.\end{split}$

be the proximal relation and regionally proximal relation of $\mathscr{X}$,
respectively.

Then $\mathscr{X}$ is equicontinuous iff
$\texttt{RP}(\mathscr{X})=\Delta_{X}$. If $\overline{Gx}=X$, then $x$ is a
distal point for $\mathscr{X}$ iff $\texttt{P}[x]=\\{x\\}$.

A point $x\in X$ is said to be a.a. under $\mathscr{X}$, denoted
$x\in\verb"A"(\mathscr{X})$, if for every net $\\{t_{n}\\}$ in $G$, $t_{n}x\to
y$ implies that $t_{n}^{-1}y\to x$. If $\mathscr{X}$ has an a.a. point that
has a dense orbit in $X$, then $\mathscr{X}$ is called an a.a. flow. See [30,
10].

Then by Def. 6.1.1a, $G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is an
a.a. flow with $f$ an a.a. point whenever
$f\in\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$ or
$f\in\verb"A"_{\textrm{c,L}}(G)$ with $G$ a LC topological group.

######  6.2.2 Lemma.

Let $\mathscr{X}=G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$, where
$f\in\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$. Then
$\verb"P"(\mathscr{X})=\verb"RP"(\mathscr{X})$; and moreover,
$\verb"P"[f]=\\{f\\}$.

###### Proof.

First by Lemma 6.1.6, we have $\texttt{P}[f]=\\{f\\}$ under $\mathscr{X}$.
Moreover, it is clear that
$\texttt{P}(\mathscr{X})\subseteq\texttt{RP}(\mathscr{X})$. For the other
inclusion, let $(x,y)\in\texttt{RP}(\mathscr{X})$. By Lemma 6.1.5, we can
choose nets $s_{n},s_{n}^{\prime},t_{n}\in G$ such that
$s_{n}f\to_{\textrm{p}}x$, $s_{n}^{\prime}f\to_{\textrm{p}}y$,
$t_{n}(s_{n}f,s_{n}^{\prime}f)\to_{\textrm{p}}(f,f)$.

Given $N\subset G$ a finite set and $\epsilon>0$, it follows from Lemmas
6.1.10 and 6.1.9 that there exists a syndetic set $B$ in $G_{\\!d}$ with
$BB\subseteq N_{G}(f,U(f;N,\epsilon))$. Then by the “two-syndetic sets”
criterion ([26, Thm. 2.3]), there exists a finite superset $M$ of $N$, a
positive real number $\delta<\epsilon$, and a syndetic set $A$ in $G_{\\!d}$
such that $AU(f;M,\delta)\subseteq U(f;N,\epsilon)$. Furthermore, we can write
$ka_{n}=t_{n}^{-1}$ with $a_{n}\in A$ and $k\in G$. Then

$a_{n}(t_{n}s_{n}f)\to_{\textrm{p}}\phi\in\overline{U(f;N,\epsilon)}^{\textrm{p}},\quad
s_{n}f\to_{\textrm{p}}x=k\phi$

and

$a_{n}(t_{n}s_{n}^{\prime}f)\to_{\textrm{p}}\phi^{\prime}\in\overline{U(f;N,\epsilon)}^{\textrm{p}},\quad
s_{n}^{\prime}f\to_{\textrm{p}}y=k\phi^{\prime}.$

Thus, $(x,y)\in\texttt{P}(\mathscr{X})$ and
$\texttt{P}(\mathscr{X})=\texttt{RP}(\mathscr{X})$. The proof is complete. ∎

######  6.2.3.

In fact, if $\mathscr{X}$ is any a.a. flow, then
$\texttt{P}(\mathscr{X})=\texttt{RP}(\mathscr{X})$, and moreover,
$\texttt{P}[x]=\\{x\\}$ iff $x\in\texttt{A}(\mathscr{X})$ (cf. [10] by
improving Lem. 6.1.10 or [26] by using Veech’s Structure Theorem of a.a.
flow—Thm. 6.2.6 below).

######  6.2.4 Lemma.

Let $\mathscr{X}$ be a compact flow and $x_{0}\in X$. Then $x_{0}$ is a.a.
under $\mathscr{X}$ iff
$\mathbb{F}(\mathscr{X},x_{0})\subseteq\verb"A"_{\textrm{c,L}}(G)$.

###### Proof.

Necessity is obvious. For sufficiency, suppose to the contrary that $x_{0}$ is
not a.a. for $\mathscr{X}$. Then we can choose a net $\\{t_{n}\\}$ in $G$ with
$t_{n}x_{0}\to y$ and $t_{n}^{-1}y\to x_{1}\not=x_{0}$. Next, we can select
$F\in\mathcal{C}(X)$ with $F(x_{0})\not=F(x_{1})$. So
$F^{x_{0}}\in\mathbb{F}(\mathscr{X},x_{0})$, $F^{x_{0}}\not=F^{x_{1}}$. Then
$t_{n}F^{x_{0}}\to_{\textrm{p}}F^{y}$ and
$t_{n}^{-1}F^{y}\to_{\textrm{p}}F^{x_{1}}\not=F^{x_{0}}$, contrary to
$F^{x_{0}}\in\verb"A"_{\textrm{c,L}}(G)$. The proof is completed. ∎

The following important result is comparable with Veech’s Structure Theorem of
a.a. functions—Thm. 6.3.11.

######  6.2.5 Lemma.

Let $f\in\verb"LUC"(G)$ or let $f\in\mathcal{C}_{b}(G)$ with $G$ a LC
topological group. Then $f\in\verb"A"_{\textrm{c,L}}(G)$ iff
$\mathscr{H}_{\textrm{p}}^{L}[f]$ is an almost 1-1 extension of
$\mathscr{H}_{\textrm{p}}^{L}[f]_{eq}$, where
$\mathscr{H}_{\textrm{p}}^{L}[f]_{eq}=G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]_{eq}$
and
$\rho\colon\verb"H"_{\textrm{p}}^{L}[f]\rightarrow\verb"H"_{\textrm{p}}^{L}[f]_{eq}$
is 1-1 at $f$.

######  Note (cf. [30, Thm. 3.3.1] for $G$ a discrete group).

So, $f\in\mathcal{C}_{b}(G)$ is left a.p. (von Neumann 1934; i.e., $Gf$ is
relatively compact in $(\mathcal{C}_{b}(G),\|\cdot\|_{\infty})$) iff
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\verb"A"_{\textrm{c,L}}(G)$.

###### Proof.

Sufficiency: By Lemma 6.2.4 and
$f\in\mathbb{F}(\mathscr{H}_{\textrm{p}}^{L}[f],f)$.

Necessity: Assume $f\in\verb"A"_{\textrm{c,L}}(G)$. Since $f\in\verb"LUC"(G)$,
it follows by Lemmas 4.2 and 6.1.5 that
$\mathscr{H}_{\textrm{p}}^{L}[f]=G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$
is a minimal compact flow with $f$ an a.a. point. By Lemma 6.2.2,
$\verb"P"(\mathscr{H}_{\textrm{p}}^{L}[f])=\verb"RP"(\mathscr{H}_{\textrm{p}}^{L}[f])$
is an invariant closed equivalence relation so that
$Y=\verb"H"_{\textrm{p}}^{L}[f]/\verb"RP"(\mathscr{H}_{\textrm{p}}^{L}[f])$ is
the maximal a.p. factor of $\mathscr{H}_{\textrm{p}}^{L}[f]$. Let
$\rho\colon\verb"H"_{\textrm{p}}^{L}[f]\rightarrow Y$ be the canonical
mapping. Then by Lemma 6.2.2 again, it follows that $\rho$ is 1-1 at the point
$f$. The proof is completed. ∎

###### Proof of Note.

If $f\in\mathcal{C}_{b}(G)$ is left a.p., then
$\texttt{H}_{\textrm{p}}^{L}[f]=\overline{Gf}^{\|\cdot\|_{\infty}}$ and
$\mathscr{H}_{\textrm{p}}^{L}[f]$ is an equicontinuous compact flow so
$f\in\texttt{LUC}(G)$ and
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\verb"A"_{\textrm{c,L}}(G)$. Now
conversely, suppose
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\verb"A"_{\textrm{c,L}}(G)$. Then
$\verb"H"_{\textrm{p}}^{L}[f]\subseteq\verb"A"_{\textrm{c,L}}(G_{\\!d})$.
Thus, for $G_{\\!d}$,
$\rho\colon\verb"H"_{\textrm{p}}^{L}[f]\rightarrow\verb"H"_{\textrm{p}}^{L}[f]_{eq}$
is 1-1 onto so that $G_{\\!d}\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is
equicontinuous. Thus, $G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is an
equicontinuous compact flow. So the pointwise topology coincides with the
$\|\cdot\|_{\infty}$-topology on $\texttt{H}_{\textrm{p}}^{L}[f]$ and $f$ is
left a.p. of von Neumann. The proof is completed. ∎

######  6.2.6 Theorem (Structure Theorem of a.a. flow).

A minimal compact flow $\mathscr{X}=G\curvearrowright X$ is a.a. iff the
canonical mapping $\rho\colon\mathscr{X}\rightarrow\mathscr{X}_{\\!eq}$ is
almost 1-1 (and, in fact, $\verb"A"(\mathscr{X})=\\{x\in
X\,|\,\\{x\\}=\rho^{-1}[\rho(x)]\\}$).

Thus, if $\mathscr{X}$ is a minimal pointwise a.a. compact flow, then it is
equicontinuous.

######  Note.

In [30] Veech only proved his structure theorem of a.a. function that is
essentially different with Lemma 6.2.5 (see Thm. 6.3.11 and Rem. 6.3.13
below). This important theorem was first claimed by Veech 1965 in [30, p.741]
and then 1977 in [31, Thm. 2.1.2] without a detailed proof. See Auslander-
Markley 2009 [5, Thm. 19] for a detailed proof in the special case that $G$ is
an abelian group and $X$ is a compact metric space. A complete proof appears
in Dai 2020 [10, Thm. 3.6] by using a variation [10, Thm. 1.9] of Lemma
6.1.10. Here we shall give an alternative self-closed simple proof based on
Lemma 6.2.5.

###### Proof (without using 6.2.3).

Sufficiency is obvious. Now conversely, suppose $\mathscr{X}$ is an a.a. flow
with $x_{0}\in\texttt{A}(\mathscr{X})$.

Given $F\in\mathcal{C}(X)$, let
$\rho_{F}\colon\verb"H"_{\textrm{p}}^{L}[F^{x_{0}}]\rightarrow
Y_{F}=\texttt{H}_{\textrm{p}}^{L}[F^{x_{0}}]_{eq}$ be the almost 1-1 extension
given by Lemma 6.2.5 for $f=F^{x_{0}}$. Let $\varrho_{F}\colon
X\rightarrow\verb"H"_{\textrm{p}}^{L}[f]$ be defined by $x\mapsto F^{x}$. Then
$\varrho_{F}$ is an extension of minimal flows by Lemma 2.4.3. Set

$y_{0,F}=\rho_{F}(F^{x_{0}}),\
y_{0}=(y_{0,F})_{F\in\mathcal{C}(X)}\in{\prod}_{F\in\mathcal{C}(X)}Y_{F},\
Y=\overline{Gy_{0}}.$

Define

$\rho\colon
X\xrightarrow[\varrho]{x\mapsto(F^{x})_{F\in\mathcal{C}(X)}}\prod_{F\in\mathcal{C}(X)}\verb"H"_{\textrm{p}}^{L}[F^{x_{0}}]\xrightarrow[\rho^{\prime}]{(F^{x})_{F\in\mathcal{C}(X)}\mapsto(\rho_{F}(F^{x}))_{F\in\mathcal{C}(X)}}\prod_{F\in\mathcal{C}(X)}Y_{F}.$

Clearly, $G\curvearrowright Y$ is a minimal a.p. compact flow and $\rho\colon
X\rightarrow Y$ is an extension of flows with $y_{0}=\rho(x_{0})$.

Next we need prove that $\rho$ is almost 1-1 at the point $x_{0}$. Indeed, for
$x\not=x_{0}$ in $X$, we can choose $\eta\in\mathcal{C}(X)$ with
$\eta(x_{0})\not=\eta(x)$ so that

$\varrho(x_{0})=(F^{x_{0}})_{F\in\mathcal{C}(X)}\not=(F^{x})_{F\in\mathcal{C}(X)}=\varrho(x).$

Moreover, since $\rho^{\prime}$ is 1-1 at the point
$(F^{x_{0}})_{F\in\mathcal{C}(X)}$, $\rho$ is 1-1 at the point $x_{0}$. Thus,
$\mathscr{X}$ is an almost 1-1 extension of $\mathscr{Y}$. This implies that
$\mathscr{X}_{\\!eq}\cong\mathscr{Y}$. The proof is completed. ∎

###### Proof of 6.2.3.

Let $x_{0}\in\texttt{A}(\mathscr{X})$. Then by Theorem 6.2.6, $\mathscr{X}$ is
locally a.p. at $x_{0}$ so $\texttt{P}(\mathscr{X})=\texttt{RP}(\mathscr{X})$
by [14, Prop. 5.27]. ∎

######  6.2.7Universal a.a. flows of $G$.

By Theorem 6.1.4 and 2.2.4, it follows that
$\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$ is a norm closed self-adjoint
right-invariant subalgebra of $\mathcal{C}_{b}(G)$ containing the constants.

Put $\verb"M"_{a}=|\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)|$, which is the
maximal ideal space of $\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$ as in
2.9. Write $\mathscr{M}_{\\!a}$ for $G\curvearrowright\verb"M"_{a}$, which
will be called the canonical a.a. flow of $G$.

Let $\mathscr{M}_{\\!eq}=G\curvearrowright\verb"M"_{eq}$ be the universal
minimal a.p./equicontinuous compact flow of $G$. Although $G$ need not be a
topological group under 6.0.1, it is easily to verify the existence of
$\mathscr{M}_{\\!eq}$.

In the category of minimal compact flows of $G$, $\mathscr{X}$ is called a
maximal almost 1-1 extension of $\mathscr{M}_{\\!eq}$ if $\mathscr{X}$ is an
almost 1-1 extension of $\mathscr{M}_{\\!eq}$ such that any other almost 1-1
extension of $\mathscr{M}_{\\!eq}$ must be a factor of $\mathscr{X}$.

######  6.2.7B Lemma.

$\mathscr{M}_{\\!a}$ is an a.a. flow with $e^{*}\in\verb"A"(\mathscr{M}_{a})$.

###### Proof.

First by 2.8.2, $(\mathscr{M}_{\\!a},e^{*})$ is a compact ambit of $G$. It
follows by (2.9a) that
$\mathbb{F}(\mathscr{M}_{\\!a},e^{*})=\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)$.
Then by Lemma 6.2.4, $e^{*}\in\texttt{A}(\mathscr{M}_{a})$ so that
$\mathscr{M}_{\\!a}$ is an a.a. flow of $G$. ∎

Note here that we cannot show that $\phi^{-1}[y]$ must contain a.a. points
when $\phi\colon\mathscr{X}\rightarrow\mathscr{Y}$ is an extension of a.a.
flows with $y\in\texttt{A}(\mathscr{Y})$.

The proof of the following lemma is an improvement of Auslander’s argument of
[4, Lem. 8.2].

######  6.2.7C Lemma (cf. [4, Lem. 8.2] for minimal flows with $A$ a maximal
a.p. subset).

Let $\mathscr{X}=(G,X)$ be an a.a. flow and $A=\verb"A"(\mathscr{X})$. Let
$\boldsymbol{z}\in X^{A}$ with $\boldsymbol{z}(a)=a\ \forall a\in A$ and
$Z=\overline{G\boldsymbol{z}}$. Suppose $\varphi$ is an endomorphism of
$G\curvearrowright Z$ and $A^{\prime}=\varphi(\boldsymbol{z})[A]$. If $z_{1}$,
$z_{2}\in Z$ with $z_{1}|_{A^{\prime}}\not=z_{2}|_{A^{\prime}}$, then
$\varphi(z_{1})\not=\varphi(z_{2})$.

###### Proof.

We first note that $\mathscr{Z}=G\curvearrowright Z$ is an a.a. flow with
$\boldsymbol{z}\in\texttt{A}(\mathscr{Z})$. Moreover, for all $z\in Z$ and all
$a_{1},a_{2}\in A$ with $a_{1}\not=a_{2}$, we have that
$z(a_{1})\not=z(a_{2})$, since $(\boldsymbol{z}(a_{1}),\boldsymbol{z}(a_{2}))$
is a distal point for $G\curvearrowright X\times X$. So, $z\colon A\rightarrow
X$ is 1-1 for all $z\in Z$.

Let $z\in Z$ be an a.a. point under $\mathscr{Z}$. Then $z[A]$ is an a.a. set
in $X$ (i.e., given any finite set $\\{a_{1},\dotsc,a_{k}\\}\subseteq A$,
$(z(a_{1}),\dotsc,z(a_{k}))\in X^{k}$ is an a.a. point for $G\curvearrowright
X^{k}$). Clearly, $z\colon A\rightarrow z[A]\subseteq A$ is 1-1.

Since $(\boldsymbol{z},\varphi(\boldsymbol{z}))$ is an a.a. point for
$G\curvearrowright Z\times Z$ by Def. 6.2.1c, hence
$\boldsymbol{z}[A]\cup\varphi(\boldsymbol{z})[A]$ is an a.a. set in $X$. But
$\boldsymbol{z}[A]$ is a maximal a.a. set in $X$, so
$\varphi(\boldsymbol{z})[A]\subseteq A$.

If $\gamma\colon A\rightarrow A$ is an injection, let $\gamma^{*}\colon
X^{A}\rightarrow X^{A}$ be a continuous mapping defined by
$z\mapsto\gamma^{*}(z)$ where $\gamma^{*}(z)(a)=z(\gamma(a))$ for all $a\in
A$. Clearly, $\gamma^{*}$ is $G$-equivariant so $\gamma^{*}$ is a self-
homomorphism of $G\curvearrowright X^{A}$.

We now define $\gamma\colon A\rightarrow A$ by
$\gamma(a)=\varphi(\boldsymbol{z})(a)$ for all $a\in A$. Since
$\varphi(\boldsymbol{z})\colon A\rightarrow X$ is 1-1 and
$A^{\prime}:=\varphi(\boldsymbol{z})[A]\subseteq A$, $\gamma$ is an injection
of $A$ to itself with $\gamma[A]=A^{\prime}$ and
$\gamma^{*}(\boldsymbol{z})=\varphi(\boldsymbol{z})$.

Since $\varphi(\boldsymbol{z})\in Z$, $\gamma^{*}[Z]\cap Z\not=\emptyset$, so
$\gamma^{*}[Z]=Z$. Now $\gamma^{*}(\boldsymbol{z})=\varphi(\boldsymbol{z})$,
so $\varphi=\gamma^{*}|_{Z}$ and thus, $\gamma^{*}|_{Z}$ is an endomorphism of
$G\curvearrowright Z$.

Let $z_{1}$, $z_{2}\in Z$ with $z_{1}|_{A^{\prime}}\not=z_{2}|_{A^{\prime}}$.
By $\varphi=\gamma^{*}|_{Z}$, it follows easily that
$\varphi(z_{1})\not=\varphi(z_{2})$. The proof is completed. ∎

######  6.2.7D Theorem.

Let $A=\verb"A"(\mathscr{M}_{\\!a})$ and let $\mathscr{Z}=(G,Z)$ be defined as
in Lemma 6.2.7C with $\mathscr{X}=\mathscr{M}_{\\!a}$. Then:

  1. (1)

$\mathscr{M}_{\\!a}$ is a universal a.a. flow of $G$.

  2. (2)

$\mathscr{Y}$ is a universal a.a. flow of $G$ iff it is a maximal almost 1-1
extension of $\mathscr{M}_{\\!eq}$.

  3. (3)

If $\phi$ is an endomorphism of $\mathscr{M}_{\\!a}$ such that
$a\in\eta(\phi(a))[A]$ for some $a\in A$ and some
$\eta\colon\mathscr{M}_{\\!a}\rightarrow\mathscr{Z}$, then $\phi$ is an
automorphism of $\mathscr{M}_{\\!a}$.

In particular, the universal a.a. flow of $G$ is unique up to almost 1-1
extensions.

###### Proof.

(1): By Lemma 6.2.7B, $\mathscr{M}_{\\!a}$ is an a.a. flow of $G$. To prove
the university, let $\mathscr{Y}$ is an a.a. flow of $G$ with
$y_{0}\in\texttt{A}(\mathscr{Y})$. Then we have by Lemma 6.2.4 and Def.
6.2.7a1 that
$\mathbb{F}(\mathscr{Y},y_{0})\subseteq\verb"A"_{\textrm{c,L}}(G)\cap\verb"LUC"(G)=\mathbb{F}(\mathscr{M}_{\\!a},e^{*})$.
Thus, $(\mathscr{Y},y_{0})$ is a factor of $(\mathscr{M}_{\\!a},e^{*})$.

(2): Suppose $\mathscr{Y}$ is a universal a.a. flow of $G$ with
$y_{0}\in\verb"A"(\mathscr{Y})$. Then by Structure Theorem of a.a. flow (Thm.
6.2.6), there exists an almost 1-1 extension
$\rho\colon(\mathscr{Y},y_{0})\rightarrow(\mathscr{Y}_{\\!eq},y_{\\!eq})$. By
the universality of $\mathscr{M}_{\\!eq}$, there is an extension
$\phi\colon(\texttt{M}_{eq},e_{\\!eq}^{*})\rightarrow(Y_{eq},y_{\\!eq})$.
Since $\mathscr{M}_{\\!eq}$ is an a.a. regular flow of $G$ with
$e_{\\!eq}^{*}$ an a.a. point, there exists an extension
$\psi\colon(Y,y_{0})\rightarrow(\texttt{M}_{eq},e_{\\!eq}^{*})$. Then
$\rho=\phi\circ\psi$ so that $\phi$ is almost 1-1. However, since
$\mathscr{M}_{\\!eq}$ is equicontinuous, $\phi$ is open and 1-1. Thus, $\phi$
is an isomorphism from $\mathscr{M}_{\\!eq}$ onto $\mathscr{Y}_{\\!eq}$. If
$\mathscr{Y}^{\prime}$ is an almost 1-1 extension of $\mathscr{M}_{\\!eq}$,
then $\mathscr{Y}^{\prime}$ is a.a. so that it is a factor of $\mathscr{Y}$.
Thus, $\mathscr{Y}$ is a maximal almost 1-1 extension of
$\mathscr{M}_{\\!eq}$.

Conversely, suppose $\mathscr{Y}$ is a maximal almost 1-1 extension of
$\mathscr{M}_{\\!eq}$. Then $\mathscr{Y}$ is an a.a. flow. To prove that
$\mathscr{Y}$ is a universal a.a. flow, it is enough to prove that
$\mathscr{M}_{\\!a}$ is a factor of $\mathscr{Y}$. Since $\mathscr{M}_{\\!a}$
is an almost 1-1 extension of $\mathscr{M}_{\\!eq}$ by (1) and the proved
necessity of (2), hence $\mathscr{M}_{\\!a}$ is a factor of $\mathscr{Y}$ by
the maximality.

(3): Let $\phi$ is an endomorphism of $\mathscr{M}_{\\!a}$ and
$\eta\colon\mathscr{M}_{\\!a}\rightarrow\mathscr{Z}$. Let $a^{\prime}\in A$
such that $a^{\prime}\in\eta(\phi(a^{\prime}))[A]$. Define $\pi^{\prime}\colon
Z\rightarrow\verb"M"_{a}$ by $z\mapsto z(a^{\prime})$, and
$\varphi=\eta\circ\phi\circ\pi^{\prime}$. Then $\varphi$ is an endomorphism of
$\mathscr{Z}$. To prove that $\phi$ is 1-1, suppose to the contrary that there
are two points $w_{1}\not=w_{2}$ in $\verb"M"_{a}$ with
$\phi(w_{1})=\phi(w_{2})$. We can choose two points $z_{1}$, $z_{2}\in Z$ such
that $z_{1}(a^{\prime})=w_{1}$ and $z_{2}(a^{\prime})=w_{2}$. By
$\varphi=\eta\circ\phi\circ\pi^{\prime}$, it follows that
$\varphi(z_{1})=\varphi(z_{2})$. However, from
$\boldsymbol{z}(a^{\prime})=a^{\prime}$ we have that
$\varphi(\boldsymbol{z})=\eta(\phi(a^{\prime}))$ and then $a^{\prime}\in
A^{\prime}=\varphi(\boldsymbol{z})[A]$. Thus,
$z_{1}|_{A^{\prime}}\not=z_{2}|_{A^{\prime}}$ and
$\varphi(z_{1})\not=\varphi(z_{2})$ by Lemma 6.2.7C, contrary to
$\varphi(z_{1})=\varphi(z_{2})$.

Finally, let $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ be two universal a.a.
flows and let $x_{1}\in\texttt{A}(\mathscr{X}_{1})$. Then there exists an
almost 1-1 extension
$\rho_{1}\colon(\mathscr{X}_{1},x_{1})\rightarrow(\texttt{M}_{eq},e_{\\!eq}^{*})$
by (2). By university of $\mathscr{X}_{1}$, there is an extension
$\phi\colon(\mathscr{X}_{1},x_{1})\rightarrow(\mathscr{X}_{2},x_{2})$.
Moreover, by (2) and regularity of $\mathscr{M}_{\\!eq}$, there also exists an
almost 1-1 extension
$\rho_{1}\colon(\mathscr{X}_{2},x_{2})\rightarrow(\texttt{M}_{eq},e_{\\!eq}^{*})$.
Since $\rho_{1}$ is 1-1 at $x_{1}$ and $\rho_{2}$ is 1-1 at $x_{2}$, $\phi$ is
1-1 at $x_{1}$ and $\phi$ is almost 1-1. The proof is completed. ∎

######  6.2.7E Theorem.

Let $G$ be an abelian group and $\mathscr{X}$ an a.a. flow of $G$. If $\phi$
is an endomorphism of $\mathscr{X}$, then $\phi$ is almost 1-1 with
$\phi^{-1}\phi(x)=\\{x\\}$ for all $x\in\verb"A"(\mathscr{X})$.

###### Proof.

Since $G$ is abelian, $X_{\\!eq}$ is a compact Hausdorff topological group and
$G<X_{\\!eq}$. Then the statement follows from Theorem 6.2.6. The proof is
complete. ∎

######  6.2.7F (An alternative construction of $\texttt{M}_{a}$).

Let

$X^{\prime}=\prod_{f\in\texttt{A}_{\textrm{c,L}}(G)\cap\texttt{LUC}(G)}\texttt{H}_{\textrm{p}}^{L}[f],x_{0}=(f)_{f\in\texttt{A}_{\textrm{c,L}}(G)\cap\texttt{LUC}(G)}\in
X^{\prime}\textrm{ and }X=\overline{Gx_{0}}\subseteq X^{\prime}.$

Then $\mathscr{X}=G\curvearrowright X$ is an a.a. flow with
$x_{0}\in\texttt{A}(\mathscr{X})$.

######  6.2.7G Theorem.

Let $(\mathscr{X},x_{0})=G\curvearrowright(X,x_{0})$ be as in Def. 6.2.7F.
Then $G\curvearrowright(\verb"M"_{a},e^{*})\cong G\curvearrowright(X,x_{0})$.

###### Proof.

By
$\mathbb{F}(\mathscr{X},x_{0})=\texttt{A}_{\textrm{c,L}}(G)\cap\texttt{LUC}(G)$
and Theorem 3.1. ∎

######  6.2.8 Theorem.

Let $G$ be an abelian group such that for each $t\in G$ with $t\not=e$, there
exists some $f\in\verb"A"_{\textrm{c}}(G)\cap\verb"LUC"(G)$ with $tf\not=f$.
Then $G\curvearrowright\verb"M"_{a}$ is free.

###### Proof.

Suppose to the contrary that $G\curvearrowright\verb"M"_{a}$ is not free. Then
there exists an element $t\in G$ with $t\not=e$ such that $tx_{0}=x_{0}$ for
some point $x_{0}\in\verb"M"_{a}$. Since $G$ is abelian and
$\overline{Gx_{0}}=\verb"M"_{a}$, it follows that $tx=x$ for all
$x\in\verb"M"_{a}$. Then by Theorem 6.2.7G, we can conclude a contradiction
that $tf=f$ for all $f\in\verb"A"_{\textrm{c}}(G)\cap\verb"LUC"(G)$. The proof
is complete. ∎

######  Corollary.

If $G=\mathbb{Z}$ or $G=\mathbb{R}$, then $G\curvearrowright\verb"M"_{a}$ is
free.

###### Proof.

First note that given $t(\not=0)\in G$ we can choose some $f\in\verb"LUC"(G)$
such that $tf=f$ and $f>0$. Then $f$ is a.a., for it is periodic. This implies
by Theorem 6.1.4 that for every $t(\not=0)\in G$ there exists some function
$f\in\verb"A"_{\textrm{c}}(G)\cap\verb"LUC"(G)$ with $tf\not=f$. Thus, by
Theorem 6.2.8, $G\curvearrowright\verb"M"_{a}$ is free. The proof is
completed. ∎

###  6.3 Veech’s hulls of a.a. functions

Let $G$ be a Hausdorff topological group in $\S$6.3, unless otherwise
specified.

######  6.3.1Two canonical bi-flows.

There are two canonical non-compact bi-flows (bi-transformation group) as
follows:

$(G_{\\!d},l^{\infty}(G),G_{\\!d})$ with continuous phase mapping

  1. $G_{\\!d}\times l^{\infty}(G)\times G_{\\!d}\xrightarrow{(\tau_{1},f,\tau_{2})\mapsto\tau_{1}f\tau_{2}}l^{\infty}(G)$

where

  1. $\tau_{1}f\tau_{2}(s)=f(\tau_{2}s\tau_{1})\ \forall s\in G$;

$(G_{\\!d},l^{\infty}(G\times G),G_{\\!d})$ with continuous phase mapping

  1. $G_{\\!d}\times l^{\infty}(G\times G)\times G_{\\!d}\xrightarrow{(\tau_{1},x,\tau_{2})\mapsto\tau_{1}x\tau_{2}}l^{\infty}(G\times G)$

where

  1. $\tau_{1}x\tau_{2}(s,t)=x(s\tau_{1},\tau_{2}t)\ \forall(s,t)\in G\times G$.

######  6.3.2Veech’s hulls of UC-functions on $G$.

Following Veech [30, p.730] we now define a continuous mapping

  1. $\xi_{\centerdot}\colon l^{\infty}(G)\rightarrow l^{\infty}(G\times G)$

by

  1. $f\mapsto\xi_{f}\colon G\times G\xrightarrow{(s,t)\mapsto f(st)}\mathbb{C}\quad\forall f\in l^{\infty}(G)$.

Clearly, $\xi_{\centerdot}\colon f\mapsto\xi_{f}$ is 1-1.

However, if $G$ is not abelian, then
$\xi_{\tau_{1}f\tau_{2}}\not=\tau_{1}\xi_{f}\tau_{2}$ for
$\tau_{1},\tau_{2}\in G$ and $f\in l^{\infty}(G)$ in general. So, $\xi$ is not
a homomorphism from $(G_{\\!d},l^{\infty}(G),G_{\\!d})$ into
$(G_{\\!d},l^{\infty}(G\times G),G_{\\!d})$.

Let $\texttt{UC}(G)=\texttt{LUC}(G)\cap\texttt{RUC}(G)$ as in 2.2.6.

Following Veech [30], for all $f\in\texttt{UC}(G)$, Veech’s hull of $f$ may be
defined as follows:

  1. $X_{\\!f}=\overline{G\xi_{f}}^{\textrm{p}}\ \left(=\overline{\xi_{f}G}^{\textrm{p}}\quad\because\tau\xi_{f}=\xi_{f}\tau\ \forall\tau\in G\right)$.

By Lemma 4.2, it follows that

  1. $(G,X_{\\!f},G)$: $G\times X_{\\!f}\times G\xrightarrow{(\tau_{1},x,\tau_{2})\mapsto\tau_{1}x\tau_{2}}X_{\\!f}$

is a compact bi-flow.

Note that $\texttt{H}_{\textrm{p}}^{L}[f]\xrightarrow{\xi}X_{\\!f}$ is
continuous 1-1 onto, yet $\xi$ is not $G$-equivariant if $G$ is not abelian.

######  6.3.3 Lemma (cf. [30, Lem. 4.1.1] for $G$ to be abelian, LC,
$\sigma$-compact and first countable).

If $G$ is LC and $f$ is Bochner a.a. on $G$ (i.e.,
$f\in\verb"A"_{\textrm{c}}(G)$), then $f\in\verb"UC"(G)$ and $(G,X_{\\!f},G)$
is a compact bi-flow.

###### Proof.

By 6.3.2c and 6. of Theorem 6.1.4. ∎

######  6.3.4.

Let $(G,X,G)$ be a compact bi-flow. A point $x\in X$ is left a.a. if $x$ is
a.a. under $G\curvearrowright X$ (cf. Def. 6.2.1c), denoted by
$x\in\verb"A"(G\curvearrowright X)$. If $x$ is a.a. under $X\curvearrowleft
G$, then it is right a.a. and denoted $x\in\verb"A"(X\curvearrowleft G)$.
Points which are both left and right a.a. are called a.a. points under
$(G,X,G)$, denoted $\verb"A"(X)$.

Clearly, if $G$ is LC and $\boldsymbol{f}\in\texttt{A}_{\textrm{c}}(G)$, then
$G\curvearrowright X_{\\!\boldsymbol{f}}$ and
$X_{\\!\boldsymbol{f}}\curvearrowleft G$ are both minimal by Lemma 6.1.5, and
$G\xi_{\boldsymbol{f}}=\xi_{\boldsymbol{f}}G\subseteq\verb"A"(X_{\\!\boldsymbol{f}})$
by Lemma 6.3.3.

If $f\in\texttt{UC}(G)$ such that $\xi_{f}\in\texttt{A}(G\curvearrowright
X_{\\!f})$, then $f\in\texttt{A}_{\textrm{c,L}}(G)$.

Then by Structure Theorem of a.a. flows (Thm. 6.2.6), $G\curvearrowright
X_{\\!\boldsymbol{f}}$ and $X_{\\!\boldsymbol{f}}\curvearrowleft G$ are both
a.a. flows so that they are almost 1-1 extensions of their maximal a.p.
factors. In fact, Veech’s Structure Theorem of a.a. function will give us more
(see Thm. 6.3.11).

######  6.3.5 Lemma.

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Let
$I=\\{z\in\mathbb{C}\,|\,|z|\leq\|f\|_{\infty}\\}$. Let $f^{\beta}\colon\beta
G\rightarrow I$ be the canonical extension of $f$ and let $p,q\in\beta G$.
Suppose $f^{\beta}(spqt)=f(st)$ for all $s,t\in G$. Then
$f^{\beta}(sqpt)=f(st)$ for all $s,t\in G$; and moreover,
$f^{\beta}(sq\\!\cdot_{\\!L}\\!pt)=f(st)=f^{\beta}(sp\\!\cdot_{\\!L}\\!qt)$
for all $s,t\in G$ under the left-topological semigroup structure
$\cdot_{\\!L}$ on $\beta G$.

###### Proof.

First, $f\in\texttt{A}_{\textrm{c,L}}(G_{\\!d})$ by Theorem 6.1.4. Given a
finite set $N\subseteq G$ and $\epsilon>0$ there exists by Lemma 6.1.10 a
finite superset $M$ of $N$ and a positive real number $\delta<\epsilon$ such
that

$C(f;M,\delta)\cdot C(f;M,\delta)\cdot C(f;M,\delta)^{-1}\subseteq
C(f;N,\epsilon).$

Let $\\{s_{i}\,|\,i\in\Lambda_{1}\\}$, $\\{t_{j}\,|\,j\in\Lambda_{2}\\}$ be
nets in $G$ with $s_{i}\to p$ and $t_{j}\to q$ in $\beta G$. By Lemma 6.1.9,
$C(f;M,\delta)$ is right-syndetic in $G_{\\!d}$, and so for each
$j\in\Lambda_{2}$ we can write $t_{j}=\tau_{j}r_{j}$ where $\tau_{j}\in
C(f;M,\delta)$ and $r_{j}\in R$ for some finite subset $R$ of $G$. By choosing
a subnet if necessary, we may assume $r_{j}=r\in R$ independent of $j$. Since
$f^{\beta}(srpqr^{-1}t)=f(st)$ for all $s,t\in G$, we can choose
$i\in\Lambda_{1}$ and $j\in\Lambda_{2}$ such that
$\tau_{ij}:=rs_{i}t_{j}r^{-1}\in C(f;M,\delta)$. Then
$t_{j}s_{i}=\tau_{j}\tau_{ij}\tau_{j}^{-1}$. Since $\tau_{ij},\tau_{j}\in
C(f;M,\delta)$, it follows that $\max_{s,t\in
N}|f(st_{j}s_{i}t)-f(st)|<\epsilon$. Passing to the limit in $i$ and then in
$j$ or in $j$ and then in $i$ gives us that

$\max_{s,t\in N}|f^{\beta}(sqpt)-f(st)|\leq\epsilon\quad\textrm{ and
}\quad\max_{s,t\in N}|f^{\beta}(sq\\!\cdot_{\\!L}\\!pt)-f(st)|\leq\epsilon.$

Since $\epsilon$, and then $N$ were arbitrary, hence
$f^{\beta}(sq\\!\cdot_{\\!R/L}\\!pt)=f(st)$ for all $s,t\in G$. By symmetry,
$f^{\beta}(sp\\!\cdot_{\\!L}\\!qt)=f(st)$ for all $s,t\in G$. The proof is
completed. ∎

######  Corollary (cf. [30, Lem. 3.1.1] for $G$ a discrete group).

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. If $x\in X_{\\!f}$ and
$\\{t_{n}\\}$ is a net in $G$ such that $\lim t_{n}x=\xi_{f}$, then $\lim
xt_{n}=\xi_{f}$. (By symmetry if $\lim xt_{n}=\xi_{f}$, then $\lim
t_{n}x=\xi_{f}$.)

###### Proof.

Choosing $q\in\beta G$ with $x=q\xi_{f}$, we have that $x(s,t)=f^{\beta}(sqt)$
for all $s,t\in G$. Let (a subnet of) $t_{n}\to p$ in $\beta G$, then we have
for all $s,t\in G$ that

$f(st)=\xi_{f}(s,t)=px(s,t)=\lim x(st_{n},t)=\lim
f^{\beta}(st_{n}qt)=f^{\beta}(spqt).$

Now by Lemma 6.3.5, it follows that $f^{\beta}(sq\cdot_{L}\\!pt)=f(st)$ for
all $s,t\in G$. So we have for $s,t\in G$ that

$\lim xt_{n}(s,t)=\lim x(s,t_{n}t)=\lim
f^{\beta}(sqt_{n}t)=f^{\beta}(sq\\!\cdot_{L}\\!pt)=f(st).$

Thus, $\lim xt_{n}=\xi_{f}$. The proof is completed. ∎

######  6.3.6Veech’s relation [30, Def. 3.1.2] and [10, Def. 5.1].

Let $(G,X,G)$ be a compact bi-flow. For $x,y\in X$ we say that
$(x,y)\in\verb"V"(G\curvearrowright X)$ if there exists a net $\\{t_{i}\\}$ in
$G$ such that $z=\lim t_{i}x$ and $\lim t_{i}^{-1}z=y$. Similarly we could
define $\verb"V"(X\curvearrowleft G)$. Write

  1. $\verb"V"(X)=\verb"V"(G\curvearrowright X)\cap\verb"V"(X\curvearrowleft G)$.

Clearly, $\texttt{V}(G\curvearrowright
X)\subseteq\texttt{RP}(G\curvearrowright X)$ and $\verb"V"(X)$ need not be a
symmetric relation on $X$ in general. However, it is symmetric in the a.a.
case as follows:

######  6.3.7 Lemma.

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Let $x,y\in X_{\\!f}$ such
that $(x,y)\in\verb"P"(G\curvearrowright X_{\\!f})$. If $\\{t_{n}\\}$ is a net
in $G$ with $t_{n}\to q\in\beta G$ such that $y=q\xi_{f}$, then $\lim
t_{n}^{-1}x=\lim xt_{n}^{-1}=\xi_{f}$. Thus, $\verb"P"(G\curvearrowright
X_{\\!f})\subseteq\verb"V"(G\curvearrowright X_{\\!f})$.

###### Proof.

First, since $G\curvearrowright X_{\\!f}$ is a compact minimal flow, we can
choose $p\in\beta G$ with $px=py=\xi_{f}$. Let (a subnet of)
$t_{n}^{-1}\to\alpha\in\beta G$ (independent of $\cdot_{\\!R}$ and
$\cdot_{\\!L}$ on $\beta G$ so that $yt_{n}^{-1}\to y\alpha$) and
$xt_{n}^{-1}\to z\in X_{\\!f}$ (i.e., $x\alpha=z$) based on
$X_{\\!f}\curvearrowleft G$. By $\xi_{f}\in\texttt{A}(X_{\\!f})$ and by
Corollary to Lemma 6.3.5, $\alpha y=\xi_{f}$ and $yt_{n}^{-1}\to\xi_{f}$ so
that $\xi_{f}=y\alpha$. By $py\alpha=\xi_{f}\alpha=px\alpha$, it follows that
$(\xi_{f},z)=(y\alpha,x\alpha)\in\texttt{P}(G\curvearrowright X_{\\!f})$. By
Lemma 6.1.6, $z=\xi_{f}$. Thus, $xt_{n}^{-1}\to\xi_{f}$; and then
$t_{n}^{-1}x\to\xi_{f}$ by Corollary to Lemma 6.3.5 again. The proof is
complete. ∎

######  Corollary (cf. [30, Lem. 3.1.2, Cor. 3.1.2] by different approaches).

Let $x,y\in X_{\\!f}$ with $(x,y)\in\verb"V"(G\curvearrowright X_{\\!f})$. If
$t_{n}\in G\to q\in\beta G$ with $x=q\xi_{f}$, then $\lim yt_{n}^{-1}=\lim
t_{n}^{-1}y=\xi_{f}$ and $(y,x)\in\verb"V"(G\curvearrowright X_{\\!f})$.

###### Proof.

By 6.2.3 and Lemma 6.3.7. ∎

######  6.3.8 Theorem.

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Then

$\verb"P"(G\curvearrowright X_{\\!f})=\verb"RP"(G\curvearrowright
X_{\\!f})=\verb"V"(G\curvearrowright
X_{\\!f})=\verb"V"(X_{\\!f}\curvearrowleft G)=\verb"V"(X_{\\!f}).$

###### Proof.

Since $G\curvearrowright X_{\\!f}$ is an a.a. flow, by Theorem 6.2.6 or 6.2.3
it follows that $\verb"P"=\verb"RP"\supseteq\verb"V"$ for $G\curvearrowright
X_{\\!f}$. Then by Lemma 6.3.7, $\verb"P"\subseteq\verb"V"$ for
$G\curvearrowright X_{\\!f}$. Thus,

$\verb"P"(G\curvearrowright X_{\\!f})=\verb"RP"(G\curvearrowright
X_{\\!f})=\verb"V"(G\curvearrowright X_{\\!f})$

and

$\verb"P"(X_{\\!f}\curvearrowleft G)=\verb"RP"(X_{\\!f}\curvearrowleft
G)=\verb"V"(X_{\\!f}\curvearrowleft G).$

Finally, we have by Lemma 6.3.5 that $\verb"P"(G\curvearrowright
X_{\\!f})=\verb"P"(X_{\\!f}\curvearrowleft G)=\verb"V"(X_{\\!f})$. The proof
is completed. ∎

######  6.3.8A Corollary (cf. [30, Thm. 3.1.1, Lem. 3.1.5]).

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Then $\verb"V"(X_{\\!f})$ is
a bi-invariant closed equivalence relation on $X_{\\!f}$.

######  6.3.8B Corollary.

If $G$ is LC, then

$\verb"A"(G\curvearrowright X_{\\!f})=\verb"A"(X_{\\!f}\curvearrowleft
G)=\verb"A"(X_{\\!f})$

for all $f\in\verb"A"_{\textrm{c}}(G)$.

######  6.3.8C Remark.

In fact, $\texttt{P}(G\curvearrowright X)\subseteq\texttt{V}(G\curvearrowright
X)$ for any minimal compact metric flow $G\curvearrowright X$, not necessarily
a.a.; see [16, Prop. 9.14] for $G=\mathbb{Z}$ and [10, Thm. 5.2] for $G$ any
semigroup. So, if $G$ is countable (or more generally, $\sigma$-compact) and
if $f\in\verb"LUC"(G)$ such that
$G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f]$ is a minimal flow, then
$\texttt{P}(G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f])\subseteq\texttt{V}(G\curvearrowright\verb"H"_{\textrm{p}}^{L}[f])$.

######  6.3.9A semigroup structure on $X_{\\!f}/\texttt{V}(X_{\\!f})$.

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Write $[x]=\verb"V"[x]$, the
$\verb"V"(X_{\\!f})$-equivalence class of $x\in X_{\\!f}$. If $x,y,z\in
X_{\\!f}$ with $x=\alpha\xi_{f}$, $y=\gamma\xi_{f}$, and
$z=\alpha\gamma\xi_{f}$ for some $\alpha,\gamma\in\beta G$, we then define a
binary operation:

$[x]\circ[y]=[z].$

If $x^{\prime}\in\verb"V"[x]$, $x^{\prime}=\alpha^{\prime}\xi_{f}$,
$y^{\prime}\in\verb"V"[y]$, $y^{\prime}=\gamma^{\prime}\xi_{f}$, and if
$z^{\prime}=\alpha^{\prime}\gamma^{\prime}\xi_{f}$, then
$z^{\prime}\in\verb"V"[z]$ and therefore, $[x]\circ[y]=[z]$ depends only on
the $\verb"V"$-classes $[x]$, $[y]$ of $x$ and $y$.

###### Proof.

Since $(y,y^{\prime})\in\verb"V"(X_{\\!f})$ and $\verb"V"(X_{\\!f})$ is a
closed invariant relation, hence
$(\alpha^{\prime}\gamma\xi_{f},z^{\prime})=(\alpha^{\prime}y,\alpha^{\prime}y^{\prime})\in\verb"V"(X_{\\!f})$.
Since $(x,x^{\prime})\in\verb"P"(G\curvearrowright X_{\\!f})$ by Theorem
6.3.8, there exists $p\in\beta G$ with
$p\alpha\xi_{f}=\xi_{f}=p\alpha^{\prime}\xi_{f}$. By
$\xi_{f}\in\texttt{A}(X_{\\!f})$ and letting $\gamma^{-1}=\lim t_{n}^{-1}$ for
a net $t_{n}\in G\to\gamma$, we have that $\gamma\gamma^{-1}\xi_{f}=\xi_{f}$
and
$\gamma\gamma^{-1}p\alpha\xi_{f}=\xi_{f}=\gamma\gamma^{-1}p\alpha^{\prime}\xi_{f}$.
Now by Lemma 6.3.5, it follows that
$\gamma^{-1}p(\alpha\gamma)\xi_{f}=\gamma^{-1}p(\alpha^{\prime}\gamma)\xi_{f}$.
Thus,
$(\alpha\gamma\xi_{f},\alpha^{\prime}\gamma\xi_{f})=(z,\alpha^{\prime}\gamma\xi_{f})\in\verb"P"(X_{\\!f})$
and $(z,z^{\prime})\in\verb"V"(X_{\\!f})$ by Theorem 6.3.8. ∎

Let $X_{0}=X_{\\!f}/\verb"V"(X_{\\!f})$, endowed with the quotient topology.
Then with phase mapping $(s,[x],t)\mapsto s[x]t=[sxt]$ for all $s,t\in G$ and
$[x]\in X_{0}$, $(G,X_{0},G)$ is a compact bi-flow.

###### Proof.

Clearly, $G\curvearrowright X_{0}$ and $X_{0}\curvearrowleft G$ are both
compact flows by Corollary 6.3.8A. Now let $s,t\in G$ and $[x]\in X_{0}$.
Since $s([x]t)=s[xt]=[sxt]$ and $(s[x])t=[sx]t=[sxt]$, then $s([x]t)=(s[x])t$
so that $G\curvearrowright X_{0}\curvearrowleft G$ is a compact bi-flow. ∎

Let $(G,X,G)$ and $(G,Y,G)$ be two compact bi-flows. Then we say that
$\rho\colon(G,X,G)\rightarrow(G,Y,G)$ is an extension if $\rho\colon
X\rightarrow Y$ is a continuous surjective map such that $\rho(sxt)=s\rho(x)t$
for all $s,t\in G$ and $x\in X$.

######  6.3.10 Theorem (cf. [30, Thm. 3.2.1]).

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. With the operation
$[x]\circ[y]$ as in Def. 6.3.9, $X_{0}$ is a compact Hausdorff topological
group with identity $[\xi_{f}]$. That is,

  1. 1)

$([x]\circ[y])\circ[z]=[x]\circ([y]\circ[z])$ for all $[x],[y],[z]\in X_{0}$.

  2. 2)

$[x]\circ[\xi_{f}]=[\xi_{f}]\circ[x]=[x]$ for all $[x]\in X_{0}$.

  3. 3)

Given $[x]\in X_{0}$ there is a point $[y]=[x]^{-1}\in X_{0}$ such that

$[x]\circ[y]=[y]\circ[x]=[\xi_{f}].$

  4. 4)

The mapping $([x],[y])\mapsto[x]\circ[y]^{-1}$ is continuous from $X_{0}\times
X_{0}$ to $X_{0}$.

###### Proof (simpler than Veech’s).

Obvious by 6.3.9 and $(\alpha\beta)\gamma=\alpha(\beta\gamma)$ for all
$\alpha,\beta,\gamma\in\beta G$.

Obvious by 6.3.9, $\xi_{f}=e\xi_{f}$ and $e\alpha=\alpha e$ for all
$\alpha\in\beta G$.

Let $x=\alpha\xi_{f}$ for some $\alpha\in\beta G$ and $t_{n}\in G\to\alpha$.
Let $\alpha^{-1}=\lim t_{n}^{-1}$ in $\beta G$. Set $y=\alpha^{-1}\xi_{f}$.
Then $\alpha\alpha^{-1}\xi_{f}=\alpha^{-1}\alpha\xi_{f}=\xi_{f}$ so that
$[x]\circ[y]=[y]\circ[x]=[\xi_{f}]$.

Thus, by 1), 2) and 3) above, it follows that $X_{0}$ is a compact Hausdorff
group.

To prove that $X_{0}$ is a topological group, according to Ellis’s Joint
Continuity Theorem it is enough to prove that $([x],[y])\mapsto[x]\circ[y]$ is
separately continuous. For this, let $[x_{i}]\to[x]$ in $X_{0}$ and $[y]\in
X_{0}$. We may assume $x_{i}\to x$ in $X_{\\!f}$ and select
$\gamma,\alpha,\alpha_{i}\in\beta G$ such that $x_{i}=\alpha_{i}\xi_{f}$,
$\alpha_{i}\to\alpha$, $x=\alpha\xi_{f}$, and $y=\gamma\xi_{f}$. Then by
$\alpha_{i}\gamma\to\alpha\gamma$ in $\beta G$ (noting $\beta G$ is a right-
topological semigroup), we have that
$[x_{i}]\circ[y]=[\alpha_{i}\gamma\xi_{f}]\to[\alpha\gamma\xi_{f}]=[x]\circ[y]$.
This shows that $X_{0}$ is a right-topological group. Using $\cdot_{\\!L}$ on
$\beta G$, we can show that $X_{0}$ is a left-topological group. Thus,
$([x],[y])\mapsto[x]\circ[y]$ is separately continuous and the proof is
completed. ∎

######  6.3.11 Theorem (Veech’s Structure Theorem of a.a. function [30]).

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Let $\rho\colon
X_{\\!f}\rightarrow X_{0}$ be the canonical mapping. Then $(G,X_{\\!f},G)$ is
an almost 1-1 extension of $(G,X_{0},G)$ such that $(G,X_{0},G)$ is a compact-
group bi-flow and $\xi_{f}\in\verb"A"(X_{\\!f})=\\{x\in
X_{\\!f}\,|\,\\{x\\}=\rho^{-1}\rho(x)\\}$.

If in addition $t\xi_{f}\not=\xi_{f}$ for all $t\in G$ and $t\not=e$, then
$G\xrightarrow{t\mapsto[t\xi_{f}]=[\xi_{f}t]}X_{0}$ is an injection and
$G\curvearrowright X_{0}$ and $X_{0}\curvearrowleft G$ are both free.

###### Proof.

First part: By Theorems 6.3.8 and 6.3.10.

Second part: Let $tx=x$ for some $t\in G$ and some point $x\in X_{0}$. Then
$txs=xs\ \forall s\in G$. Further $tx=x$ for all $x\in X$ so that
$t\xi_{f}=\xi_{f}$ and $t=e$. Thus, $G\curvearrowright X_{0}$ is free. By
symmetry, $X_{0}\curvearrowleft G$ is also free. The proof is completed. ∎

######  6.3.12 Theorem.

Let $G$ be LC and $f\in\verb"A"_{\textrm{c}}(G)$. Then every endomorphism of
$G\curvearrowright X_{\\!f}$ (resp. $X_{\\!f}\curvearrowleft G$) is almost
1-1.

###### Proof.

See Proof of Theorem 6.2.7E. ∎

######  6.3.13 Remark.

Lemma 6.2.5 and Theorem 6.3.11 indicates that if $G$ is not abelian, then
$\verb"H"_{\textrm{p}}^{L}[f]$ need not be isomorphic to $X_{\\!f}$ for any
$f\in\verb"A"_{\textrm{c}}(G)\cap\verb"UC"(G)$. In fact, there exist a.a.
functions $f$ on non-abelian discrete groups such that the maximal a.p. factor
of $\verb"H"_{\textrm{p}}^{L}[f]$ is not a compact group. So Lemma 6.2.5 and
Theorem 6.3.11 are two essentially different structure theorems of a.a.
functions.

######  6.3.14A group structure of a.p. functions.

Recall that a function $f$ on $G$ is left a.p. in the sense of von Neumann
(1934) if $Gf$ is relatively compact in $(\mathbb{C}^{G},\|\cdot\|_{\infty})$.
Clearly, if $f$ is left a.p., then $f\in l^{\infty}(G)$. If $f$ is both left
a.p. and right a.p., then $f$ is said to be a.p. and denoted
$f\in\texttt{AP}(G)$. This is independent of the topology on $G$.

######  6.3.14A Lemma (J. von Neumann; cf. [20, Thm. 18.1]).

Let $f\in\mathbb{C}^{G}$. Then the following conditions are pairwise
equivalent:

  1. 1.

$Gf$ is relatively compact in $(\mathbb{C}^{G},\|\cdot\|_{\infty})$.

  2. 2.

$fG$ is relatively compact in $(\mathbb{C}^{G},\|\cdot\|_{\infty})$.

  3. 3.

Given $\varepsilon>0$, there are two finite sets $\\{a_{1},\dotsc,a_{n}\\}$
and $\\{b_{1},\dotsc,b_{m}\\}$ in $G$ such that for all $a,b\in G$ there are
$i\in\\{1,\dotsc,n\\}$ and $j\in\\{1,\dotsc,m\\}$ with $\|bfa-
b_{j}fa_{i}\|_{\infty}<\varepsilon$.

  4. 4.

Given $\varepsilon>0$, there exists a finite set $\\{b_{1},\dotsc,b_{m}\\}$ in
$G$ such that for all $b\in G$ there is some $k\in\\{1,\dotsc,m\\}$ with
$\|b\xi_{f}-b_{k}\xi_{f}\|_{\infty}<\varepsilon$.

Consequently, $f$ is left a.p. iff it is right a.p. iff
$\overline{Gf}^{\|\cdot\|_{\infty}}$ is compact iff
$\overline{fG}^{\|\cdot\|_{\infty}}$ is compact.

###### Proof.

See Appendix B ∎

######  6.3.14B.

Let $G$ be LC and $f\in\texttt{UC}(G)$. If $x=\alpha\xi_{f}$,
$y=\gamma\xi_{f}$, and $z=\alpha\gamma\xi_{f}$ in $X_{\\!f}$ where
$\alpha,\gamma\in\beta G$, then define

$x\circ y=z.$

Note that $(X_{\\!f},\circ)$ is not a group in general. However, we can obtain
the following for a.p. functions:

######  6.3.14C Theorem.

Let $G$ be LC and let $f\in\texttt{UC}(G)$. Then $f\in\verb"AP"(G)$ iff
$(X_{\\!f},\circ)$ is a compact Hausdorff topological group with $e=\xi_{f}$.

###### Proof.

Necessity: Suppose $f\in\verb"AP"(G)$. Then $f\in\texttt{A}_{\textrm{c}}(G)$
by Note of Lemma 6.2.5. By Lemma 6.3.14A, $G\curvearrowright X_{\\!f}$ and
$X_{\\!f}\curvearrowleft G$ are both compact a.p./equicontinuous minimal
flows. Thus, $\rho\colon X_{\\!f}\rightarrow X_{0}$, as in Theorem 6.3.11, is
continuous 1-1 onto so that $(X_{\\!f},\circ)$ is a compact Hausdorff
topological group with $e=\xi_{f}$.

Sufficiency: Suppose $(X_{\\!f},\circ)$ is a compact Hausdorff topological
group with $e=\xi_{f}$. Then that $s\xi_{f}\circ t\xi_{f}=st\xi_{f}\ \forall
s,t\in G$ implies that $\alpha\gamma\xi_{f}=\alpha\xi_{f}\circ\gamma\xi_{f}$
for all $\alpha,\gamma\in\beta G$. Now given $t\in G$, $tx=t\xi_{f}\circ x$
for all $x\in X_{\\!f}$. Thus, $G\curvearrowright X_{\\!f}$ is equicontinuous.
Further, $G\curvearrowright X_{\\!f}$ is an a.p. flow so that
$G\curvearrowright\texttt{H}_{\textrm{p}}^{L}[f]$ is an a.p. flow. Thus,
$G\curvearrowright\texttt{H}_{\textrm{p}}^{L}[f]$ is equicontinuous so
$f\in\texttt{AP}(G)$ by Note of Lemma 6.2.5. The proof is completed. ∎

Note that if $G$ is non-abelian, then there need not exist a natural group
structure on
$\texttt{H}_{\textrm{p}}^{L}[f]=\overline{Gf}^{\|\cdot\|_{\infty}}$ for all
$f\in\texttt{AP}(G)$. In view of this, Theorem 6.3.14C is of interest.

###  6.4 Bohr a.a. functions

Let $G$ be a topological group in $\S$6.4, unless otherwise specified. We
shall characterize $\verb"A"_{\textrm{c}}(G)$ in terms of the so-called Bohr
a.a. functions on $G$ (Thm. 6.4.6).

######  6.4.1bUC-functions on $G\times G$.

A bounded function $\Phi\colon G\times G\rightarrow\mathbb{C}$ is said to be
bi-uniformly continuous, denoted $\Phi\in\verb"bUC"(G\times G)$, if for every
$\epsilon>0$ there exists some $V\in\mathfrak{N}_{e}(G)$ such that
${\sup}_{s,t\in G}|\Phi(v_{1}s,tv_{2})-\Phi(s,t)|<\epsilon$ for all
$v_{1},v_{2}\in V$.

######  6.4.1B Lemma.

Let $(G,X,G)$ be a compact bi-flow. Given $x\in X$ and $F\in\mathcal{C}(X)$,
let

$F_{\\!x}^{b}\colon G\times G\rightarrow\mathbb{C},\quad(s,t)\mapsto
F_{\\!x}^{b}(s,t)=F(sxt).$

Then $F_{\\!x}^{b}\in\verb"bUC"(G\times G)$. If $x\in\verb"A"(X)$, then
$F_{\\!x}^{b}$ is an a.a. point under $(G_{\\!d},l^{\infty}(G\times
G),G_{\\!d})$.

###### Proof.

Let $\epsilon>0$. There exists some $V\in\mathfrak{N}_{e}(G)$ such that

${\max}_{y\in X}|F(v_{1}yv_{2})-F(y)|<\epsilon\quad\forall v_{1},v_{2}\in V.$

Thus,

${\sup}_{s,t\in G}|F(v_{1}sxtv_{2})-F(sxt)|<\epsilon\quad\forall
v_{1},v_{2}\in V.$

This implies that

${\sup}_{s,t\in
G}|F_{\\!x}^{b}(v_{1}s,tv_{2})-F_{\\!x}^{b}(s,t)|<\epsilon\quad\forall
v_{1},v_{2}\in V.$

Finally, we note that if $t_{n}x\to y$ then
$t_{n}F_{\\!x}^{b}\to_{\textrm{p}}F_{\\!y}^{b}$, and if $xt_{n}\to y$ then
$F_{\\!x}^{b}t_{n}\to_{\textrm{p}}F_{\\!y}^{b}$. The proof is completed. ∎

######  6.4.2bi-hulls of UC-functions on $G$.

Define a continuous mapping

  1. $\theta\colon l^{\infty}(G)\rightarrow l^{\infty}(G\times G)$

by

  1. $f\mapsto\theta_{\\!f}\colon G\times G\xrightarrow{(s,t)\mapsto f(ts)}\mathbb{C}$.

It is different with $\xi_{f}$ in Def. 6.3.2a.

As in Def. 6.3.1b, for all $x\in l^{\infty}(G\times G)$ and
$\tau_{1},\tau_{2}\in G$, let

  1. $\tau_{1}x\tau_{2}\colon G\times G\xrightarrow{(s,t)\mapsto x(s\tau_{1},\tau_{2}t)}\mathbb{C}$.

Then $\tau_{1}x\tau_{2}\in l^{\infty}(G\times G)$ such that
$(\tau_{1}x)\tau_{2}=\tau_{1}(x\tau_{2})$.

Given $f\in\texttt{UC}(G)$, we can define a bi-hull of $f$ as follows:

  1. $W_{\\!f}=\overline{G\theta_{\\!f}G}^{\textrm{p}}$.

By Lemma 4.2, it follows that

  1. $G\times W_{\\!f}\times G\xrightarrow{(\tau_{1},x,\tau_{2})\mapsto\tau_{1}x\tau_{2}}W_{\\!f}$

is compact bi-flow.

It should be noted that if $G$ is non-abelian, then
$t\theta_{\\!f}\not=\theta_{\\!f}t$ for $t\not=e$ in general. So
$G\curvearrowright W_{\\!f}$ and $W_{\\!f}\curvearrowleft G$ need not be
minimal.

######  6.4.3 Lemma.

If $f\in\verb"UC"(G)$, then $\theta_{\\!f}\in\verb"bUC"(G\times G)$; that is,
for every $\varepsilon>0$ there exists some $V\in\mathfrak{N}_{e}(G)$ such
that

${\sup}_{s,t\in G}|f(tvs)-f(ts)|<\varepsilon\quad\forall v\in V.$

###### Proof.

Define $F\colon W_{\\!f}\rightarrow\mathbb{C}$ by $w\mapsto F(w)=w(e,e)$. Then
$F\in\mathcal{C}(W_{\\!f})$. Then by 6.4.2c and Lemma 6.4.1B, it follows that
given $\varepsilon>0$ there exists $V\in\mathfrak{N}_{e}(G)$ such that

$\sup_{s,t\in
G}|F_{\\!\theta_{\\!f}}^{b}(v_{1}s,tv_{2})-F_{\\!\theta_{\\!f}}^{b}(s,t)|<\varepsilon\quad\forall
v_{1},v_{2}\in V.$

Then

$\sup_{s,t\in
G}|F(v_{1}s\theta_{\\!f}tv_{2})-F(s\theta_{\\!f}t)|<\varepsilon\quad\forall
v_{1},v_{2}\in V.$

Thus,

$None$ $\sup_{s,t\in
G}|\theta_{\\!f}(v_{1}s,tv_{2})-\theta_{\\!f}(s,t)|<\varepsilon\quad\forall
v_{1},v_{2}\in V.$

So

$None$ $\sup_{s,t\in G}|f(tv_{2}v_{1}s)-f(ts)|<\varepsilon\quad\forall
v_{1},v_{2}\in V.$

The proof is completed. ∎

######  6.4.4.

We say that a function $f$ on $G$ is a Bohr a.a. function on $G$ if for each
finite set $N\subset G$ and all real number $\varepsilon>0$ there exists a set
$B=B(N,\varepsilon)\subseteq G$ such that:

  1. (a)

$B$ is syndetic in $G$ (Caution: not necessarily in $G_{\\!d}$);

  2. (b)

$B$ is symmetric (i.e., $B=B^{-1}$);

  3. (c)

If $b\in B$, then $\max_{s,t\in N}|f(sbt)-f(st)|<\varepsilon$;

  4. (d)

If $b_{1},b_{2}\in B$, then $\max_{s,t\in
N}|f(sb_{1}^{-1}b_{2}t)-f(st)|<2\varepsilon$;

  5. (e)

$f\in\verb"UC"(G)$.

######  Note.

A function $f$ on $G$ is called a continuous Bohr a.a. function on $G$ in the
sense of Veech (cf. [30, Def. 4.1.2]) if for each finite set $N\subset G$ and
all real number $\varepsilon>0$ there exists a set
$B=B(N,\varepsilon)\subseteq G$ such that

  1. (a)′

$B$ is relatively dense in $G$

and such that (b), (c), (d), and (e) above.

If $G$ is discrete, then $l^{\infty}(G)=\verb"UC"(G)$ so that Def. 6.4.1
coincides with Veech [30, Def. 2.1.1] in this case.

Clearly, a continuous Bohr a.a. function on $G$ in the sense of Veech is a
Bohr a.a. function on $G$ in our sense. It turns out that vice versa (Thm.
6.4.7). For this, we need a lemma.

######  6.4.5 Lemma.

Let $K$ be a compact subset of $G$ and $\\{k_{j}\,|\,j\in\Lambda\\}$ a net in
$K$ with $k_{j}\to k\in K$. Given $j\in\Lambda$ let $j^{\prime}\in\Lambda$
such that $j^{\prime}\geq j$. Then
$\\{k_{j^{\prime}}\,|\,j^{\prime}\in\Lambda\\}$ is a subnet of $\\{k_{j}\\}$
and $k_{j^{\prime}}\to k$ such that $k_{j}^{-1}k_{j^{\prime}}\to e$.

###### Proof.

For $j_{1}^{\prime},j_{2}^{\prime}\in\Lambda$, there exists some $j\in\Lambda$
with $j_{1}^{\prime}\leq j$ and $j_{2}^{\prime}\leq j$. So $j_{1}^{\prime}\leq
j^{\prime}$ and $j_{2}^{\prime}\leq j^{\prime}$. This shows that
$\\{k_{j^{\prime}}\,|\,j^{\prime}\in\Lambda\\}$ is a subnet of $\\{k_{j}\\}$.
Thus, $k_{j^{\prime}}\to k$. Since $G$ is a topological group, hence
$k_{j}^{-1}\to k^{-1}$ and $k_{j}^{-1}k_{j^{\prime}}\to e$. The proof is
completed. ∎

######  6.4.6 Theorem.

Let $G$ be LC and $f$ a function on $G$. Then $f$ is a Bohr a.a. function on
$G$ (cf. Def. 6.4.4) iff $f\in\verb"A"_{\textrm{c}}(G)$.

######  Note.

See Veech [30, Thm. 2.2.1] for $G$ to be a discrete group, and, [30, Thm.
4.1.1] for $G$ to be an abelian, LC, $\sigma$-compact and first countable
group. Theorem 6.4.6 gives us a positive solution to [10, Question 2.7] for
a.a. functions instead of a.a. points of flows.

###### Proof.

Necessity: Assume $f$ is a Bohr a.a. function on $G$ (cf. Def. 6.4.4). Let
$\\{t_{j}\\}_{j\in\Lambda}$ be a net in $G$ such that
$t_{j}f\to_{\textrm{p}}\phi\in l^{\infty}(G)$ and
$t_{j}^{-1}\phi\to_{\textrm{p}}h\in l^{\infty}(G)$. To prove that
$f\in\verb"A"_{\textrm{c,L}}(G)$, we need only prove that
$\phi\in\mathcal{C}(G)$ and $h=f$. Indeed, by Def. 6.4.4(e) and Lemma 4.2,
$\phi\in\verb"LUC"(G)\subseteq\mathcal{C}(G)$. It remains to show $h=f$.
Suppose to the contrary that $h\not=f$. Then there exist some $t\in G$ and
$\varepsilon>0$ such that $|f(t)-h(t)|\geq 5\varepsilon$. Let
$B=B(N,\varepsilon)$, where $N=\\{e,t\\}$, be given by Def. 6.4.4. Passing to
a subnet if necessary, we may assume $t_{j}=k_{j}s_{j}$ such that $s_{j}\in B$
and $k_{j}\to k\in G$. Let $j$ and then $j^{\prime}\geq j$ be chosen from
$\Lambda$ so large that $|t_{j}^{-1}t_{j^{\prime}}f(t)-h(t)|<\varepsilon$.
However, by Lemma 6.4.5 and Lemma 6.4.3, it follows that

$\begin{split}|t_{j}^{-1}t_{j^{\prime}}f(t)-f(t)|&=|s_{j}^{-1}k_{j}^{-1}k_{j^{\prime}}s_{j^{\prime}}f(t)-f(t)|\\\
&=|f(ts_{j}^{-1}k_{j}^{-1}k_{j^{\prime}}s_{j^{\prime}})-f(t)|\\\
&\leq|f(ts_{j}^{-1}k_{j}^{-1}k_{j^{\prime}}s_{j^{\prime}})-f(ts_{j}^{-1}s_{j^{\prime}})|+|f(ts_{j}^{-1}s_{j^{\prime}})-f(t)|\\\
&\leq 3\varepsilon\end{split}$

eventually. So $|f(t)-h(t)|\leq 4\varepsilon$, contrary to $|f(t)-h(t)|\geq
5\varepsilon$. Thus, $f\in\verb"A"_{\textrm{c,L}}(G)$. By symmetry,
$f\in\verb"A"_{\textrm{c,R}}(G)$, and, $f\in\verb"A"_{\textrm{c}}(G)$.

Sufficiency: Suppose $f\in\verb"A"_{\textrm{c}}(G)$. By Lemma 6.3.3,
$f\in\verb"UC"(G)$; and so Def. 6.4.4(e) holds. Finally, it is easy to check
that conditions (a)$\sim$(d) in Def. 6.4.4 are all fulfilled by Lemmas 6.1.9
and 6.1.10.

The proof of Theorem 6.4.6 is therefore completed. ∎

######  6.4.7 Theorem.

Let $G$ be LC and $f$ a function on $G$. Then $f$ is a Bohr a.a. function on
$G$ (Def. 6.4.4) iff it is a continuous Bohr a.a. function on $G$ in the sense
of Veech.

###### Proof.

Sufficiency is obvious. Necessity follows easily from Theorems 6.4.6 and
6.1.4. ∎

##  Appendix A Appendix: Proof of Lemma 6.1.10

Lemma 6.1.10 is the most technical tool for proving the structure theorem of
a.a. functions. We now present its proof here for reader’s convenience.

######  6.1.10 Lemma (cf. [30, Lem. 2.1.2] or [10, Thm. 1.9]).

Let $f\in l^{\infty}(G)$ be a.a. on $G_{\\!d}$. Given $\epsilon>0$ and a
finite set $N\subset G$, there exist $\delta>0$ and a finite superset $M$ of
$N$ such that $\sigma^{-1}\tau\in C(f;N,\epsilon)$ for all $\sigma,\tau\in
C(f;M,\delta)$.

###### Proof (Veech 1965 [30]).

Suppose to the contrary that, for some finite set $N\subset G$ and some
$\epsilon>0$, for every finite superset $M$ of $N$ and all $\delta>0$ with
$\delta<\epsilon$ there must exist $\sigma,\tau\in C(f;M,\delta)$ with
$\sigma^{-1}\tau\notin C(f;N,\epsilon)$.

Choose a sequence of positive real numbers $\\{\delta_{n}\\}_{n=1}^{\infty}$
with $\epsilon>\delta_{n}\searrow 0$ and $\sum_{n}\delta_{n}<\infty$. Let a
sequence $\\{M_{n}\\}$ of finite supersets of $N$ together with a sequence
$(\sigma_{n},\tau_{n})$ of pairs of elements of $G$ be chosen as follows:

Let $M_{1}^{\prime}=N\cup\\{e\\}$ and set
$M_{1}=M_{1}^{\prime}\cup(M_{1}^{\prime})^{-1}$. $M_{1}$ is a finite superset
of $N$, so by assumption there exists $(\sigma,\tau)=(\sigma_{1},\tau_{1})$
such that $\sigma,\tau\in C(f;M_{1},\delta_{1})$ but $\sigma^{-1}\tau\notin