Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters

TL;DR

This paper investigates the size and structure of topologically invariant means on function spaces related to locally compact groups, proving new results about their cardinality and convergence properties using ultrafilter theory.

Contribution

It introduces a unified approach to analyze invariant means on $L_ ext{infty}(G)$ and $VN(G)$, proving the existence of orthogonal nets converging to invariance and confirming Paterson's conjecture.

Findings

Cardinality of invariant means is maximally large, as determined by ultrafilter theory.

Existence of orthogonal nets in $L_1(G)$ and $A(G)$ converging to invariance.

Proof of Paterson's conjecture relating invariant means to conjugacy class precompactness.

Abstract

In 1970, Chou showed there are $|\mathbb{N}^*| = 2^{2^\mathbb{N}}$ topologically invariant means on $L_\infty(G)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_\infty(G)$ and $VN(G)$ were determined for any locally compact group. Each paper on a new case reached the same conclusion -- "the cardinality is as large as possible" -- but a unified proof never emerged. In this paper, I show $L_1(G)$ and $A(G)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\Gamma$ has $|\Gamma^*|$ accumulation points, where $|\Gamma^*|$ is determined by ultrafilter theory. Among a smattering of other results, I prove Paterson's conjecture that left and right topologically invariant means on $L_\infty(G)$ coincide iff $G$ has precompact conjugacy classes.

Full text

Among a smattering of other results, I prove Paterson’s conjecture that left and right topologically invariant means on $L_{\infty}(G)$ coincide iff $G$ has precompact conjugacy classes.

The present paper will form part of the author’s PhD thesis under supervision of Ching Chou.

Chapter 1 Background

1. History

1.1**.**

Følner’s condition for amenable discrete groups says, for all finite $K\subset G$ and $\epsilon>0$ there exists finite $F_{(K,\epsilon)}\subset G$ which is $(K,\epsilon)$-invariant. The set $\Gamma$ of all ordered pairs $\gamma=(K,\epsilon)$ is a directed set, ordered by increasing $K$ and decreasing $\epsilon$. Define $m_{\gamma}\in\ell_{\infty}^{*}(G)$ by $m_{\gamma}(f)=\frac{1}{|F_{\gamma}|}\sum_{x\in F_{\gamma}}f(x)$. Then the net $\{m_{\gamma}\}_{\gamma\in\Gamma}$ converges to invariance, and any limit point is an invariant mean on $\ell_{\infty}(G)$.

The analogue of Følner’s condition for locally compact amenable groups says, for all compact $K\subset G$ and $\epsilon>0$, there exists compact $F\subset G$ which is $(K,\epsilon)$-invariant. The analogous definition of $m_{\gamma}\in L_{\infty}^{*}(G)$ is $m_{\gamma}(f)=\frac{1}{|F_{\gamma}|}\int_{F_{\gamma}}f(x)\>\mathrm{d}{x}$. If $m$ is any limit point of $\{m_{\gamma}\}_{\gamma\in\Gamma}$, it is not only invariant but topologically invariant. That is, $m(\phi)=m(f*\phi)$ for any $\phi\in L_{\infty}(G)$ and $f\in L_{1}(G)$ with $\|f\|_{1}=\int_{G}f=1$.

1.2**.**

Non-topologically invariant means are harder to come by. All invariant means on a discrete group are topologically invariant, and it was not until 1972 that [Rud72] and [Gra73] independently discovered a construction of non-topologically invariant means for any non-discrete $G$ that is amenable-as-discrete. Several papers have discussed the implications of this construction. Notably, [Ros76] combined it with a Baire category argument to construct $2^{2^{\mathbb{N}}}$ invariant means that are not topologically invariant.

The general problem of enumerating non-topologically invariant means seems intractable. For instance, the famously difficult Banach-Ruziewicz problem boils down to whether $SO(n,\mathbb{R})$ admits any non-topologically invariant means, c.f. [CLR85, Proposition 1.3].

1.3**.**

Let $\operatorname{TLIM}(G)$ denote the topologically left-invariant means on $L_{\infty}(G)$, and $\operatorname{TIM}(G)$ the (two-sided) topologically invariant means. In [Pat79], Paterson conjectured that $\operatorname{TLIM}(G)=\operatorname{TIM}(G)$ iff $G$ has precompact conjugacy classes, and proved it assuming $G$ is compactly generated. The short and insightful paper [Mil81] proved Paterson’s conjecture assuming $G$ is $\sigma$-compact, and gave me the ideas to prove it in full generality.

1.4**.**

For any set $S$, let $|S|$ denote its cardinal number – that is, the first ordinal $\alpha$ such that there exists a bijection from $\alpha$ to $S$. (There exists an ordinal of each cardinality by the axiom of choice.) By definition, an ordinal is the set of all previous ordinals. Thus the cardinal usually called $\aleph_{0}$ is none other than $\mathbb{N}=\{0,1,\ldots\}$.

1.5**.**

Let $\kappa=\kappa(G)$ be the first ordinal such that there is a family $\mathcal{K}$ of compact subsets of $G$ with $|\mathcal{K}|=\kappa$ and $G=\bigcup\mathcal{K}$. It’s not hard to prove $|\operatorname{TIM}(G)|\leq|\operatorname{TLIM}(G)|\leq 2^{2^{\kappa}}$. Of course when $\kappa=1$, the unique topologically invariant mean is Haar measure. But when $\kappa\geq\mathbb{N}$, $|\operatorname{TIM}(G)|$ actually equals $2^{2^{\kappa}}$. Here is an abbreviated history of this surprising result, which took almost 20 years to establish:

When $\kappa=\mathbb{N}$, [Cho70] defined $\pi:L_{\infty}(G)\rightarrow\ell_{\infty}(\mathbb{N})$ by $\pi(f)(n)=\frac{1}{|U_{n}|}\int_{U_{n}}f$, where $\{U_{n}\}$ is a Følner sequence of mutually disjoint sets. Thus $\pi^{*}:\ell_{\infty}^{*}(\mathbb{N})\rightarrow L_{\infty}^{*}(G)$ is an embedding. Let $c_{0}=\{f\in\ell_{\infty}(\mathbb{N}):\lim_{n}f(n)=0\}$ and $\mathcal{F}=\{m\in\ell_{\infty}^{*}(\mathbb{N}):\|m\|=1,m\geq 0,m|_{c_{0}}\equiv 0\}$, so that $\pi^{*}[\mathcal{F}]\subset\operatorname{TLIM}(G)$. Regarding the nonprincipal ultrafilters on $\mathbb{N}$ as elements of $\mathcal{F}$, we see $|\operatorname{TLIM}(G)|\geq|\beta\mathbb{N}-\mathbb{N}|=2^{2^{\mathbb{N}}}$.

When $G$ is discrete and $\kappa\geq\mathbb{N}$, [Cho76] proved $|\operatorname{TIM}(G)|\geq 2^{2^{\kappa}}$. When $G$ is non-discrete and $\kappa\geq\mathbb{N}$, [LP86] proved $|\operatorname{TLIM}(G)|\geq 2^{2^{\kappa}}$. These papers take the more direct approach of constructing disjoint, translation-invariant subsets of $\beta G$.

The full result $|\operatorname{TIM}(G)|\geq 2^{2^{\kappa}}$ was finally proved by [Yan88]. Yang realized that the trick to generalizing Chou’s embedding argument is to replace $\mathbb{N}$ by the indexing set of a Følner net.

1.6**.**

When $G$ is discrete abelian, $\kappa(\widehat{G})=1$ and Haar measure is the unique topologically invariant mean on $L_{\infty}(\widehat{G})$. More generally, let $\mu=\mu(G)$ be the first ordinal such that $G$ has a neighborhood basis $\mathcal{U}$ at the origin with $|\mathcal{U}|=\mu$. When $G$ is abelian, $\kappa(\widehat{G})=\mu(G)$ by [HR79, (24.48)]. Thus when $G$ is non-discrete abelian, $\kappa(\widehat{G})\geq\mathbb{N}$ and $|\operatorname{TIM}(\widehat{G})|=2^{2^{\mu}}$.

When $G$ is non-abelian, the group von Neumann algebra $VN(G)$ is the natural analogue of $L_{\infty}(\widehat{G})$.

If $\operatorname{TIM}(\widehat{G})$ denotes the set of topologically invariant means on $VN(G)$, then the analogous results hold:

When $\mu=1$, $\operatorname{TIM}(\widehat{G})$ is the singleton comprising the point-measure $\delta_{e}$, as proved by [Ren72].

When $\mu\geq\mathbb{N}$, $|\operatorname{TIM}(\widehat{G})|=2^{2^{\mu}}$. This is proved by [Cho82] when $\mu=\mathbb{N}$, using an embedding $\pi^{*}:\ell_{\infty}^{*}(\mathbb{N})\rightarrow VN(G)^{*}$, and by [Hu95] when $\mu>\mathbb{N}$, using a family $\{\pi_{\gamma}^{*}:\ell_{\infty}^{*}(\mu)\rightarrow VN(G)^{*}\}_{\gamma<\mu}$ of embeddings!

2. Ultrafilters

The following exposition is sparse. For more details, see [HS12, Chapter 3].

2.1**.**

Suppose $\{x_{\gamma}\}=\{x_{\gamma}\}_{\gamma\in\Gamma}$ is an infinite subset of the compact Hausdorff space $X$. Obviously $\{x_{\gamma}\}$ has limit points in $X$, but how do we “name” them? Regarding $\Gamma$ as a discrete topological space, let $f:\Gamma\rightarrow X$ be the continuous function $\gamma\mapsto x_{\gamma}$. Let $\tilde{f}:\beta\Gamma\rightarrow X$ be the unique continuous extension of $f$ to the Stone-Čech compactification of $\Gamma$. Since $\tilde{f}[\beta\Gamma]$ is compact and has $f[\Gamma]$ as a dense subset, the limit points of $\{x_{\gamma}\}$ are precisely the points $\tilde{f}(p)$ with $p\in\beta\Gamma$. We usually write $\operatorname*{\mathit{p}-lim}_{\gamma}x_{\gamma}$ instead of $\tilde{f}(p)$.

2.2**.**

In functional analysis, the Stone-Čech compactification $\beta\Gamma$ of a completely regular space $\Gamma$ is realized as the Gelfand spectrum of the Banach algebra $C(\Gamma)$. In particular, suppose $\Gamma$ is discrete, and $p\in\beta\Gamma$ is a nonzero multiplicative functional on $C(\Gamma)=\ell_{\infty}(\Gamma)$. For any $S\subset\Gamma$, we have $\langle p,1_{S}\rangle=\langle p,1_{S}\cdot 1_{S}\rangle=\langle p,1_{S}\rangle^{2}$. Thus $p$ may be regarded as a $\{0,1\}$-valued measure on $\Gamma$.

But, by definition, an ultrafilter on $\Gamma$ is nothing more than a set of the form $\{S\subset\Gamma:p(S)=1\}$ where $p$ is a nonzero $\{0,1\}$-valued measure. (Equivalently, it is a maximal collection of subsets of $\Gamma$ that is closed under finite intersections and does not contain $\varnothing$.) In this way, the combinatoric construction of $\beta\Gamma$ as the set of all ultrafilters on $\Gamma$ is identical to the Gelfand spectrum construction.

2.3**.**

In functional analysis, we embed $\Gamma$ in $\beta\Gamma$ by sending $\gamma$ to the point-measure $\delta_{\gamma}$, which corresponds to the principal ultrafilter $p_{\gamma}=\{S\subset\Gamma:\gamma\in S\}$. The defining feature of a principal ultrafilter is that its smallest element has cardinality 1. For any cardinal $\kappa$, we can define the $\kappa$-uniform ultrafilters as those whose smallest elements have cardinality $\kappa$. As we shall see, there are $2^{2^{|\Gamma|}}$ $|\Gamma|$-uniform ultrafilters on $\Gamma$. Of course, since $\beta\Gamma\subset\mathscr{P}(\mathscr{P}(\Gamma))$, there can’t be more than $2^{2^{|\Gamma|}}$ ultrafilters in total.

2.4**.**

Consider the case when $\Gamma$ is a directed set. Let $\Gamma^{*}$ denote the set of ultrafilters in $\beta\Gamma$ that include every tail $T_{\alpha}=\{\gamma\in\Gamma:\gamma>\alpha\}$, and suppose $|T_{\alpha}|=|\Gamma|=\kappa$ for each $\alpha\in\Gamma$. Notice that any finite intersection $T_{\gamma_{1}}\cap\ldots\cap T_{\gamma_{n}}$ contains $T_{\alpha}$, where $\alpha\geq\gamma_{1},\ldots,\gamma_{n}$, hence $|T_{\gamma_{1}}\cap\ldots\cap T_{\gamma_{n}}|=\kappa$.

Lemma 2.5**.**

$\Gamma^{*}$ has cardinality $2^{2^{\kappa}}$, the same as $\beta\Gamma$.

Proof.

This follows immediately from [HS12, Theorem 3.62], which actually says something a bit stronger:

Let $\Gamma$ be an infinite set with cardinality $\kappa$, and let $\mathcal{T}$ be a collection of at most $\kappa$ subsets of $\Gamma$ such that $|\bigcap F|=\kappa$ for any finite $F\subset\mathcal{T}$. Then there are $2^{2^{\kappa}}$ $\kappa$-uniform ultrafilters containing $\mathcal{T}$. ∎

3. Projections and Means

3.1**.**

A positive unital functional on a von Neumann algebra $X$ is called a state. A state $u$ is called normal if $\langle u,\sup_{\alpha}P_{\alpha}\rangle=\sup_{\alpha}\langle u,P_{\alpha}\rangle$ for any family $\{P_{\alpha}\}\subset X$ of (orthogonal) projections. Let $\mathcal{P}_{1}$ denote the set of normal states on $X$. Sakai famously proved that the linear span of $\mathcal{P}_{1}$ forms the predual of $X$ – that is, $\mathrm{span}(\mathcal{P}_{1})^{*}=X$. For each $u\in\mathcal{P}_{1}$, we can define a projection $S(u)$ called the support of $u$, which is the inf of all projections $P$ such that $\langle P,u\rangle=\langle I,u\rangle=1$. Now $u,v\in\mathcal{P}_{1}$ are called orthogonal if $S(u)S(v)=0$.

Lemma 3.2**.**

Suppose $\{u_{\gamma}\}_{\gamma\in\Gamma}\subset\mathcal{P}_{1}$ are mutually orthogonal. In other words, $S(u_{\gamma})S(u_{\beta})=0$ when $\gamma\neq\beta$. Then the map $\beta\Gamma\rightarrow X^{*}$ given by $p\mapsto\operatorname*{\mathit{p}-lim}_{\gamma}u_{\gamma}$ is one-to-one.

Proof.

Suppose $p,q$ are distinct ultrafilters, say $E\in p$ and $E^{C}\in q$. Let $P=\sup_{\gamma\in E}S(u_{\gamma})$. Clearly $\langle\operatorname*{\mathit{p}-lim}_{\gamma}u_{\gamma},P\rangle=1$. For any $\beta\in E^{C}$, $P\leq 1-S(u_{\beta})$, hence $\langle u_{\beta},P\rangle=0$, hence $\langle\operatorname*{\mathit{q}-lim}_{\gamma}u_{\gamma},P\rangle=0$. ∎

3.3**.**

For example, $L_{\infty}(G)$ is a von Neumann algebra of multipliers on $L_{2}(G)$, and its predual is $L_{1}(G)$. Let $\mathcal{P}_{1}(G)$ denote the normal states on $L_{\infty}(G)$. By Sakai’s result, the linear span of $\mathcal{P}_{1}(G)$ is $L_{1}(G)$, so $\mathcal{P}_{1}(G)$ itself must be $\{f\in L_{1}(G):f\geq 0,\|f\|_{1}=1\}$. Given $f\in\mathcal{P}_{1}(G)$, let $\operatorname{supp}(f)=\{x\in G:f(x)\neq 0\}$. Clearly $S(f)$ is the indicator function $1_{\operatorname{supp}(f)}$. From this, we conclude $f,g\in\mathcal{P}_{1}(G)$ are orthogonal if $\operatorname{supp}(f)\cap\operatorname{supp}(g)=\varnothing$.

3.4**.**

Let $X$ be any von Neumann algebra. Endow $X^{*}$ with the $w^{*}$-topology, and let $\mathcal{M}\subset X^{*}$ be the set of all states. In the context of amenability, $\mathcal{M}$ is traditionally called the set of means on $X$. Notice $\|m\|=1$ for each $m\in\mathcal{M}$, since $T\leq\|T\|I$ for any self-adjoint $T$. Notice $\mathcal{P}_{1}$ is convex and $\mathcal{M}$ is compact convex.

Lemma 3.5** (Hahn-Banach).**

Suppose $A,B\subset X^{*}$ are disjoint compact convex sets. Then they are separated by some $T\in X$, in the sense $\inf_{a\in A}\Re\langle a,T\rangle>\sup_{b\in B}\Re\langle b,T\rangle$. If $A,B$ consist of positive functionals, decompose $T$ into self-adjoint parts as $T_{1}+iT_{2}$. Then $\inf_{a\in A}\langle a,T_{1}\rangle>\sup_{b\in B}\langle b,T_{1}\rangle$.

Proof.

See any text on functional analysis, for example [Rud91, Theorems 3.4 and 3.10]. ∎

Lemma 3.6**.**

$\mathcal{P}_{1}$ is dense in $\mathcal{M}$.

Proof.

Suppose to the contrary that a mean $m$ lies outside $\operatorname{c\ell}(\mathcal{P}_{1})$. Applying Lemma 3.5 to the sets $\{m\}$ and $\operatorname{c\ell}(\mathcal{P}_{1})$, obtain self-adjoint $T\in X$ such that $\langle m,T\rangle>\sup_{u\in\mathcal{P}_{1}}\langle u,T\rangle$. Letting $S=T+\|T\|I\geq 0$, we have $\langle m,S\rangle>\sup_{u\in\mathcal{P}_{1}}\langle u,S\rangle=\|S\|$, contradicting $\|m\|=1$. ∎

Chapter 2 Topologically Invariant Means on $L_{\infty}(G)$

4. TI-nets in $\mathcal{P}_{1}(G)$

4.1**.**

In previous sections, $|E|$ has denoted the cardinal number of $E$. But when $E\subset G$, $|E|$ is understood to denote its left Haar measure. Integrals are always taken with respect to left Haar measure. The modular function $\triangle:G\rightarrow\mathbb{R}^{\times}$ is a continuous homomorphism defined by $|Ut|=|U|\triangle(t)$. The map $f\mapsto f^{\dagger}$ defined by $f^{\dagger}(x)=f(x^{-1})\triangle(x^{-1})$ is an involution of $L_{1}(G)$ that sends $\mathcal{P}_{1}(G)$ to itself. Left and right translation are defined by $l_{t}f(x)=f(t^{-1}x)$ and $r_{t}f(x)=f(xt)$, so that $l_{xy}=l_{x}l_{y}$ and $r_{xy}=r_{x}r_{y}$. Additionally, we define $R_{t}f(x)=f(xt^{-1})\triangle(t^{-1})$, so that $R_{t}\frac{1_{U}}{|U|}=\frac{1_{Ut}}{|Ut|}$. Notice $R_{xy}=R_{y}R_{x}$.

4.2**.**

For $T\in L_{\infty}(G)$ and $f,g\in L_{1}(G)$, define $fT$ by $\langle fT,g\rangle=\langle T,f*g\rangle$, and $Tf$ by $\langle Tf,g\rangle=\langle T,g*f\rangle$. Equivalently, $fT(t)=\langle R_{t}f,T\rangle$, and $Tf(t)=\langle l_{t}f,T\rangle$.

A mean $m$ is said to be topologically left invariant if $\langle m,fT\rangle=\langle m,T\rangle$ for all $f\in\mathcal{P}_{1}(G)$ and $T\in L_{\infty}(G)$. (This is equivalent to the traditional definition, $\langle m,f*T\rangle=\langle m,T\rangle$.) Likewise, $m$ is topologically right invariant if $\langle m,Tf\rangle=\langle m,T\rangle$, and topologically two-sided invariant if it is both of the above. The sets of topologically left/right/two-sided invariant means on $L_{\infty}(G)$ are denoted $\operatorname{TLIM}(G)/\operatorname{TRIM}(G)/\operatorname{TIM}(G)$.

4.3**.**

A net $\{f_{\gamma}\}\subset\mathcal{P}_{1}(G)$ is called a weak left TI-net if $\langle g*f_{\gamma}-f_{\gamma},T\rangle\rightarrow 0$ for all $g\in\mathcal{P}_{1}(G)$ and $T\in L_{\infty}(G)$. By Lemma 3.6, topologically left invariant means are precisely the limit points of weak left TI-nets. $\{f_{\gamma}\}$ is simply called a left TI-net if $\|g*f_{\gamma}-f_{\gamma}\|_{1}\rightarrow 0$ for all $g\in\mathcal{P}_{1}(G)$. The method of [RW01] can be used to construct weak TI-nets that are not TI-nets, which makes Corollary 4.7 interesting.

The analogous definitions and statements for right/two-sided TI-nets are obvious.

4.4**.**

As in 1.5, suppose $\mathcal{K}$ is a collection of compact sets covering $G$ with $|\mathcal{K}|=\kappa$. Let $\mathcal{K}^{\prime}$ be the set of all finite unions in $\mathcal{K}$. Pick any compact $U\subset G$ with nonempty interior. Then $\mathcal{K}^{\prime\prime}=\{UK:K\in\mathcal{K}^{\prime}\}$ is a collection of compact sets satisfying: (1) $|\mathcal{K}^{\prime\prime}|=\kappa$. (2) $\mathcal{K}^{\prime\prime}$ is closed under finite unions. (3) $\bigcup_{K\in\mathcal{K}^{\prime\prime}}K^{\circ}=G$.

Hence there is no loss of generality in supposing $\mathcal{K}$ itself satisfies (1)-(3). It follows that every compact $C\subset G$ is contained in some $K\in\mathcal{K}$.

4.5**.**

Suppose $G$ is noncompact amenable. Let $\Gamma=\{(K,n):K\in\mathcal{K},\ n\in\mathbb{N}\}$. Følner’s condition says that for each $\gamma=(K,n)$, we can pick a compact set $F_{\gamma}$ that is $(K,\frac{1}{n})$-left-invariant. In other words, letting $\lambda_{\gamma}=\frac{1_{F_{\gamma}}}{|F_{\gamma}|}$, we have $\|l_{x}\lambda_{\gamma}-\lambda_{\gamma}\|_{1}<\frac{1}{n}$ for each $x\in K$. Now $\Gamma$ becomes a directed set when endowed with the following partial-order: $[(K,m)\preceq(J,n)]\iff[K\subseteq J$ and $m\leq n]$. By condition (2) above, notice that each tail of $\Gamma$ has cardinality $\kappa\cdot\mathbb{N}=\kappa=|\Gamma|$, as required by Lemma 2.5.

Lemma 4.6**.**

$\{\lambda_{\gamma}\}$ is a left TI-net.

Proof.

Pick $f\in\mathcal{P}_{1}(G)$. Since compactly supported functions are dense in $\mathcal{P}_{1}(G)$, we may suppose $f$ has compact support $C$. Pick $K\in\mathcal{K}$ containing $C$.

Notice $\|f*\lambda_{\gamma}-\lambda_{\gamma}\|_{1}\leq\int\int f(t)|\lambda_{\gamma}(t^{-1}x)-\lambda_{\gamma}(x)|\>\mathrm{d}{x}\>\mathrm{d}{t}=\int f(t)\|l_{t}\lambda_{\gamma}-\lambda_{\gamma}\|_{1}\>\mathrm{d}{t}\leq\sup_{t\in C}\|l_{t}\lambda_{\gamma}-\lambda_{\gamma}\|_{1}$.

In particular, $\|f*\lambda_{\gamma}-\lambda_{\gamma}\|_{1}<1/n$ whenever $\gamma\succeq(K,n)$. ∎

Corollary 4.7**.**

Every topologically left invariant mean is the limit of a left TI-net. (Likewise for right/two-sided invariant means, although we omit the proof for those cases.)

Proof.

Suppose $m$ is not the limit of any left TI-net. Then there exist $T_{1},\ldots,T_{n}\in L_{\infty}(G)$, $f\in\mathcal{P}_{1}(G)$, and $\epsilon>0$ such that for any $g\in\mathcal{P}_{1}(G)$ with $\|f*g-g\|_{1}<\epsilon$, $\max_{i=1\ldots n}|\langle m-g,T_{i}\rangle|>\epsilon$.

Let $\gamma$ be large enough that $\|f*\lambda_{\gamma}-\lambda_{\gamma}\|_{1}<\epsilon$, and let $\lambda=\lambda_{\gamma}$.

By Lemma 3.6, pick $g\in\mathcal{P}_{1}(G)$ with $\max_{i=1\ldots n}|\langle m-g,T_{i}\rangle|<\epsilon$ and $\max_{i=1\ldots n}|\langle m-g,\lambda T_{i}\rangle|<\epsilon$.

Now $\|f*(\lambda*g)-(\lambda*g)\|_{1}<\epsilon$, so by hypothesis $|\langle m-\lambda*g,T_{i}\rangle|>\epsilon$ for some $i$.

But by choice of $g$, $|\langle m-\lambda*g,T_{i}\rangle|=|\langle m,T_{i}\rangle-\langle g,\lambda T_{i}\rangle|=|\langle m-g,\lambda T_{i}\rangle|<\epsilon$, a contradiction. ∎

Corollary 4.8**.**

Let $\rho_{\gamma}=\lambda_{\gamma}^{\dagger}$. Then $\{\rho_{\gamma}\}$ is a right TI-net.

Proof.

For any $f\in\mathcal{P}_{1}(G)$, $\|\rho_{\gamma}*f-\rho_{\gamma}\|_{1}=\|f^{\dagger}*\lambda_{\gamma}-\lambda_{\gamma}\|_{1}\rightarrow 0$. ∎

Lemma 4.9**.**

If $\{f_{\gamma}\}$ is a left TI-net, and $\{g_{\gamma}\}$ is a right TI-net, then $\{f_{\gamma}*g_{\gamma}\}$ is a two-sided TI-net.

Proof.

First of all, $f_{\gamma}*g_{\gamma}\geq 0$ and $\|f_{\gamma}*g_{\gamma}\|_{1}=1$, so $\{f_{\gamma}*g_{\gamma}\}\subset\mathcal{P}_{1}(G)$. Now for each $h\in\mathcal{P}_{1}(G)$, $\|h*f_{\gamma}*g_{\gamma}-f_{\gamma}*g_{\gamma}\|\leq\|h*f_{\gamma}-f_{\gamma}\|\cdot\|g_{\gamma}\|\rightarrow 0$ and $\|f_{\gamma}*g_{\gamma}*h-f_{\gamma}*g_{\gamma}\|\leq\|f_{\gamma}\|\cdot\|g_{\gamma}*h-g_{\gamma}\|\rightarrow 0$. ∎

4.10**.**

The next lemma generalizes [Cho70, Theorem 3.2]. Intuitively it says, “we can prove facts about the entire set $\operatorname{TLIM}(G)$, simply by proving them about the right-translates of a single left TI-net.” After that, we have generalizations to $\operatorname{TRIM}(G)$ and $\operatorname{TIM}(G)$.

Lemma 4.11**.**

For any $p\in\Gamma^{*}$, let $X_{p}=\{\operatorname*{\mathit{p}-lim}_{\gamma}[R_{t_{\gamma}}\lambda_{\gamma}]:\{t_{\gamma}\}\in G^{\Gamma}\}.$ Then $\operatorname{c\ell}(\operatorname{conv}(X_{p}))=\operatorname{TLIM}(G)$.

Proof.

Suppose $m\in\operatorname{TLIM}(G)$ lies outside the closed convex hull of $X_{p}$.

As in the proof of Lemma 3.6, there exists $T\in L_{\infty}(G,\mathbb{R})$ with $\langle m,T\rangle>\sup_{\nu\in X_{p}}\langle\nu,T\rangle$.

For each $\gamma$, $\langle m,T\rangle=\langle m,\lambda_{\gamma}T\rangle\leq\|\lambda_{\gamma}T\|_{\infty}=\sup_{t}[\lambda_{\gamma}T(t)]=\sup_{t}\langle R_{t}\lambda_{\gamma},\ T\rangle$.

In particular, if $\gamma=(K,n)$, choose $t_{\gamma}$ so that $\langle m,T\rangle<\langle R_{t_{\gamma}}\lambda_{\gamma},T\rangle+1/n$.

Now $\langle m,T\rangle\leq\langle\operatorname*{\mathit{p}-lim}_{\gamma}[R_{t_{\gamma}}\lambda_{\gamma}],T\rangle$, contradicting our choice of $m$. ∎

Lemma 4.12**.**

For any $p\in\Gamma^{*}$, let $X_{p}=\{\operatorname*{\mathit{p}-lim}_{\gamma}[l_{t_{\gamma}}\rho_{\gamma}]:\{t_{\gamma}\}\in G^{\Gamma}\}.$ Then $\operatorname{c\ell}(\operatorname{conv}(X_{p}))=\operatorname{TRIM}(G)$.

Proof.

Essentially the same as the proof of Lemma 4.11 but the third line becomes:

For each $\gamma$, $\langle m,T\rangle=\langle m,T\rho_{\gamma}\rangle\leq\|T\rho_{\gamma}\|_{\infty}=\sup_{t}[T\rho_{\gamma}(t)]=\sup_{t}\langle l_{t}\rho_{\gamma},\ T\rangle$. ∎

Lemma 4.13**.**

For any $p\in\Gamma^{*}$, let $X_{p}=\{\operatorname*{\mathit{p}-lim}_{\gamma}[\lambda_{\gamma}*(l_{t_{\gamma}}\rho_{\gamma})]:\{t_{\gamma}\}\in G^{\Gamma}\}$. Then $\operatorname{c\ell}(\operatorname{conv}(X_{p}))=\operatorname{TIM}(G)$.

Proof.

Essentially the same as the proof of Lemma 4.11 but the third line becomes:

For each $\gamma$, $\langle m,T\rangle=\langle m,\lambda_{\gamma}T\rho_{\gamma}\rangle\leq\|\lambda_{\gamma}T\rho_{\gamma}\|_{\infty}=\sup_{t}[\lambda_{\gamma}T\rho_{\gamma}(t)]=\sup_{t}\langle\lambda_{\gamma}*(l_{t}\rho_{\gamma}),\ T\rangle$. ∎

Lemma 4.14**.**

There exists $\{t_{\gamma}\}\in G^{\Gamma}$ such that $\{F_{\gamma}t_{\gamma}F_{\gamma}^{-1}\}$ are mutually disjoint. Since $\operatorname{supp}(\lambda_{\gamma}*(l_{t}\rho_{\gamma}))\subset F_{\gamma}tF_{\gamma}^{-1}$, it follows from 3.3 and Lemma 4.9 that $\{\lambda_{\gamma}*(l_{t}\rho_{\gamma})\}$ is an orthogonal TI-net.

Proof.

Since $|\Gamma|=\kappa$, let $(\Gamma,<)$ denote the well-ordering of $\Gamma$ induced by some bijection with $\kappa$. Let $t_{0}=e$. As induction hypothesis, suppose $\{F_{\gamma}t_{\gamma}F_{\gamma}^{-1}\}_{\gamma<\alpha}$ are disjoint, where $\alpha\in\Gamma$. If it is impossible to find $t_{\alpha}$ such that $\{F_{\gamma}t_{\gamma}F_{\gamma}^{-1}\}_{\gamma\leq\alpha}$ are disjoint, then $\{F_{\alpha}^{-1}F_{\gamma}t_{\gamma}F_{\gamma}^{-1}F_{\alpha}\}_{\gamma<\alpha}$ is a collection of compact sets of cardinality less than $\kappa$ covering $G$, a contradiction. ∎

Lemma 4.15**.**

Let $K\subset G$ be any compact set with nonempty interior, and $\{Kt_{\gamma}\}_{\gamma<\kappa}$ be a covering of $G$ by translates of $K$. For each $n$, let $\lambda_{n}\in\mathcal{P}_{1}(G)$ be $(K^{-1},\frac{1}{n})$-left-invariant, and let $\rho_{n}=\lambda_{n}^{\dagger}$.

Finally, let $X=\left\{R_{t_{\gamma}}\rho_{n}:n\in\mathbb{N},\ \gamma<\kappa\right\}$. Then $\operatorname{TLIM}(G)\subset\operatorname{c\ell}(\operatorname{conv}(X))$, and $|\operatorname{c\ell}(\operatorname{conv}(X))|\leq 2^{2^{\kappa}}$.

Proof.

Suppose $m\in\operatorname{TLIM}(G)$ lies outside the closed convex hull of $X$. As in Lemma 3.6, there exists $T\in L_{\infty}(G,\mathbb{R})$ and $\epsilon>0$ with $\langle m,T\rangle>2\epsilon+\langle R_{t_{\gamma}}\rho_{n},T\rangle$ for all $n\in\mathbb{N}$ and all $\gamma<\kappa$.

Let $n$ be large enough that $\|T\|_{\infty}/n<\epsilon$.

Let $s$ be chosen so that $\sup_{t}\langle R_{t}\rho_{n},T\rangle<\epsilon+\langle R_{s}\rho_{n},T\rangle$. Say $s\in Kt_{\alpha}$.

By definition of $\rho_{n}$, $\langle R_{s}\rho_{n},T\rangle<\langle R_{t_{\alpha}}\rho_{n},T\rangle+\|T\|_{\infty}/n$.

Now $\langle m,T\rangle=\langle m,\rho_{n}T\rangle\leq\|\rho_{n}T\|_{\infty}=\sup_{t}\langle R_{t}\rho_{n},T\rangle<2\epsilon+\langle R_{t_{\alpha}}\rho_{n},T\rangle$ contradicting the choice of $m$.

This proves $\operatorname{TLIM}(G)\subset\operatorname{c\ell}(\operatorname{conv}(X))$.

Let $\operatorname{conv}_{\mathbb{Q}}(X)$ denote the set of all finite convex combinations in $X$ with rational coefficients. Evidently $|\operatorname{conv}_{\mathbb{Q}}(X)|=|X|=\kappa$, and $\operatorname{c\ell}(\operatorname{conv}(X))=\operatorname{c\ell}(\operatorname{conv}_{\mathbb{Q}}(X))$. Since $\operatorname{c\ell}(\operatorname{conv}(X))$ is a regular Hausdorff space with dense subset $\operatorname{conv}_{\mathbb{Q}}(X)$, it has cardinality at most $2^{2^{|\operatorname{conv}_{\mathbb{Q}}(X)|}}=2^{2^{\kappa}}$. ∎

Theorem 4.16**.**

$|\operatorname{TIM}(G)|=|\operatorname{TLIM}(G)|=2^{2^{\kappa}}$.

Proof.

By Lemma 4.14, there exists be an orthogonal TI-net $\{\varphi_{\gamma}\}$. By Lemma 3.2, the map $p\mapsto\operatorname*{\mathit{p}-lim}_{\gamma}\varphi_{\gamma}$ is one-to-one from $\Gamma^{*}$ to $\operatorname{TIM}(G)$, so $|\operatorname{TIM}(G)|\geq|\Gamma^{*}|$. By Lemma 2.5, $|\Gamma^{*}|=2^{2^{\kappa}}$. Of course, $\operatorname{TIM}(G)\subset\operatorname{TLIM}(G)$, so $2^{2^{\kappa}}\leq|\operatorname{TIM}(G)|\leq|\operatorname{TLIM}(G)|$. Lemma 4.15 yields the opposite inequality. ∎

5. A Proof of Paterson’s Conjecture

Let $t^{g}$ be shorthand for $gtg^{-1}$. We write $G\in[\operatorname{FC}]^{-}$ to signify that each conjugacy class $t^{G}=\{t^{g}:g\in G\}$ is precompact. When $G\in[FC]^{-}$ is furthermore discrete, each conjugacy class must be finite. In this case, we write $G\in\operatorname{FC}$.

Lemma 5.1**.**

If $G\in\operatorname{FC}$, then $G$ is amenable.

Proof.

It suffices to show that each finitely-generated subgroup is amenable. Suppose $K\subset G$ is finite, and let $\langle K\rangle$ denote the subgroup generated by $K$. For any $x\in\langle K\rangle$, let $C(x)=\{y\in\langle K\rangle:x^{y}=x\}$. Evidently $C(x)$ is a subgroup, whose right cosets correspond to the (finitely many) distinct values of $x^{y}$. Therefore $[\langle K\rangle:C(x)]$ is finite. Let $Z$ denote the center of $\langle K\rangle$. Clearly $Z=\bigcap_{x\in K}C(x)$. Thus $[\langle K\rangle:Z]\leq\prod_{x\in K}[\langle K\rangle:C(x)]$ is finite. Since $\langle K\rangle$ is finite-by-abelian, it is amenable. ∎

Theorem 5.2**.**

$G\in[\operatorname{FC}]^{-}$ iff $G$ is a compact extension of $\mathbb{R}^{n}\times D$, for some $D\in\operatorname{FC}$ and $n\in\mathbb{N}$.

Proof.

This is [Liu73, Theorem 2.2]. ∎

Corollary 5.3**.**

If $G\in[\operatorname{FC}]^{-}$, then $G$ is amenable.

Proof.

In light of Theorem 5.1, any group of the form $\mathbb{R}^{n}\times D$ is amenable.

Compact extensions of amenable groups are amenable. ∎

Corollary 5.4**.**

If $G\in[\operatorname{FC}]^{-}$, then $G$ is unimodular.

Proof.

Clearly groups of the form $\mathbb{R}^{n}\times D$ are unimodular, since $D$ is discrete.

Compact extensions of unimodular groups are unimodular. ∎

Lemma 5.5**.**

If $G\in[\operatorname{FC}]^{-}$, and $C\subset G$ is compact, then $C^{G}=\{c^{g}:c\in C,g\in G\}$ is precompact.

In [Mil81], Milnes obverves that this lemma would imply Theorem 5.7. He is unable to prove it, apparently because he is unaware of Theorem 5.2.

Proof.

By Theorem 5.2, let $G/K=\mathbb{R}^{n}\times D$, where $K\vartriangleleft G$ is a compact normal subgroup.

Let $\pi:G\rightarrow G/K$ denote the canonical epimorphism.

Pick $C\subset G$ compact. Notice $C^{G}$ is precompact if $\pi(C^{G})$ is, because the kernel of $\pi$ is compact.

Since $\pi(C)$ is compact, $\pi(C)\subset B\times F$ for some box $B\subset\mathbb{R}^{n}$ and finite $F\subset D$.

Now $\pi(C^{G})=\pi(C)^{\pi(G)}\subset(B\times F)^{\mathbb{R}^{n}\times D}=B\times F^{D}$.

Evidently $F^{D}$ is finite, hence $B\times F^{D}$ is compact, proving $\pi(C^{G})$ is precompact. ∎

5.6**.**

As in 4.5, let $\{F_{\gamma}\}_{\gamma\in\Gamma}$ be a Følner net for $G$, and $\lambda_{\gamma}=1_{F_{\gamma}}/|F_{\gamma}|$ the corresponding left TI-net. Assuming $G$ is unimodular, by [Cho70, Theorem 4.4] we can choose $F_{\gamma}$ to be symmetric. Hence $\lambda_{\gamma}(x)=\lambda_{\gamma}(x^{-1})=\lambda_{\gamma}^{\dagger}(x)$, and $\{\lambda_{\gamma}\}$ is a two-sided TI-net.

Theorem 5.7** (Following Milnes).**

If $G\in[\operatorname{FC}]^{-}$ then $\operatorname{TLIM}(G)\subset\operatorname{TIM}(G)$.

Proof.

By Corollary 5.4, $G$ is unimodular. As above, take $\{\lambda_{\gamma}\}=\{\lambda_{\gamma}^{\dagger}\}$ to be a TI-net.

Recall Lemma 4.11, which says $\operatorname{c\ell}(\operatorname{conv}(X_{p}))=\operatorname{TLIM}(G)$. So it suffices to prove $X_{p}\subset\operatorname{TIM}(G)$.

To this end, we will show $\{R_{t_{\gamma}}\lambda_{\gamma}\}$ is a right TI-net for any $\{t_{\gamma}\}\in G^{\Gamma}$. Let $x_{\gamma}$ be shorthand for $t_{\gamma}xt_{\gamma}^{-1}$.

Now $\|r_{x}R_{t_{\gamma}}\lambda_{\gamma}-R_{t_{\gamma}}\lambda_{\gamma}\|_{1}=\frac{|F_{\gamma}t_{\gamma}x^{-1}\Delta F_{\gamma}t_{\gamma}|}{|F_{\gamma}t_{\gamma}|}=\frac{|F_{\gamma}x_{\gamma}^{-1}\Delta F_{\gamma}|}{|F_{\gamma}|}=\|r_{x_{\gamma}}\lambda_{\gamma}-\lambda_{\gamma}\|_{1}=\|l_{x_{\gamma}}\lambda_{\gamma}-\lambda_{\gamma}\|_{1}$.

For any compact $C\subset G$, notice $\{x_{\gamma}:x\in C,\gamma\in\Gamma\}\subset C^{G}$, which is precompact by Lemma 5.5.

Hence $\sup_{x\in C}\|r_{x}R_{t_{\gamma}}\lambda_{\gamma}-R_{t_{\gamma}}\lambda_{\gamma}\|_{1}\leq\sup_{y\in C^{G}}\|l_{y}\lambda_{\gamma}-\lambda_{\gamma}\|\rightarrow 0$.

By the same argument as Lemma 4.6, $\|(R_{t_{\gamma}}\lambda_{\gamma})*f-R_{t_{\gamma}}\lambda_{\gamma}\|_{1}\rightarrow 0$ for any $f\in\mathcal{P}_{1}(G)$. ∎

Lemma 5.8**.**

If $x^{G}$ is not precompact, then there is a $\sigma$-compact open subgroup $H\leq G$ such that $x^{H}$ is not precompact.

Proof.

Let $U$ be any compact neigborhood of $t_{0}=e$. Inductively pick $t_{n+1}$ such that $x^{t_{n+1}}U\cap\{x^{t_{0}},\ldots,x^{t_{n}}\}U=\varnothing$. This is possible, otherwise $x^{G}$ is contained in the compact set $\{x^{t_{0}},\ldots,x^{t_{n}}\}UU^{-1}$, a contradiction. Now the sequence $\{x^{t_{0}},x^{t_{1}},\ldots\}$ does not accumulate anywhere, so it escapes any compact set. Take $H$ to be the subgroup generated by $\{x^{t_{0}},x^{t_{1}},\ldots\}U$. ∎

Theorem 5.9**.**

If $G$ has an element $x$ such that $x^{G}$ is not precompact, then $\operatorname{TLIM}(G)\neq\operatorname{TIM}(G)$.

Proof.

Let $H\leq G$ be any $\sigma$-compact open subgroup such that $x^{H}$ is not precompact. Pick a sequence of compact sets $\{H_{n}\}$ such that $\forall n$ $H_{n}\subset H_{n+1}^{\circ}$, and $\bigcup_{n}H_{n}=H$. It follows that any compact $K\subset H$ is contained in some $H_{n}$.

Inductively construct a sequence $\{c_{n}\}\subset H$ satisfying the following properties:

(1) $c_{n}\in H\setminus K_{n}$, where $K_{n}=\bigcup_{m<n}H_{n}^{-1}H_{m}c_{m}\{x,x^{-1}\}$.

(2) $(c_{n}xc_{n}^{-1})\not\in H_{n}^{-1}H_{n}$.

Since $K_{n}$ is compact, $x^{H\setminus K_{n}}$ is not precompact, and in particular escapes $H_{n}^{-1}H_{n}$.

Thus it is possible to satisfy (1) and (2) simultaneously.

Now $A=\bigcup_{n}H_{n}c_{n}$ and $B=\bigcup_{n}H_{n}c_{n}x$ are easily seen to be disjoint:

If they are not disjoint, then $H_{n}c_{n}x\cap H_{m}c_{m}\neq\varnothing$ for some $n,m$.

If $n>m$, then $c_{n}\in H_{n}^{-1}H_{m}c_{m}x^{-1}$, violating (1).

If $n<m$, then $c_{m}\in H_{m}^{-1}H_{n}c_{n}x$, violating (1).

If $n=m$, then $c_{n}xc_{n}^{-1}\in H_{n}^{-1}H_{n}$, violating (2).

Let $T$ be a transversal for $G/H$. Notice $TA\cap TB=\varnothing$, since $A,B$ are disjoint subsets of $H$.

Define $\pi:TH\rightarrow H$ by $\pi(th)=h$, which is continuous since $H$ is open.

Let $\{F_{\gamma}\}$ be a Følner net for $G$. Each $F_{\gamma}$ is compact, hence $\pi[F_{\gamma}]\subset H_{n(\gamma)}$ for some $n(\gamma)$. Let $c_{\gamma}=c_{n(\gamma)}$. Now $\pi[F_{\gamma}c_{\gamma}]=\pi[F_{\gamma}]c_{\gamma}\subset A$, hence $F_{\gamma}c_{\gamma}\subset\pi^{-1}[A]=TA$. Likewise $F_{\gamma}c_{\gamma}x\subset TB$.

Since this holds for each $\gamma$, $C=\bigcup_{\gamma}F_{\gamma}t_{\gamma}\subset TA$ and $D=\bigcup_{\gamma}F_{\gamma}t_{\gamma}x\subset TB$. Thus $C\cap D=\varnothing$.

Let $m=\operatorname*{\mathit{p}-lim}_{\gamma}\left[R_{t_{\gamma}}\frac{1_{F_{\gamma}}}{|F_{\gamma}|}\right]$ for some $p\in\Gamma^{*}$.

Now $m(1_{C})=1$, but $m(r_{x}1_{C})=m(1_{D})=0$, hence $m\in\operatorname{TLIM}(G)\setminus\operatorname{TIM}(G)$. ∎

Chapter 3 Topologically Invariant means on $VN(G)$

6. Background on Fourier Algebra

The algebras $A(G)$ and $VN(G)$ are defined and studied in the influential paper [Eym64]. Chapter 2 of [KL18] gives a modern treatment of the same material in English.

6.1**.**

Let $\lambda$ denote the left-regular representation. By definition, $VN(G)=\lambda[G]^{\prime\prime}=\lambda[L_{1}(G)]^{\prime\prime}$. $\mathcal{P}_{1}(\widehat{G})$ denotes the set of normal states on $VN(G)$, and the linear span of $\mathcal{P}_{1}(\widehat{G})$ is denoted by $A(G)$. It turns out that $\mathcal{P}_{1}(\widehat{G})$ consists of all vector states of the form $\omega_{f}(T)=\langle Tf,f\rangle$ where $\|f\|_{2}=1$. By the polarization identity, $A(G)$ consists of all functionals of the form $\omega_{f,g}(T)=\langle Tf,g\rangle$, with $f,g\in L_{2}(G)$.

6.2**.**

$A(G)$ can also be characterized as the set of all functions $u_{f,g}(x)=\langle\lambda(x)f,g\rangle=\overline{g}*\check{f}(x)$ with $f,g\in L_{2}(G)$. As such, the elements of $A(G)$ are continuous and vanish at infinity. The duality with $VN(G)$ is given by $\langle T,u_{f,g}\rangle=\langle Tf,g\rangle$. In particular, $\langle\lambda(h),u\rangle=\int h(x)u(x)\>\mathrm{d}{x}$ for all $h\in L_{1}(G)$.

$A(G)$ is a commutative Banach algebra with pointwise operations and norm $\|u\|=\sup_{\|T\|=1}|\langle T,u\rangle|$. Equivalently, $\|u\|$ is the sup of $\left|\int h(x)u(x)\>\mathrm{d}{x}\right|$ over all $h\in L_{1}(G)$ such that $\|\lambda(h)\|\leq 1$. Since $\|\lambda(h)\|\leq\|h\|_{1}$, we conclude $\|u\|\geq\|u\|_{\infty}$.

6.3**.**

If $u\in A(G)$ is positive as a functional on $VN(G)$, we write $u\geq 0$. It’s easy to see $u\geq 0$ iff $u=u_{f,f}$ for some $f\in L_{2}(G)$ iff $\|u\|_{\infty}=u(e)=\langle I,u\rangle=\|u\|$. In particular, $\mathcal{P}_{1}(\widehat{G})=\{u\in A(G):\|u\|=u(e)=1\}$ is closed under multiplication. Suppose $\|f\|_{2}=\|g\|_{2}=1$. Then $\|u_{f,f}-u_{g,g}\|=\sup_{\|T\|=1}|\langle Tf,f\rangle-\langle Tg,g\rangle|\leq\sup_{\|T\|=1}|\langle T(f-g),f\rangle|+|\langle Tg,(f-g)\rangle|\leq 2\|f-g\|_{2}$.

7. TI-nets in $\mathcal{P}_{1}(\widehat{G})$

7.1**.**

For $T\in VN(G)$ and $u,v\in A(G)$, define $uT$ by $\langle uT,v\rangle=\langle T,uv\rangle$. A mean $m$ is said to be topologically invariant if $\langle m,uT\rangle=\langle m,T\rangle$ for all $u\in\mathcal{P}_{1}(\widehat{G})$ and $T\in VN(G)$. $\operatorname{TIM}(\widehat{G})$ denotes the set of topologically invariant means on $VN(G)$. There is no distinction between left/right topologically invariant means, since multiplication in $\mathcal{P}_{1}(\widehat{G})$ is commutative.

7.2**.**

A net $\{u_{\gamma}\}\subset\mathcal{P}_{1}(\widehat{G})$ is called a weak TI-net if $\lim_{\gamma}\langle vu_{\gamma}-u_{\gamma},T\rangle=0$ for all $v\in\mathcal{P}_{1}(\widehat{G})$ and $T\in VN(G)$. By Lemma 3.6, topologically invariant means are precisely the limit points of weak TI-nets. $\{u_{\gamma}\}$ is simply called a TI-net if $\lim_{\gamma}\|vu_{\gamma}-u_{\gamma}\|=0$ for all $v\in\mathcal{P}_{1}(\widehat{G})$.

Lemma 7.3**.**

Pick any net $\{u_{\gamma}\}\subset\mathcal{P}_{1}(\widehat{G})$. Suppose that $\operatorname{supp}(u_{\gamma})$ is eventually small enough to fit in any $V\in\mathscr{C}(G)$. Then $\{u_{\gamma}\}$ is a TI-net.

Proof.

This is [Ren72, Proposition 3]. I’ll repeat the proof for convenience.

Pick $u\in\mathcal{P}_{1}(\widehat{G})$. Since compactly supported functions are dense in $A(G)$, it suffices to consider the case when $u$ has compact support $C$. $A(G)$ is regular by [KL18, Proposition 2.3.2], so we can pick $v\in A(G)$ with $v\equiv 1$ on $C$. Since $u-v(e)=0$, by [KL18, Lemma 2.3.7] we can find some $w\in A(G)$ with $\|(u-v)-w\|<\epsilon$ and $w\equiv 0$ on some neighborhood $W$ of $e$. Suppose $\gamma$ is large enough that $\operatorname{supp}(u_{\gamma})\subset W\cap C$. Then $vu_{\gamma}=u_{\gamma}$, and $wu_{\gamma}=0$, hence $\|uu_{\gamma}-u_{\gamma}\|=\|(u-v)u_{\gamma}\|\leq\|(u-v-w)u_{\gamma}\|+\|wu_{\gamma}\|\leq\|(u-v-w)\|<\epsilon$. ∎

7.4**.**

Let $\mathscr{C}(G)$ denote the compact, symmetric neighborhoods of $e$. As a consequence of Lemma 7.3, we see that $\mathcal{P}_{1}(\widehat{G})$ has TI-nets for any $G$, in contrast to $\mathcal{P}_{1}(G)$ which has TI-nets only when $G$ is amenable: If $\{U_{\gamma}\}\subset\mathscr{C}(G)$ is a neighborhood basis at $e$, directed by inclusion, let $u_{\gamma}=(1_{U_{\gamma}}*1_{U_{\gamma}})/|U_{\gamma}|$. Since $\operatorname{supp}(u_{\gamma})\subset U_{\gamma}^{2}$, and $\{U_{\gamma}^{2}\}$ is a neighborhood basis at $e$, $\{u_{\gamma}\}$ is a TI-net.

Lemma 7.3 has a sort of converse, given by the following lemma.

Lemma 7.5**.**

Every $m\in\operatorname{TIM}(\widehat{G})$ is the limit of a net $\{u_{\gamma}\}$ such that $\operatorname{supp}(u_{\gamma})$ is eventually small enough to fit in any $V\in\mathscr{C}(G)$.

Proof.

Suppose to the contrary that there exist $V\in\mathscr{C}(G)$, $T\in VN(G)$, and $\epsilon>0$ such that $|\langle m-u,T\rangle|>\epsilon$ for all $u\in\mathcal{P}_{1}(\widehat{G})$ with $\operatorname{supp}(u)\subset V$. Fix any $u\in\mathcal{P}_{1}(\widehat{G})$ with $\operatorname{supp}(u)\subset V$. By Lemma 3.6, pick $v\in\mathcal{P}_{1}(\widehat{G})$ with $|\langle m-v,T\rangle|<\epsilon$, and $|\langle m-v,uT\rangle|<\epsilon$. Now $vu\in\mathcal{P}(\widehat{G})$ with $\operatorname{supp}(vu)\subset V$. But $|\langle m-vu,T\rangle|=|\langle m,T\rangle-\langle vu,T\rangle|=|\langle m,uT\rangle-\langle v,uT\rangle|=|\langle m-v,uT\rangle|<\epsilon$, a contradiction. ∎

Corollary 7.6**.**

Every $m\in\operatorname{TIM}(\widehat{G})$ is the limit of a TI-net.

Corollary 7.7**.**

$|\operatorname{TIM}(\widehat{G})|\leq 2^{2^{\mu}}$

Proof.

Recall that $\mu$ is the minimal cardinality of a neighborhood basis at $e$. Pick any $V\in\mathscr{C}(G)$. By the previous lemma, $\operatorname{TIM}(\widehat{G})$ is in the closure of $X=\{u\in\mathcal{P}_{1}(\widehat{G}):\operatorname{supp}(u)\subset V\}$. Therefore, it suffices to find a set $D$ with $|D|=\mu$ and $X\subset\operatorname{c\ell}(D)$.

Let $B=\{f\in L_{2}(G):\|f\|_{2}=1,\ \operatorname{supp}(f)\subset V\}$, and let $C$ comprise the continuous functions in $B$. Obviously $C$ is dense in $B$, and it is routine to show that $C$ has a dense subset $C^{\prime}$ of cardinality $\mu$. Now $D=\{u_{f,f}:f\in C^{\prime}\}$ is the desired set. ∎

7.8**.**

Let $u_{f,f}$ be a normal state on $VN(G)$, and $P$ the projection onto the linear span of $f$. Since $P$ has rank one, it is clearly the inf of all projections $Q$ such that $1=\langle Qf,f\rangle=\langle Q,u_{f,f}\rangle$. In the terminology of 3.1, $P$ is the support of $u_{f,f}$. Thus a family $\{u_{f_{\alpha},f_{\alpha}}\}\subset A(G)$ is orthogonal iff $\{f_{\alpha}\}\subset L_{2}(G)$ is orthogonal.

Apparently neither Chou nor Hu made this observation. If it seems obvious, it is because we took care to express normal states as $u_{f,f}$. For example, in order to construct an orthogonal TI-sequence, [Cho82, pages 210-213] takes three pages to describe a generalized Gram-Schmidt procedure for normal states on a von Neumann algebra. This procedure does not generalize beyond sequences, and [Hu95] is unable to construct an orthogonal net when $\mu>\mathbb{N}$.

Lemma 7.9**.**

When $\mu=\mathbb{N}$, $\mathcal{P}_{1}(\widehat{G})$ contains an orthogonal TI-sequence.

Proof.

Let $\{U_{n}\}\subset\mathscr{C}(G)$ be a descending neighborhood basis at $e$, such that $|U_{n}\setminus U_{n+1}|>0$. Let $V_{n}=U_{n}\setminus U_{n+1}$, and $f_{n}=1_{V_{n}}/|V_{n}|^{1/2}$. In light of Lemma 7.3 and 7.8, $\{u_{f_{n},f_{n}}\}$ is the desired sequence. ∎

Lemma 7.10**.**

Let $\mathcal{U}\subset\mathscr{C}(G)$. If $\bigcap\mathcal{U}=\{e\}$, then $\mathcal{U}$ is a neighborhood sub-basis at $e$ (hence $|\mathcal{U}|\geq\mu$).

Proof.

Suppose $\bigcap\mathcal{U}=\{e\}$, and let $V$ be any open neighborhood of $e$. Since $\varnothing=\{e\}\cap V^{C}=\bigcap_{U\in\mathcal{U}}[U\cap V^{C}]$ is an intersection of compact sets, it follows that some finite sub-intersection is also empty. In other words, $U_{1}\cap\ldots\cap U_{n}\subset V$ for some $U_{1},\ldots,U_{n}\in\mathcal{U}$. ∎

Lemma 7.11** (Kakutani-Kodaira).**

Let $\mathcal{U}\subset\mathscr{C}(G)$. Suppose each $U\in\mathcal{U}$ has a “successor” $V\in\mathcal{U}$, such that $V^{2}\subset U$. Then $H=\bigcap\mathcal{U}$ is a compact subgroup of $G$.

Proof.

$H$ is compact, because it is the intersection of compact sets in a Hausdorff space.

Suppose $x,y\in H$. Suppose $U,V\in\mathcal{U}$ with $V^{2}\subset U$. Since $x,y\in H\subset V$, $xy\in V^{2}\subset U$.

Since $U$ was arbitrary, we conclude $xy\in H$. Likewise, $x^{-1}\in H$ because each $U$ is symmetric. ∎

7.12**.**

Suppose $H$ is a compact subgroup of $G$, with normalized Haar measure $\nu$. Let $L^{2}(H\backslash G)$ be the set of functions in $L^{2}(G)$ that are constant on right-cosets of $H$. Define $Pf(x)=\int f(hx)\>\mathrm{d}\nu(h)$. Now it is routine to check that $P$ is the orthogonal projection onto $L^{2}(H\backslash G)$. The only interesting detail, needed to prove $P=P^{*}$, is that $\int f(hx)\>\mathrm{d}\nu(h)=\int f(h^{-1}x)\>\mathrm{d}\nu(h)$ because $H$ is unimodular.

7.13**.**

Suppose $\{H_{\gamma}\}=\{H_{\gamma}\}_{\gamma<\mu}$ is a descending chain of compact subgroups. Let $P_{\gamma}$ denote the orthogonal projection onto $L^{2}(H_{\gamma}\backslash G)$. Then $\{P_{\gamma}\}$ is an ascending chain of projections, and $\{P_{\gamma+1}-P_{\gamma}\}$ is a chain of mutually orthogonal projections. Suppose we construct functions $\{f_{\gamma}\}\subset L_{2}(G)$ with $f_{\gamma}\in L^{2}(H_{\gamma+1}\backslash G)-L^{2}(H_{\gamma}\backslash G)$, so that $(P_{\gamma+1}-P_{\gamma})f_{\gamma}\neq 0$. Letting $g_{\gamma}=(P_{\gamma+1}-P_{\gamma})f_{\gamma}$, we see $\{u_{\gamma}\}=\{u_{g_{\gamma},g_{\gamma}}/\|g_{\gamma}\|_{2}^{2}\}$ is a chain of mutually orthogonal functions in $\mathcal{P}_{1}(\widehat{G})$, since $S(u_{\gamma})\leq P_{\gamma+1}-P_{\gamma}$.

Consider the condition $f_{\gamma}\in L^{2}(H_{\gamma+1}\backslash G)-L^{2}(H_{\gamma}\backslash G)$. How can we achieve this? Let $\nu_{\gamma}$ denote the Haar measure of $H_{\gamma}$. Pick $U_{\gamma}\in\mathscr{C}(G)$ small enough that $\nu_{\gamma}(U_{\gamma}^{4}\cap H_{\gamma})<1$. Construct $H_{\gamma+1}$ small enough that $H_{\gamma+1}\subset U_{\gamma}$, and let $f_{\gamma}=1_{H_{\gamma+1}U_{\gamma}}$. Obviously $f_{\gamma}\in L^{2}(H_{\gamma+1}\backslash G)$. On the other hand, $P_{\gamma}f_{\gamma}(x)=\nu_{\gamma}(\{h\in H_{\alpha}:hx\in H_{\gamma+1}U_{\gamma}\})<1=f_{\gamma}(x)$ for any $x\in H_{\gamma+1}U_{\gamma}$. Hence $P_{\gamma}f_{\gamma}\neq f_{\gamma}$, and $f_{\gamma}\not\in L^{2}(H_{\gamma}\backslash G)$.

We need to add one more detail to our construction, to make the chain of orthogonal functions $\{u_{\gamma}\}$ into a TI-net. Namely, suppose $\{V_{\gamma}\}\subset\mathscr{C}(G)$ is a neighborhood basis at $e$, and $\{U_{\gamma},H_{\gamma}\}$ satisfy $H_{\gamma}U_{\gamma}^{2}\subset V_{\gamma}$. Then $\operatorname{supp}(g_{\gamma})\subset H_{\gamma}H_{\gamma+1}U_{\gamma}\subset V_{\gamma}$, hence $\operatorname{supp}(u_{\gamma})\subset V_{\gamma}^{2}$. Let $\Gamma$ denote the set of ordinals less than $\mu$, ordered by $\alpha\prec\gamma\iff V_{\gamma}\subset V_{\alpha}$. By Lemma 7.3, $\{u_{\gamma}\}_{\gamma\in\Gamma}$ is a TI-net. Notice that each tail of $\Gamma$ has cardinality $|\Gamma|=\mu$, as required by Lemma 2.5.

Lemma 7.14**.**

When $\mu>\mathbb{N}$, $\mathcal{P}_{1}(\widehat{G})$ contains an orthogonal TI-net of cardinality $\mu$.

Proof.

Fix $\{V_{\gamma}\}_{\gamma<\mu}\subset\mathscr{C}(G)$, a well-ordered neighborhood basis at $e$. Of course this well-ordering has no topological meaning, but it’s necessary for transfinite induction. The purpose of our induction is to select $\{U_{\gamma},H_{\gamma}\}_{\gamma<\mu}$ as in 7.13, from which the lemma clearly follows.

For $0\leq\beta<\mu$, suppose we have picked $\{U_{\gamma},H_{\gamma}\}_{\gamma<\beta}$ such that (1)-(4) hold for all $\gamma<\beta$:

(1) $H_{\gamma}$ is the intersection of $(\gamma+1)\cdot\mathbb{N}$ elements of $\mathscr{C}(G)$, and is a subgroup of each previous $H_{\alpha}$.

(2) If $\gamma=\alpha+1$ is a successor ordinal, then $H_{\gamma}\subset U_{\alpha}$.

(3) $H_{\gamma}U_{\gamma}^{2}\subset V_{\gamma}$.

(4) $\nu_{\gamma}(H_{\gamma}\cap U_{\gamma}^{4})<1$.

Let $\{W_{n}\}\subset\mathscr{C}(G)$ be any chain with $W_{0}\subset V_{\beta}$ and $W_{n+1}^{2}\subset W_{n}$ for all $n$. If $\beta=\alpha+1$, we may suppose $W_{0}\subset U_{\alpha}$. Let $H_{\beta}=\bigcap(\{W_{n}\}\cup\{H_{\gamma}\}_{\gamma<\beta})$. Notice $H_{\beta}W_{2}^{2}\subset W_{0}\subset V_{\beta}$. Since $H_{\beta}$ is the intersection of $(\beta+1)\cdot\mathbb{N}<\mu$ elements of $\mathscr{C}(G)$, it follows from Lemma 7.10 that $H_{\beta}\neq\{e\}$. In particular, it is possible to pick $U\in\mathscr{C}(G)$ with $\nu_{\beta}(H_{\beta}\cap U^{4})<1$. Let $U_{\beta}=U\cap W_{2}$. Now $U_{\beta},H_{\beta}$ satisfy (1)-(4). ∎

Theorem 7.15**.**

$|\operatorname{TIM}(\widehat{G})|=2^{2^{\mu}}$.

Proof.

Let $\{u_{\gamma}\}_{\gamma\in\Gamma}$ be the orthogonal TI-net of of Lemma 7.9 or Lemma 7.14, depending on $\mu$.

By Lemma 3.2, $p\mapsto\operatorname*{\mathit{p}-lim}_{\gamma}u_{\gamma}$ is an injection of $\Gamma^{*}$ into $\operatorname{TIM}(\widehat{G})$.

Thus $|\operatorname{TIM}(\widehat{G})|\geq|\Gamma^{*}|=2^{2^{\mu}}$ by Lemma 2.5. The opposite inequality is Corollary 7.7. ∎