# Gauge-Invariant Uniqueness and
Reductions of Ordered Groups

Robert Huben

(March 2021)

###### Abstract

A reduction $\varphi$ of an ordered group $(G,P)$ to another ordered group is
an order homomorphism which maps each interval $[1,p]$ bijectively onto
$[1,\varphi(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and
reduces to an amenable ordered group, then there is a gauge-invariant
uniqueness theorem for $P$-graph algebras. We also consider the class of
ordered groups which reduce to an amenable ordered group, and show this class
contains all amenable ordered groups and is closed under direct products, free
products, and hereditary subgroups.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Ordered Groups
    2. 2.2 $C^{*}$-algebras
      1. 2.2.1 The Factors Through Theorem
      2. 2.2.2 Universal Algebras
      3. 2.2.3 Conditional Expectations
      4. 2.2.4 Tensor Products
    3. 2.3 Connections between Groups and $C^{*}$-Algebras
      1. 2.3.1 Group $C^{*}$-algebras and Amenable Groups
      2. 2.3.2 Fell Bundles, Topologically Graded $C^{*}$-algebras, and Coactions
      3. 2.3.3 Coactions as Actions of the Dual Group
    4. 2.4 Graphs and $P$-graphs
    5. 2.5 Representations of $P$-graphs and $P$-graph Algebras
      1. 2.5.1 Gauge coactions and Coactionization
    6. 2.6 The Classical Gauge-Invariant Uniqueness Theorem
  3. 3 Reductions of Ordered Groups
    1. 3.1 Definition and Basic Properties
    2. 3.2 Constructions
      1. 3.2.1 Composition
      2. 3.2.2 Hereditary Subgroups
      3. 3.2.3 Direct Products
      4. 3.2.4 Free Products
    3. 3.3 Summary
  4. 4 Gauge-Invariant Uniqueness for $P$-graphs
    1. 4.1 A Co-Universal Algebra
      1. 4.1.1 The Ultrafilter Representation
      2. 4.1.2 Balanced Algebras and Balanced Coverings
      3. 4.1.3 The Kernel of $\psi^{\mathcal{T}}_{t}$
      4. 4.1.4 Normalizations of Coactions
      5. 4.1.5 The Co-Universal Algebra
    2. 4.2 The Tight Algebra
    3. 4.3 Reductions and Representations
    4. 4.4 Gauge-Invariant Uniqueness for Ordered Groups which Reduce to Amenable Groups
    5. 4.5 Additional Results in the case of a Strong Reduction
    6. 4.6 Applications to Kirchberg Algebras in the UCT Class
  5. 5 Appendix 1: Tight Representations
    1. 5.1 Background
    2. 5.2 Building our Semilattice
    3. 5.3 Technical Lemmas
    4. 5.4 The Proof of Equivalence

##  1 Introduction

Cuntz-Krieger algebras and their generalizations (Exel-Laca algebras, graph
algebras, higher-rank graph algebras, etc) are all, broadly speaking, the
universal algebras generated by partial isometries whose range and source
projections satisfy certain combinatorial relations. But defining these
algebras by their universality comes at a cost, since it becomes difficult to
check if any particular collection of partial isometries (a “representation”)
is universal. Mathematicians responded with uniqueness theorems, conditions on
the representation which guarantee that it is universal. A classic example is
the _gauge-invariant uniqueness theorem_ for graph algebras, which states that
so long as the canonical generators of the algebra are nonzero and there is a
gauge action (meaning an action $\alpha$ of the circle satisfying
$\alpha_{z}(T_{\gamma})=z^{\text{length}(\gamma)}T_{\gamma}$ for any
$z\in\mathbb{T}$ and path $\gamma$ in the graph), then any other
representation of the graph is a quotient of this representation. This
construction and uniqueness theorem has been generalized to higher rank graphs
by Kumjian and Pask [15], where paths are given lengths in $\mathbb{N}^{k}$
instead of $\mathbb{N}$, and an action of
$\mathbb{T}^{k}=\widehat{\mathbb{Z}^{k}}$ replaces the gauge action of
$\mathbb{T}=\widehat{\mathbb{Z}}$.

One might then ask that paths be allowed to have “lengths” in any positive
cone $P$ of a group $G$. The first attempt at such a generalization was by
Brownlowe, Sims, and Vittadello [3], who studied $P$-graphs when $(G,P)$ is
quasi-lattice ordered, but surprisingly the authors construct a $P$-graph
algebra which is not universal but _co-_ universal, meaning it is a quotient
of any sufficiently large representation.

In Section 4 of this paper, we build on their work to show that if the grading
group $(G,P)$ is weakly quasi-lattice ordered and has a certain kind of
quotient map called a reduction such that the quotient is amenable, this
algebra is both universal and co-universal. This allows us to generalize the
gauge-invariant uniqueness theorems for graphs and $k$-graphs (such as [18,
Theorem 2.2] or [15, Theorem 3.4]) to make a gauge-invariant uniqueness
theorem for $P$-graphs:

theoremuniquenesstheorem Let $(G,P)$ be a weakly quasi-lattice ordered group
which reduces to an amenable ordered group, and let $\Lambda$ be a finitely-
aligned $P$-graph. Then there is exactly one representation (up to canonical
isomorphism) of $\Lambda$ which is $\Lambda$-faithful, tight, and has a gauge
coaction of $G$. This representation is universal for tight representations
and co-universal for representations which are $\Lambda$-faithful and have a
gauge coaction by $G$.

This diagram provides a visual summary of the result:

$\Lambda$-faithful gauge coactingrepresentationstight
representations$C^{*}_{min}(\Lambda)$$C^{*}_{tight}(\Lambda)$$C^{*}_{min}(\Lambda)$no
gap if $(G,P)$ reduces to an amenable groupbigger representationssmaller
representations

We believe that the algebra generated by this simultaneously universal and co-
universal representation deserves the title of _the_ Cuntz-Krieger algebra of
the $P$-graph.

The notion of a reduction of ordered groups is introduced in Section 3,
culminating in this theorem showing that several natural operations on ordered
groups preserve the existence of a reduction onto an amenable group:

theoremAmenableReductionTheorem The class of ordered groups which have
(strong) reductions onto amenable groups contains all amenable ordered groups
and is closed under hereditary subgroups, finite direct products, and finite
free products.

In Theorem 1, being reducible to an amenable group plays the role of
amenability in guaranteeing a unique representation, but by Theorem 1, this
condition is, in at least one sense, more robust than amenability. In
particular, _being reducible to_ an amenable group is preserved under free
products, whereas _being_ amenable is almost always destroyed by a free
product (for instance if $G$ and $H$ contain copies of $\mathbb{Z}$, then
$G*H$ is not amenable).

The notion of being reducible to an amenable group is sufficiently robust
that, as corollary to Theorem 1, the important example
$(\mathbb{Z}^{2}*\mathbb{Z},\mathbb{N}^{2}*\mathbb{N})$ from the literature
([21, 3]) reduces to an amenable group. In Proposition 4.43, we use this fact
in combination with the results of [3] to show that every Kirchberg algebra in
the UCT class is stably isomorphic to the simultaneously universal and co-
universal algebra of a $(\mathbb{N}^{2}*\mathbb{N})$-graph.

Besides the results listed in Theorem 1, a great deal is not yet known about
how reductions interact with other group constructions. The limits of our
understanding are discussed in Section 3.3.

We hope that this new approach will provide a useful tool for the analysis of
$P$-graphs and their algebras, and possibly have implications for product
systems and Fell bundles.

##  2 Background

###  2.1 Ordered Groups

Ordered groups have been studied in both a context of group theory and
operator theory, and we largely follow the conventions of [3] and [10, Chapter
32]. The reader should note that other authors, such as [4], use the term
“ordered group” to refer to a group with a total order, but we follow the
convention that the group has a partial order.

######  Definition 2.1.

A positive cone in a group $G$ is a submonoid $P$ such that $P\cap
P^{-1}=\\{1\\}$.

A left-invariant partial order on a group $G$ is a partial order $\leq$ such
that for all $a,b,c\in G$, $a\leq b$ if and only if $ca\leq cb$.

Every left-invariant partial order on a group arises naturally from a positive
cone $P$ by saying that $a\leq b$ if and only if $a^{-1}b\in P$. We will say
that an ordered group $(G,P)$ is a group along with a positive cone, with the
implication that the group takes on a left-invariant partial order in this
way.

A group is totally ordered if its partial order is a total order. This is
equivalent to the condition that $P\cup P^{-1}=G$.

Although our theory of reductions does not require any additional structure on
our ordered groups, the relators for a graph algebra will ask for an
additional property on our orderings from [10, Chapter 32]:

######  Definition 2.2.

We say $(G,P)$ is weakly quasi-lattice ordered (WQLO) if whenever $x,y\in P$
have a common upper bound in $P$, they have a supremum (least common upper
bound), denoted $x\vee y$. Note that any upper bound of a positive element is
itself positive by transitivity.

While many mathematicians (including[3, 20, 22]) worked in the context of
quasi-lattice ordered groups, we have found that the slightly more general
condition of weak quasi-lattice order suffices. The condition of weak quasi-
lattice order is also friendlier than quasi-lattice order since it is entirely
a property of the submonoid $P$, and therefore one can “forget” the ambient
group. The same is not true for quasi-lattice order. For a more thorough
introduction to quasi-lattice order, weak quasi-lattice order, and their
connections, the reader is directed to [10, Chapter 32].

######  Notation 2.3.

In this work, we will make use of the following notation:

  * •

$\mathbb{N}=\\{0,1,2,3...\\}$ includes $0$.

  * •

Given a group $G$ and subset $S\subseteq G$, $\langle S\rangle$ will denote
the group generated by the elements of $S$, and $S^{*}$ will denote the monoid
generated by the elements of $S$.

  * •

The identity element in a multiplicative group $G$ will be denoted by $1_{G}$,
or $1$ if the group is clear from context. Less commonly we will denote the
identity element by $e$.

######  Example 2.4.

Let $G=\mathbb{Z}^{k}$, and let $P=\mathbb{N}^{k}$. Then $(G,P)$ is WQLO.
Every pair of elements $p=(p_{1},...,p_{k})$ and $q=(q_{1},...,q_{k})$ have a
supremum $p\vee q=(\max(p_{1},q_{1}),...,\max(p_{k},q_{k}))$. (In fact, this
ordering is a “lattice order”, meaning any two elements have both a supremum
and an infimum.)

Let $G=F_{k}=\langle a_{1},...,a_{k}\rangle$ be the free group on $k$
generators, and let $P=\\{a_{1},...,a_{k}\\}^{*}$ be the free monoid generated
by the generators of $G$. Then $(G,P)$ is WQLO, since any $p,q\in P$ have a
common upper bound if and only if $p\leq q$ or $q\leq p$, in which case $p\vee
q=\max\\{p,q\\}$.

######  Notation 2.5.

Let $(G,P)$ be an ordered group. We use the following standard notation for
intervals: for $a,b\in G$, write

$[a,b]:=\\{g\in G|a\leq g\leq b\\}.$

It is immediate that such an interval is nonempty if and only if $a\leq b$. We
will most often be interested in intervals of the form $[1,p]$ for some $p\in
P$.

######  Definition 2.6.

Let $(G,P)$ and $(H,Q)$ be ordered groups. We say that $\varphi:G\rightarrow
H$ is an order homomorphism if it is a group homomorphism with
$\varphi(P)\subseteq Q$. We will write $\varphi:(G,P)\rightarrow(H,Q)$ to
denote an order homomorphism.

Order homomorphisms preserve both the group structure and the order structure,
as the following lemma shows:

######  Lemma 2.7.

Let $(G,P)$ and $(H,Q)$ be ordered groups, and $\varphi:G\rightarrow H$ a
group homomorphism. Then $\varphi$ is an order homomorphism if and only if
$x\leq y$ implies $\varphi(x)\leq\varphi(y)$ for all $x,y\in G$.

In particular, if $\varphi$ is an order homomorphism and $p\in P$, then
$\varphi([1,p])\subseteq[1,\varphi(p)]$.

###### Proof.

If $\varphi$ is an order homomorphism and $x\leq y$, then $x^{-1}y\in P$, so
$\varphi(x^{-1}y)=\varphi(x)^{-1}\varphi(y)\in Q$. Thus
$\varphi(x)\leq\varphi(y)$.

Conversely, if $\varphi$ is not an order homomorphism, then there is $p\in P$
such that $\varphi(p)\not\in Q$. Then $1_{G}\leq p$, and
$1_{H}=\varphi(1_{G})\not\leq\varphi(p)$, as desired.

For the “in particular”, if $s\in[1,p]$, then $1\leq s\leq p$, so
$1=\varphi(1)\leq\varphi(s)\leq\varphi(p)$, so $\varphi(s)\in[1,\varphi(p)]$
as desired. ∎

###  2.2 $C^{*}$-algebras

####  2.2.1 The Factors Through Theorem

The following result is an elementary notion from algebra, which we state here
for completeness since we will make frequent use of it:

######  Lemma 2.8 (Factors Through Theorem).

Let $A,B$, and $C$ be sets, let $f:A\rightarrow B$ and $g:A\rightarrow C$ be
functions, and suppose that $f$ is surjective.

${A}$${B}$${C}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\exists h}$

  1. 1.

Suppose that whenever $f(a)=f(a^{\prime})$, then $g(a)=g(a^{\prime})$. Then
there is a “bonding map” $h:B\rightarrow C$ such that $h\circ f=g$. In
particular, $h$ is (well-)defined by $h(f(a))=g(a)$ for all $f(a)\in B$.

  2. 2.

If $A,B,$ and $C$ are groups, $f$ and $g$ are group homomorphisms, and $\ker
f\subseteq\ker g$, then hypothesis of (1) that $f(a)=f(a^{\prime})$ implies
$g(a)=g(a^{\prime})$ is satisfied. Therefore, the conclusion of (1) is true.

  3. 3.

If $A,B,C$ have a binary operation $\cdot$ (respectively a unary operation ∗)
which is preserved by $f$ and $g$, then $\cdot$ (respectively ∗) is also
preserved by $h$.

###### Proof.

For (1), for all $b\in B$, since $f$ is surjective, there is some $a\in A$
such that $f(a)=b$. Now, define $h(b)=g(a)$, which is well-defined since if
there were some other $a^{\prime}\in A$ with $f(a^{\prime})=b$, then
$g(a)=g(a^{\prime})$ by hypothesis. That is, we have defined $h$ by
$h(f(a))=g(a)$, so $h\circ f=g$.

For (2), if $f(a)=f(a^{\prime})$, then $a^{-1}a^{\prime}\in\ker f\subseteq\ker
g$, so $g(a)=g(a^{\prime})$, as desired.

For (3), let $b_{1},b_{2}\in B$. Since $f$ is surjective, there is some
$a_{1},a_{2}\in A$ such that $f(a_{1})=b_{1},f(a_{2})=b_{2}$. Then

$\displaystyle h(b_{1}\cdot b_{2})$ $\displaystyle=$ $\displaystyle
h(f(a_{1})\cdot f(a_{2}))=h(f(a_{1}\cdot a_{2}))=g(a_{1}\cdot a_{2})$
$\displaystyle=$ $\displaystyle g(a_{1})\cdot g(a_{2})=h(f(a_{1}))\cdot
h(f(a_{2}))=h(b_{1})\cdot h(b_{2})$

and

$h(b_{1}^{*})=h(f(a_{1})^{*})=h(f(a_{1}^{*}))=g(a_{1}^{*})=g(a_{1})^{*}=h(f(a_{1}))^{*}=h(b_{1})^{*}$

as desired.

∎

####  2.2.2 Universal Algebras

The following construction of a “universal algebra” is a slight simplification
of the one outlined by Blackadar in [2]. We state it here for completeness.

######  Lemma 2.9 (Construction of a Universal Algebra, [2]).

Let $\mathcal{G}=\\{x_{\alpha}\\}_{\alpha\in I}$ denote a set of formal
symbols, $\mathcal{G}^{*}=\\{x_{\alpha}^{*}\\}_{\alpha\in I}$ another set of
formal symbols over the same indexing set, and $\mathcal{R}$ a set of relators
of the form
$p(x_{\alpha_{1}},...,x_{\alpha_{n}},x_{\alpha_{1}}^{*},...,x_{\alpha_{n}}^{*})=0$
where $p$ is some polynomial in $2n$ non-commuting variables and complex
coefficients.

We define a representation of $(\mathcal{G},\mathcal{R})$ to be a collection
of elements $\\{y_{\alpha}\\}_{\alpha\in I}$ in a $C^{*}$-algebra that satisfy
$p(y_{\alpha_{1}},...,y_{\alpha_{n}},y_{\alpha_{1}}^{*},...,y_{\alpha_{n}}^{*})=0$
for each relator in $\mathcal{R}$.

Suppose that the relators on $(\mathcal{G},\mathcal{R})$ imply that in any
representation, the images of the generators are partial isometries. Then
there is a representation $\\{z_{\alpha}\\}_{\alpha\in I}$ such that for any
other representation $\\{y_{\alpha}\\}_{\alpha\in I}$ there is a surjective
$*$-homomorphism $\pi:C^{*}(\\{z_{\alpha}\\}_{\alpha\in I})\rightarrow
C^{*}(\\{y_{\alpha}\\}_{\alpha\in I})$ satisfying $z_{\alpha}\mapsto
y_{\alpha}$.

The existence of this surjective $*$-homomorphism is called the universal
property of $C^{*}(\\{z_{\alpha}\\}_{\alpha\in I})$. We call this
representation the universal representation and say that
$C^{*}(\\{z_{\alpha}\\}_{\alpha\in I})$ is the universal algebra for
$(\mathcal{G},\mathcal{R})$.

###### Proof.

Let $\mathcal{F}(\mathcal{G})$ denote the free $*$-algebra over $\mathbb{C}$
generated by $\mathbb{C}$ and $\mathcal{G}\cup\mathcal{G}^{*}$ where the
$*$-map is given by $(x_{\alpha})^{*}=x_{\alpha}^{*}$ and
$(x_{\alpha}^{*})^{*}=x_{\alpha}$ and extended to the entire algebra via
conjugate-linearity and $(ab)^{*}=b^{*}a^{*}$. Note that any representation
$\varphi:x_{\alpha}\mapsto y_{\alpha}$ of $(\mathcal{G},\mathcal{R})$ extends
uniquely to a $*$-algebra homomorphism
$\bar{\varphi}:\mathcal{F}(\mathcal{G})\rightarrow
C^{*}(\\{y_{\alpha}\\}_{\alpha\in I})$.

For $X\in\mathcal{F}(\mathcal{G})$, let
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=\sup\\{\left\|{\bar{\varphi}(X)}\right\|:\varphi\text{
a rep. of }(\mathcal{G},\mathcal{R})\\}$. Note that since
$(\mathcal{G},\mathcal{R})$ can be represented by the zero map, this supremum
is bounded below by 0, and since each generator must be represented by a
partial isometry, the image of each generator has norm 1, and thus the
supremum is bounded above by the sum of the absolute values of its
$\mathbb{C}$-coefficients. Thus
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is a well-defined, non-negative finite number.

Let
$J=\\{X\in\mathcal{F}(\mathcal{G}):{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=0\\}$,
and note that $J=\displaystyle\bigcap_{\varphi\text{ a
rep.}}\ker\bar{\varphi}$ where the intersection is taken over all
representations. Thus $J$ is the intersection of $*$-ideals, so it is a
$*$-ideal. Now let
$\psi:\mathcal{F}(\mathcal{G})\rightarrow\mathcal{F}(\mathcal{G})/J$ denote
the quotient map, and for all $\alpha$, let $z_{\alpha}=\psi(x_{\alpha})$. For
any relator
$p(x_{\alpha_{1}},...,x_{\alpha_{n}},x_{\alpha_{1}}^{*},...,x_{\alpha_{n}}^{*})=0\in\mathcal{R}$,
since this relator is satisfied for every representation, then
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(x_{\alpha_{1}},...,x_{\alpha_{n}},x_{\alpha_{1}}^{*},...,x_{\alpha_{n}}^{*})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=0$,
and thus
$p(x_{\alpha_{1}},...,x_{\alpha_{n}},x_{\alpha_{1}}^{*},...,x_{\alpha_{n}}^{*})\in
J$, so
$p(z_{\alpha_{1}},...,z_{\alpha_{n}},z_{\alpha_{1}}^{*},...,z_{\alpha_{n}}^{*})=0$.
That is, the $\\{z_{\alpha}\\}_{\alpha\in I}$ satisfy the relators.

Now note that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is a $C^{*}$-seminorm, so the norm
$\left\|{X+J}\right\|:={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is a well-defined $C^{*}$-norm on $\mathcal{F}(\mathcal{G})/J$, and thus the
completion of $\mathcal{F}(\mathcal{G})/J$ with respect to this norm is a
$C^{*}$-algebra. Thus $\psi:x_{\alpha}\rightarrow z_{\alpha}$ is a
representation of $(\mathcal{G},\mathcal{R})$.

Now, given any other presentation $\varphi:x_{\alpha}\mapsto y_{\alpha}$, note
that
$\ker\bar{\varphi}=\\{X\in\mathcal{F}(\mathcal{G}):\left\|{\bar{\varphi}(X)}\right\|=0\\}\subseteq\\{X\in\mathcal{F}(\mathcal{G}):{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|X\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=0\\}=\ker\bar{\psi}$,
so by Lemma 2.8(2), there is a function on the $*$-algebras generated by $y$
and $z$ given by $y_{\alpha}\mapsto z_{\alpha}$, and by Lemma 2.8(3) applied
to $+,\times,$ and ∗, this function is a $*$-homomorphism, so it extends to a
$*$-homomorphism $\pi:C^{*}(\\{z_{\alpha}\\}_{\alpha\in I})\rightarrow
C^{*}(\\{y_{\alpha}\\}_{\alpha\in I})$ as desired. ∎

####  2.2.3 Conditional Expectations

We will also make frequent use of conditional expectations, whose definition
we include here. For a more thorough treatment of conditional expectations,
the reader is directed to [23, Chapter III.3], where they are called
projections of norm one.

######  Definition 2.10.

Let $C$ be a $C^{*}$-algebra, and $D\subseteq C$ a $C^{*}$-subalgebra. We say
a conditional expectation or projection of norm one is a linear map
$\Phi:C\rightarrow D$ such that:

  * •

$\Phi$ is contractive, meaning
$\left\|{\Phi(c)}\right\|\leq\left\|{c}\right\|$ for all $c\in C$ (and is thus
continuous).

  * •

$\Phi$ is idempotent, meaning $\Phi\circ\Phi=\Phi$.

  * •

$\Phi(d)=d$ for all $d\in D$.

If additionally $c>0$ implies $\Phi(c)>0$, we say $\Phi$ is faithful.

######  Example 2.11.

In a matrix algebra $M_{n}$, there is a faithful conditional expectation
$\Phi$ given by “restricting to the diagonal”, meaning
$\Phi([a_{ij}])=[b_{ij}]$ where

$b_{ij}=\begin{cases}a_{ij}&\text{ if }i=j\\\ 0&\text{ if }i\neq
j\end{cases}.$

The following result is [24, Theorem 1] and is known as Tomiyama’s Theorem. It
says that a conditional expectation has some additional properties “for free”.
Some authors will define conditional expectations as requiring these
properties, and some will define them to only require the shorter list of
conditions (as we have done here).

######  Theorem 2.12 (Tomiyama’s Theorem).

If $\Phi:C\rightarrow D$ is a conditional expectation, then $\Phi$ is
positive, meaning that $x\geq 0$ implies $\Phi(x)\geq 0$. Furthermore, $\Phi$
is a $D$-bimodule map, meaning $\Phi(d_{1}cd_{2})=d_{1}\Phi(c)d_{2}$ for all
$d_{1},d_{2}\in D$ and $c\in C$.

####  2.2.4 Tensor Products

Here we will give a very brief introduction to the minimal tensor product. A
more thorough introduction can be found in [23, Chapter 4]. Throughout this
paper, as in much of the coaction literature, we will use unadorned $\otimes$
for the minimal tensor product of $C^{*}$-algebras.

######  Definition 2.13.

Given two vector spaces $V$ and $W$, we let $V\odot W$ denote the algebraic
tensor product of $V$ and $W$, meaning the $\mathbb{C}$-vector space spanned
by formal symbols $\\{v\otimes w:v\in V,w\in W\\}$ with the relations of
linearity over either coordinate.

Let $A$ and $B$ be $C^{*}$-algebras. Then one can give $A\odot B$ a
$*$-algebra structure by

$(a\otimes b)(a^{\prime}\otimes
b^{\prime})=(aa^{\prime})\otimes(bb^{\prime})\text{ and }(a\otimes
b)^{*}=a^{*}\otimes b^{*}.$

If $\pi:A\rightarrow\mathcal{B}(H)$ and
$\rho:B\rightarrow\mathcal{B}(\mathcal{K})$ are representations of $A$ and $B$
on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ respectively, then there
exists a $*$-algebra representation $\pi\otimes\rho:A\odot
B\rightarrow\mathcal{B}(\mathcal{H}\otimes\mathcal{K})$ satisfying
$\left[(\pi\otimes\rho)(a\otimes b)\right](h\otimes k)=\pi(a)h\otimes\rho(b)k$
for all $a\in A,b\in B,h\in\mathcal{H},k\in\mathcal{K}$. The minimal norm on
$A\odot B$ is defined by

$\left|\left|\displaystyle\sum_{i=1}^{n}a_{i}\otimes
b_{i}\right|\right|_{min}=\sup\left\\{\left|\left|\displaystyle\sum_{i=1}^{n}\pi(a_{i})\otimes\rho(b_{i})\right|\right|\right\\}$

where the supremum is taken over all representations $\pi$ and $\rho$ of $A$
and $B$, respectively (and this is indeed a norm which is minimal in an
appropriate sense). Note that $\left\|{a\otimes
b}\right\|_{min}=\left\|{a}\right\|\cdot\left\|{b}\right\|$ (that is,
$\left\|{\cdot}\right\|_{min}$ is a “$C^{*}$-cross norm”). The minimal tensor
product $A\otimes_{min}B$ is the $C^{*}$-algebra created by completing $A\odot
B$ with respect to this norm. Since all of our tensor products will be
minimal, we will henceforth write $\left\|{\cdot}\right\|$ for
$\left\|{\cdot}\right\|_{min}$ and $A\otimes B$ for $A\otimes_{min}B$.

The following result will be used occasionally, and it is a minor variant of
[6, Lemma A.1]:

######  Lemma 2.14.

Let $\varphi:A\rightarrow C$ and $\psi:B\rightarrow D$ be $*$-homomorphisms of
$C^{*}$-algebras. Then there is a homomorphism $\varphi\otimes\psi:A\otimes
B\rightarrow C\otimes D$ satisfying $(\varphi\otimes\psi)(a\otimes
b)=\varphi(a)\otimes\psi(b)$. If $\varphi$ and $\psi$ are nondegenerate
(respectively, faithful), then so is $\varphi\otimes\psi$.

###### Proof.

Consider $C$ and $D$ as subalgebras (respectively) of their multiplier
algebras $M(C)$ and $M(D)$, and therefore consider $\varphi$ and $\psi$ as
mapping into $M(C)$ and $M(D)$, respectively. By [6, Lemma A.1], there is a
$*$-homomorphism $\varphi\otimes\psi:A\otimes B\rightarrow M(C\otimes D)$
satisfying $(\varphi\otimes\psi)(a\otimes b)=\varphi(a)\otimes\psi(b)$, and if
$\varphi$ and $\psi$ are nondegenerate (respectively, faithful), then so is
$\varphi\otimes\psi$.

It then suffices to show that $\varphi\otimes\psi$ maps into $C\otimes D$
properly, instead of mapping into $M(C\otimes D)$. But for all $a\in A,b\in
B$, we have $\varphi(a)\in C$ and $\varphi(b)\in D$, so
$(\varphi\otimes\psi)(a\otimes b)\in C\otimes D$ for all $a\in A,b\in B$.
Taking closed spans, we get that for any $x\in A\otimes B$, we have
$(\varphi\otimes\psi)(x)\in C\times D$, as desired. ∎

###  2.3 Connections between Groups and $C^{*}$-Algebras

In this section we will remind the reader of some of the many connections
between groups and $C^{*}$-algebras, including group $C^{*}$-algebras,
amenability, Fell bundles, gradings, coactions, and actions. Our context is
relatively simple since we are limiting our attention to discrete groups.

####  2.3.1 Group $C^{*}$-algebras and Amenable Groups

The following discussion of group $C^{*}$-algebras and amenability is adapted
from [5, Chapter VII], to which the reader is directed for a more thorough
treatment.

######  Definition 2.15.

Let $G$ be a discrete group. Then a representation of $G$ in a $C^{*}$-algebra
is a collection of unitary operators $\\{u_{g}\\}_{g\in G}$ such that
$u_{g}u_{h}=u_{gh}$ and $u_{g}^{*}=u_{g^{-1}}$ for all $g,h\in G$.

Since these relators are all polynomials, then by Lemma 2.9 there is a
representation of $G$ which is universal for all representations. We denote it
by $\\{U_{g}\\}_{g\in G}$, and call $C^{*}(G):=C^{*}(\\{U_{g}\\}_{g\in G})$
the full group $C^{*}$-algebra.

A particularly nice representation of a group $G$, called the left-regular
representation of a group, comes from the natural action of $G$ multiplying on
itself. In particular, let $\\{e_{g}\\}_{g\in G}$ denote the standard
orthonormal basis for $\ell^{2}(G)$, and define an operator $L_{g}$ by
$L_{g}e_{g^{\prime}}=e_{gg^{\prime}}$ for $g,g^{\prime}\in G$. Then
$\\{L_{g}\\}_{g\in G}$ is a representation of $G$ in
$\mathcal{B}(\ell^{2}(G))$. We call $C_{r}^{*}(G):=C^{*}(\\{L_{g}\\}_{g\in
G})$ the reduced group $C^{*}$-algebra.

The left-regular representation is more commonly denoted by $\lambda$ instead
of $L$, but our choice of notation will help avoid ambiguity with paths
$\lambda\in\Lambda$ later in the text.

Note that $C^{*}(G)$ and $C^{*}_{r}(G)$ are the closed spans of
$\\{U_{g}\\}_{g\in G}$ and $\\{L_{g}\\}_{g\in G}$, respectively.

Famously, there are many equivalent definitions of amenability, but for our
purposes, this one is the most convenient:

######  Definition 2.16.

By the universal property of $C^{*}(G)$, there is a surjective
$*$-homomorphism $\pi^{U}_{L}:C^{*}(G)\rightarrow C^{*}_{r}(G)$ given by
$U_{g}\mapsto L_{g}$. We say that $G$ is amenable if and only if $\pi^{U}_{L}$
is injective (and hence an isomorphism).

Readers more familiar with another definition of amenability may wish to read
[5, Theorem VII.2.5] which proves the equivalence of this definition with a
more common one.

The following remark summarizes some of the well-known results about
amenability of groups.

######  Remark 2.17.

By [5, Proposition VII.2.3], for discrete groups amenability is preserved
under:

  1. 1.

Subgroups

  2. 2.

Quotients

  3. 3.

Direct Limits

  4. 4.

Extensions (meaning that if $1\rightarrow N\rightarrow G\rightarrow
H\rightarrow 1$ is a short exact sequence of groups, with $N$ and $H$
amenable, then $G$ is amenable).

  5. 5.

Finite direct products (which is immediate from being closed under
extensions).

Furthermore, by [5, Proposition VII.2.2], every abelian group is amenable.
Every finite group is amenable since $C^{*}(G)$ and $C^{*}_{r}(G)$ have the
same finite dimension $|G|$, so by the rank-nullity theorem the surjective map
$\pi^{U}_{L}$ is injective.

Notably, the free group on two generators is not amenable by [5, Example
VII.2.4]. By the subgroup property, any group containing a free group on two
(or more) generators is also not amenable.

####  2.3.2 Fell Bundles, Topologically Graded $C^{*}$-algebras, and Coactions

In this section, we will give a short introduction to Fell bundles,
topologically graded $C^{*}$-algebras, and coactions, and show some of the
ways these closely-related structures overlap. For a more detailed treatment,
the reader is directed to [10, 9] for Fell bundles, [10] for topological
gradings, and [6, Appendix A] for coactions.

Throughout this section, all of our groups will be discrete, which simplifies
some definitions. The following is from [10, Definition 16.1]:

######  Definition 2.18.

Let $G$ be a discrete group. Let $\mathcal{B}=\\{B_{g}\\}_{g\in G}$ be a
collection of Banach spaces, and write $\mathscr{B}$ for the disjoint union of
the $\\{B_{g}\\}_{g\in G}$, called the total space. Suppose $\mathscr{B}$ has
a binary operation $\cdot$ called multiplication, and an involution $*$ which
satisfy the following properties for all $g,h\in G$ and $b,c\in\mathscr{B}$:

  1. a.

$B_{g}B_{h}\subseteq B_{gh}$,

  2. b.

Multiplication is bilinear from $B_{g}\times B_{h}$ to $B_{gh}$,

  3. c.

Multiplication on $\mathscr{B}$ is associative,

  4. d.

$\left\|{bc}\right\|\leq\left\|{b}\right\|\cdot\left\|{c}\right\|$,

  5. e.

$(B_{g})^{*}\subseteq B_{g^{-1}}$,

  6. f.

Involution is conjugate-linear from $B_{g}$ to $B_{g^{-1}}$,

  7. g.

$(bc)^{*}=c^{*}b^{*}$,

  8. h.

$b^{**}=b$,

  9. i.

$\left\|{b^{*}}\right\|=\left\|{b}\right\|$,

  10. j.

$\left\|{b^{*}b}\right\|=\left\|{b}\right\|^{2}$,

  11. k.

$b^{*}b\geq 0$ in $B_{1}$.

Then we say that $\mathcal{B}$ is a Fell bundle over $G$. We call each $B_{g}$
a fiber.

The following is from [10, Definitions 16.2 and 19.2]:

######  Definition 2.19.

Let $A$ be a $C^{*}$-algebra, and let $G$ be a (discrete) group. We say that a
($C^{*}$-) grading for $A$ is a collection $\\{A_{g}\\}_{g\in G}$ of linearly
independent closed subspaces such that $\bigoplus_{g\in G}A_{g}$ is dense in
$A$, $A_{g}A_{h}\subseteq A_{gh}$, and $A_{g}^{*}\subseteq A_{g^{-1}}$ for
$g,h\in G$. Each $A_{g}$ is called a graded subspace or graded component.

If there is also a conditional expectation $\Phi:A\rightarrow A_{e}$
satisfying

$\Phi(a)=\begin{cases}a&\text{ if }a\in A_{e}\\\ 0&\text{ if }a\in A_{g}\text{
for }g\neq e\end{cases},$

then we say that $\\{A_{g}\\}_{g\in G}$ is a topological grading.

######  Remark 2.20.

It is straightforward to verify that if $A$ is a topologically graded
$C^{*}$-algebra, then its graded components form a Fell bundle. Most of the
Fell bundles we will use arise in this way.

######  Remark 2.21.

Given a Fell bundle $\mathcal{B}$, one can construct a “reduced” and “full”
cross sectional algebra representing this bundle, respectively denoted
$C^{*}_{r}(\mathcal{B})$ and $C^{*}(\mathcal{B})$ (see Definition 2.3 of [9]
and the comment following it).

By [9, Theorem 3.3], if $B$ is a topologically graded $C^{*}$-algebra, and
$\mathcal{B}$ denotes its associated Fell bundle, then there is a
$*$-homomorphism $L:B\rightarrow C^{*}_{r}(\mathcal{B})$ called the left-
regular representation of the Fell bundle. Combining this result with their
Proposition 3.7, $L$ is an isomorphism if and only if the conditional
expectation from the topological grading is faithful.

In a Fell bundle $\mathcal{B}$, the full cross sectional algebra
$C^{*}(\mathcal{B})$ has a topological grading, and the bundle is called
amenable if its left-regular representation $L:C^{*}(\mathcal{B})\rightarrow
C^{*}_{r}(\mathcal{B})$ is injective (and hence an isomorphism). The reader
may notice the parallel with our definition of amenability for a group
(Definition 2.16).

The following two results are respectively [9, Theorem 4.7] and [9,
Proposition 4.2] :

######  Lemma 2.22.

Let $G$ be a discrete amenable group. Then every Fell bundle over $G$ is
amenable.

######  Lemma 2.23.

If $\mathcal{B}$ is an amenable Fell bundle, then all topologically graded
$C^{*}$-algebras whose associated Fell bundles coincide with $\mathcal{B}$ are
isomorphic to each other.

Finally, we will give a brief introduction to coactions. The following is both
the simplest example of a coaction, and a necessary component of its
definition.

######  Example 2.24.

If $G$ is a discrete group, then the operators $\\{U_{g}\otimes U_{g}\\}_{g\in
G}\subset C^{*}(G)\otimes C^{*}(G)$ are a representation of $G$. Therefore, by
the universal property of $C^{*}(G)$, there is a $*$-homomorphism
$\delta_{G}:C^{*}(G)\rightarrow C^{*}(G)\otimes C^{*}(G)$ given by
$U_{g}\mapsto U_{g}\otimes U_{g}$.

Note that letting $\operatorname{id}_{G}:=\operatorname{id}_{C^{*}(G)}$, then
$(\delta_{G}\otimes\operatorname{id}_{G})\circ\delta_{G}=(\operatorname{id}_{G}\otimes\delta_{G})\circ\delta_{G}$,
which can be easily verified by checking that both send $U_{g}\mapsto
U_{g}\otimes U_{g}\otimes U_{g}\in C^{*}(G)\otimes C^{*}(G)\otimes C^{*}(G)$.

The following definition of coactions for discrete groups is taken from [7,
Section 2]:

######  Definition 2.25.

Let $G$ be a discrete group. A coaction of $G$ on a $C^{*}$-algebra $A$ is an
injective, nondegenerate homomorphism $\delta:A\rightarrow A\otimes C^{*}(G)$
satisfying the coaction identity that
$(\delta\otimes\operatorname{id}_{G})\circ\delta=(\operatorname{id}_{A}\otimes\delta_{G})\circ\delta$
as maps from $A$ into $A\otimes C^{*}(G)\otimes C^{*}(G)$, summarized in this
diagram:

${A}$${A\otimes C^{*}(G)}$${A\otimes C^{*}(G)}$${A\otimes C^{*}(G)\otimes
C^{*}(G)}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta\otimes\operatorname{id}_{G}}$$\scriptstyle{\operatorname{id}_{A}\otimes\delta_{G}}$

In this context, nondegeneracy means that
$\overline{\operatorname{span}}\left[\delta(A)(A\otimes
C^{*}(G))\right]=A\otimes C^{*}(G)$.

We call the triple $(A,G,\delta)$ a cosystem.

If $G$ is discrete, a cosystem has a nice topological grading as described in
[20, Proposition A.3]:

######  Lemma 2.26.

Let $(A,G,\delta)$ be a discrete cosystem, and for $g\in G$ let

$A_{g}=\\{a\in A|\delta(a)=a\otimes U_{g}\\}.$

Then the collection $\\{A_{g}\\}_{g\in G}$ is a topological grading of $A$. We
will write $\Phi_{A}:A\rightarrow A_{e}$ for the conditional expectation.

Some coactions are particularly nice, having a condition called normality.
There are many definitions of normality, but we will find a definition
depending on this conditional expectation to be the most convenient:

######  Definition 2.27.

We say a discrete coaction $(A,G,\delta)$ is normal if the conditional
expectation $\Phi_{A}$ is faithful.

A more common definition of normality of a coaction (such as [6, Definition
A.50] or the comments preceding Definition 1.1 of [17]) is that the coaction
is normal if and only if the map
$j_{A}:=(\operatorname{id}_{A}\otimes\pi^{U}_{L})\circ\delta$ is injective.
Our definition is proved equivalent to the more common one in [17, Lemma 1.4].

For a coaction over a discrete amenable group, this property is automatic:

######  Lemma 2.28.

Let $(A,G,\delta)$ be a cosystem. If $G$ is amenable and discrete, then
$\delta$ is normal.

###### Proof.

For a cosystem $(A,G,\delta)$ over an amenable group, by Lemma 2.26 there is a
topological grading $\\{A_{g}\\}_{g\in G}$ of $A$. By Remark 2.20, these
graded components form a Fell bundle $\mathcal{B}$, and this Fell bundle is
amenable by Lemma 2.22.

Since $\mathcal{B}$ is amenable, by Lemma 2.23, $A\cong
C^{*}_{r}(\mathcal{B})$. But for any Fell bundle, $C^{*}_{r}(\mathcal{B})$ has
a conditional expectation by [9, Proposition 2.9], which is faithful by [9,
Proposition 2.12]. ∎

Finally, we will show that coactions, topological gradings, and Fell bundles
are equivalent over amenable groups.

######  Lemma 2.29.

Let $G$ be an amenable group, and let $A$ be a $C^{*}$-algebra. Then the
following are equivalent:

  1. 1.

$A$ has a coaction by $G$.

  2. 2.

$A$ has a topological grading $\\{A_{g}\\}_{g\in G}$.

  3. 3.

There is a Fell bundle $\mathcal{B}=\\{B_{g}\\}_{g\in G}$ whose fibers are
linearly independent closed subspaces of $A$ such that $A\cong
C^{*}(\mathcal{B})$.

And under these conditions, the structures coincide in the sense that the
fibers $\\{B_{g}\\}_{g\in G}$ of the Fell bundle are equal to the graded
components $\\{A_{g}\\}_{g\in G}$, and the graded components arise from the
coaction via $A_{g}=\\{a\in A:\delta(a)=a\otimes U_{g}\\}$.

###### Proof.

By Lemma 2.26, (1) implies (2).

To show that (2) implies (3), suppose that $A$ has a topological grading
$\\{A_{g}\\}_{g\in G}$. By Remark 2.20, these fibers form a Fell bundle. To
see that the fibers are linearly independent, fix a finite sum
$\displaystyle\sum_{i=1}^{n}a_{g_{i}}$ where each $a_{g_{i}}\in A_{g_{i}}$ for
some distinct $g_{i}\in G$, and suppose this sum equals 0. Then by [10,
Corollary 19.6], there are contractive linear maps $F_{g}:A\rightarrow A_{g}$
satisfying $F_{g_{j}}(\displaystyle\sum_{i=1}^{n}a_{g_{i}})=a_{g_{j}}$. But
since $0=\displaystyle\sum_{i=1}^{n}a_{g_{i}}$, then for each $g_{j}$, we have
$0=F_{g_{j}}(\displaystyle\sum_{i=1}^{n}a_{g_{i}})=a_{g_{j}}$, so each summand
is 0, so the graded components are linearly independent. Finally, we must show
that $A\cong C^{*}(\mathcal{B})$. By Lemma 2.22, $\mathcal{B}$ is amenable,
and by [10, Proposition 19.3], $C^{*}(\mathcal{B})$ is a topologically graded
$C^{*}$-algebra. Since this amenable bundle coincides with the bundle in $A$,
then by Lemma 2.23, $A\cong C^{*}(\mathcal{B})$, as desired.

To see that (3) implies (1), suppose that $\mathcal{B}=\\{B_{g}\\}_{g\in G}$
is a Fell bundle of linearly independent closed subspaces of $A$ such that
$A\cong C^{*}(\mathcal{B})$. Since $G$ is amenable, $C^{*}(G)\cong
C^{*}_{r}(G)$, and by Lemma 2.22, $\mathcal{B}$ is an amenable Fell bundle, so
$C^{*}(\mathcal{B})\cong C^{*}_{r}(\mathcal{B})$. Combining these
simplifications with [10, Proposition 18.7], there is an injective
$*$-homomorphism $\delta:C^{*}(\mathcal{B})\rightarrow
C^{*}(\mathcal{B})\otimes C^{*}(G)$ satisfying $\delta(b_{g})=b_{g}\otimes
U_{g}$ for all $g\in G$ and $b_{g}\in B_{g}$. It is routine to verify that
this is a coaction, and that the graded parts of this coaction coincide with
the original Fell bundle $\mathcal{B}$.

∎

####  2.3.3 Coactions as Actions of the Dual Group

In this section we will show that if $G$ is discrete and abelian, then a
coaction by $G$ is equivalent to an action by its dual group $\widehat{G}$.
For a non-abelian group, this correspondence breaks down because there is no
dual group.

First, we will remind the reader of actions by groups and dual groups. The
following definition is well-known, and can be found for instance in the
remarks preceding Proposition 2.1 in [18]:

######  Definition 2.30.

Given a locally compact group $G$, an action of $G$ on a $C^{*}$-algebra $A$
is a strongly continuous homomorphism
$\alpha:G\rightarrow\operatorname{Aut}(A)$, the group of automorphisms of $A$.
(Here, strong continuity means that for all $a\in A$, the map
$g\mapsto\alpha_{g}(a)$ is continuous as a function from $G$ to $A$.) The trio
$(A,G,\alpha)$ is called a ($C^{*}$-dynamical) system.

We will now define the dual group. A more detailed treatment of dual groups
can be found in [12, Chapter 1.7].

######  Definition 2.31.

Given a abelian locally compact group $G$, we say that a character of $G$ is a
continuous homomorphism $\chi:G\rightarrow\mathbb{T}$, where
$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$ denotes the unit circle in the complex
numbers. The set of characters $\widehat{G}$ forms an abelian locally compact
group called the dual group under the operation
$(\chi_{1}*\chi_{2})(g):=\chi_{1}(g)\chi_{2}(g)$ and the topology of uniform
convergence on compacta.

Some examples of dual groups are that
$\widehat{\mathbb{Z}}^{k}\cong\mathbb{T}^{k}$,
$\widehat{\mathbb{R}}^{k}\cong\mathbb{R}^{k}$, and that if $G$ is finite and
abelian then $\widehat{G}\cong G$.

The following two theorems are well-known results about the duals of abelian
locally compact groups. They can be found in [12, Theorem 1.85] and [12,
Theorem 1.88] respectively.

######  Theorem 2.32 (Pontryagin duality theorem).

An abelian locally compact group is naturally isomorphic to its double dual by
the map $g\mapsto[\chi\mapsto\chi(g)]$. That is,
$G\cong\widehat{\widehat{G}}$.

######  Theorem 2.33.

If $G$ is a locally compact abelian group, then $G$ is discrete if and only if
$\widehat{G}$ is compact.

We are now ready to prove the equivalence between coactions and actions by a
dual group.

######  Lemma 2.34.

Let $G$ be a discrete abelian group and let $A$ be a $C^{*}$-algebra. Then for
every $C^{*}$-dynamical system $(A,\widehat{G},\alpha)$, there is a unique
cosystem $(A,G,\delta)$ which satisfies

$A_{g}=\\{a\in A:\alpha_{\chi}(a)=\chi(g)\cdot a\text{ for all
}\chi\in\widehat{G}\\}$

for all $g\in G$. All cosystems by discrete abelian groups arise in this way.

###### Proof.

Since $G$ is abelian, it is amenable. Thus by Lemma 2.29, a cosystem is
equivalent to a topological grading whose graded components are the
$\\{A_{g}\\}_{g\in G}$. By [19, Theorem 3], such a topological grading is
equivalent to the desired group action. ∎

###  2.4 Graphs and $P$-graphs

######  Definition 2.35.

Let $(G,P)$ be an ordered group, and consider $P$ as a category with one
object where the morphisms are the elements of $P$ under their multiplication
structure. A $P$-graph is a countable category $\Lambda$ along with a functor
$d:\Lambda\rightarrow P$ with the unique factorization property: if
$\lambda\in\Lambda$ and $p_{1},p_{2}\in P$ with $p_{1}p_{2}=d(\lambda)$, then
there exists unique $\lambda_{1},\lambda_{2}\in\Lambda$ with
$\lambda=\lambda_{1}\lambda_{2}$ and $d(\lambda_{1})=p_{1}$ (and hence
necessarily $d(\lambda_{2})=p_{2}$).

We refer to the morphisms of $\Lambda$ as paths, and identity morphisms in
$\Lambda$ as vertices. We let $\Lambda^{0}$ denote the set of vertices in
$\Lambda$, and for $p\in P$, we write
$\Lambda^{p}=\\{\mu\in\Lambda:d(\mu)=p\\}$. As is common in the literature, we
identify the objects in $\Lambda$ with the identity morphism at those objects,
so we will refer to an object $v$ with its identity morphism
$\operatorname{id}_{v}$ interchangeably. Given a path $\lambda\in\Lambda$, let
$r(\lambda)$ denote its range vertex and $s(\lambda)$ its source vertex. We
write the composition of paths “working backwards” so that given
$\alpha,\beta\in\Lambda$, the product $\alpha\beta$ is defined if and only if
$s(\alpha)=r(\beta)$, in which case $s(\alpha\beta)=s(\beta)$ and
$r(\alpha\beta)=r(\alpha)$. This can be summarized in the following diagram:

${r(\alpha)}$${s(\alpha)=r(\beta)}$${s(\beta)\par}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\alpha\beta}$

######  Remark 2.36.

The existence of the degree functor and the unique factorization property
gives several nice properties to the graph.

First, $\Lambda^{0}=\\{\lambda\in\Lambda:d(\lambda)=1_{G}\\}$, since for a
vertex $v$, we have $v^{2}=v$, so $d(v)^{2}=d(v)$, so $d(v)=1_{G}$, and
conversely if $d(v)=1_{G}$, then $r(v)v=v=vs(v)$ are two factorizations with
the same degrees, so $r(v)=v=s(v)$ and hence $v$ is a vertex.

Second, having $\alpha\beta=s(\beta)$ implies $\alpha=\beta=s(\beta)$, since
then $d(\alpha)d(\beta)=d(s(\beta))=1$, so $d(\alpha),d(\beta)\in P\cap
P^{-1}=\\{1\\}$.

Finally, the category has both left- and right-cancellation. For left
cancellation, if $\alpha\beta=\alpha\gamma$, then this provides two
factorizations, so by the uniqueness of the factorizations, we have
$\beta=\gamma$, and a nearly identical argument shows right-cancellation.

These properties together imply that a $P$-graph is a category of paths in the
sense of [22]. Our representation of a $P$-graph will be a special case of
their representation of a category of paths, and the reader is invited to
compare our Definitions 2.41 and 2.44c to Theorems 6.3 and 8.2 of [22],
respectively.

######  Definition 2.37.

Let $(G,P)$ be an ordered group, let $\Lambda$ be a $P$-graph. For
$\alpha\in\Lambda$, we write $\alpha\Lambda=\\{\alpha\mu:\mu\in\Lambda\\}$.

We can give $\Lambda$ a partial order $\leq$ by saying $\alpha\leq\beta$ if
there exists some $\alpha_{1}\in\Lambda$ such that $\alpha\alpha_{1}=\beta$.
That is, $\alpha\leq\beta$ if and only if $\beta\in\alpha\Lambda$.

We say that $\alpha$ and $\beta$ have a common extension if there is a
$\mu\in\Lambda$ such that $\alpha\leq\mu$ and $\beta\leq\mu$.

Given $\alpha,\beta\in\Lambda$, we say that $\mu\in\Lambda$ is a minimal
common extension of $\alpha$ and $\beta$ if $\mu$ is a common extension of
$\alpha$ and $\beta$, and for all other common extensions $\nu$, $\nu\leq\mu$
implies $\nu=\mu$. We let $MCE(\alpha,\beta)$ denote the set of minimal common
extensions of $\alpha$ and $\beta$.

For general ordered groups, the ordering on $\Lambda$ may be sufficiently
poorly behaved that there are no _minimal_ common extensions, even if there
are common extensions. However, the humble hypothesis of weak quasi-lattice
ordering on $(G,P)$ prevents this catastrophe, and therefore will recur as a
hypothesis in almost all results relating to $P$-graphs. The following lemma
is the main “entry-point” for the hypothesis of weak quasi-lattice ordering
into $P$-graphs:

######  Lemma 2.38.

Let $(G,P)$ be an ordered group, let $\Lambda$ be a $P$-graph, and
$\alpha,\beta\in\Lambda$. If $(G,P)$ is weakly quasi-lattice ordered (WQLO),
then:

  1. 1.

Let
$MDCE(\alpha,\beta)=\\{\mu\in\alpha\Lambda\cap\beta\Lambda:d(\mu)=d(\alpha)\vee
d(\beta)\\}$ denote the set of minimal degree common extensions. For every
common extension $\lambda$ of $\alpha$ and $\beta$, there is a $\mu\in
MDCE(\alpha,\beta)$ with $\mu\leq\lambda$.

  2. 2.

$MCE(\alpha,\beta)=MDCE(\alpha,\beta)$.

  3. 3.

For every common extension $\lambda$ of $\alpha$ and $\beta$, there is a
$\mu\in MCE(\alpha,\beta)$ with $\mu\leq\lambda$.

  4. 4.

$\displaystyle\alpha\Lambda\cap\beta\Lambda=\bigsqcup_{\mu\in
MCE(\alpha,\beta)}\mu\Lambda$, where $\bigsqcup$ denotes a disjoint union.

###### Proof.

(1) Suppose that $\lambda$ is a common extension of $\alpha$ and $\beta$, so
$\lambda=\alpha\alpha_{1}=\beta\beta_{1}$.

Note that $d(\lambda)\geq d(\alpha)$ and $d(\lambda)\geq d(\beta)$, so
$d(\lambda)\geq d(\alpha)\vee d(\beta)$. Thus by factorization, we may write
$\lambda=\mu\mu_{1}$ where $d(\mu)=d(\alpha)\vee d(\beta)$. Now, $d(\mu)\geq
d(\alpha)$, so $\mu$ may be additionally factored as
$\mu=\alpha^{\prime}\alpha_{2}$, where $d(\alpha^{\prime})=d(\alpha)$. Then we
have $\lambda=\alpha\alpha_{1}=\mu\mu_{1}=\alpha^{\prime}\alpha_{2}\mu_{1}$,
meaning we have two factorizations of $\lambda$ whose initial segments are of
equal length $d(\alpha^{\prime})=d(\alpha)$. Then by uniqueness of
factorizations we have $\alpha^{\prime}=\alpha$, so $\mu$ is an extension of
$\alpha$. Similarly, $\mu$ will be an extension of $\beta$, so $\mu$ is a
common extension of $\alpha$ and $\beta$ of length $d(\alpha)\vee d(\beta)$,
so $\mu\in MDCE(\alpha,\beta)$ as desired.

(2) If $\nu\in MCE(\alpha,\beta)$, then by (1) there is some $\mu\in
MDCE(\alpha,\beta)$ with $\mu\leq\nu$. Then by minimality of $\nu$, we have
$\mu=\nu$, so $\nu\in MDCE(\alpha,\beta)$ and thus $MCE(\alpha,\beta)\subseteq
MDCE(\alpha,\beta)$.

Conversely, if $\mu\in MDCE(\alpha,\beta)$, suppose there were some
$\nu\in\Lambda$ with $\alpha\leq\nu,\beta\leq\nu,\nu\leq\mu$. Then by (1),
there would be some $\mu^{\prime}\in MDCE(\alpha,\beta)$ with
$\mu^{\prime}\leq\nu\leq\mu$. Then $\mu^{\prime}\leq\mu$, so we may write $\mu
s(\mu)=\mu=\mu^{\prime}\mu_{1}$ for some $\mu_{1}\in\Lambda$, and since
$d(\mu)=d(\alpha)\vee d(\beta)=d(\mu^{\prime})$, then by the uniqueness of the
factorization we have $\mu=\mu^{\prime}$. Then $\mu=\nu$, so $\mu$ is
“minimal” in the appropriate sense such that $\mu\in MCE(\alpha,\beta)$. Thus
$MDCE(\alpha,\beta)=MCE(\alpha,\beta)$.

(3) Immediately follows from (1) and (2).

(4) It is immediate that $\bigcup_{\mu\in
MCE(\alpha,\beta)}\mu\Lambda\subseteq\alpha\Lambda\cap\beta\Lambda$, and by
(3) we have $\alpha\Lambda\cap\beta\Lambda\subseteq\bigcup_{\mu\in
MCE(\alpha,\beta)}\mu\Lambda$, so it suffices to show the union is disjoint.
To this end, if $\mu,\nu\in MCE(\alpha,\beta)$ and
$\lambda\in\mu\Lambda\cap\nu\Lambda$, then we would have
$\lambda=\mu\mu_{1}=\nu\nu_{1}$, but this presents two factorizations with
equal degrees $d(\mu)=d(\alpha)\vee d(\beta)=d(\nu)$, so we must have
$\mu=\nu$. Thus the $\\{\mu\Lambda:\mu\in MCE(\alpha,\beta)\\}$ are pairwise
disjoint, as desired. ∎

######  Remark 2.39.

It should be noted that much of the literature (e.g. [3, Definition 2.3])
defines $MCE(\alpha,\beta)$ as we have defined $MDCE(\alpha,\beta)$, which the
previous lemma shows are equivalent. We have chosen to emphasize the
definition arising purely from the category structure because later we will be
considering a category as a $P$-graph and as a $Q$-graph, and this definition
clarifies that the category structure does not depend on the choice of the
grading (as long as one exists).

###  2.5 Representations of $P$-graphs and $P$-graph Algebras

Two additional notions will be needed in our study of $P$-graph algebras.

######  Definition 2.40.

Let $(G,P)$ be a WQLO group, and let $\Lambda$ be a $P$-graph. We say that
$\Lambda$ is finitely-aligned if for all $\mu,\nu\in\Lambda$, $MCE(\mu,\nu)$
is finite.

We say that a set $A\subseteq\Lambda$ is exhaustive for a set
$B\subseteq\Lambda$ if for all $\beta\in B$ there is an $\alpha\in A$ such
that $\alpha$ and $\beta$ have a common extension.

We are at last ready to define our main object of study, the representations
of $P$-graphs in a $C^{*}$-algebra. The following definition is due to [3],
with the slight modification to apply to WQLO groups:

######  Definition 2.41.

Let $(G,P)$ be a WQLO group and $\Lambda$ a finitely-aligned $P$-graph. A
representation of $\Lambda$ in a $C^{*}$-algebra $B$ is a function
$t:\Lambda\rightarrow B,\lambda\mapsto t_{\lambda}$ such that:

  1. (T1)

$t_{v}=t_{v}^{*}$ and $t_{v}t_{w}=\delta_{v,w}t_{v}$ for all
$v,w\in\Lambda^{0}$ (here $\delta$ denotes the Kronecker delta). That is,
$\\{t_{v}:v\in\Lambda^{0}\\}$ is a collection of pairwise orthogonal
projections.

  2. (T2)

$t_{\mu}t_{\nu}=t_{\mu\nu}$ whenever $s(\mu)=r(\nu)$.

  3. (T3)

$t_{\mu}^{*}t_{\mu}=t_{s(\mu)}$ for all $\mu\in\Lambda$.

  4. (T4)

$t_{\mu}t_{\mu}^{*}t_{\nu}t_{\nu}^{*}=\displaystyle\sum_{\lambda\in
MCE(\mu,\nu)}t_{\lambda}t_{\lambda}^{*}$ for all $\mu,\nu\in\Lambda$.

We denote by $C^{*}(t)$ the $C^{*}$-algebra generated by the
$\\{t_{\lambda}\\}_{\lambda\in\Lambda}$, and $C^{*}(t)$ is called a $P$-graph
$C^{*}$-algebra or just $P$-graph algebra.

We require that our graphs be finitely-aligned so that the expression in the
(T4) relator is a finite sum. The fact that an infinite sum of projections
does not (usually) converge in the norm topology is a perennial problem in the
field of generalizing Cuntz-Krieger algebras, and our solution is to limit our
attention to the finitely-aligned case where the issue does not arise.

We will often want to check the properties of $C^{*}(t)$ on a dense spanning
set, so the following lemma is useful. This result is widely known (see for
instance [3, Remark 5.2]).

######  Lemma 2.42.

Let $(G,P)$ be a WQLO group, let $\Lambda$ a finitely aligned $P$-graph, and
let $t$ be a representation of a $P$-graph. Then
$C^{*}(t)=\overline{\operatorname{span}}\\{t_{\alpha}t_{\beta}^{*}:\alpha,\beta\in\Lambda\\}$.

###### Proof.

Fix some $\alpha,\beta,\mu,\nu\in\Lambda$. If $s(\alpha)\neq s(\beta)$ or
$s(\mu)\neq s(\nu)$, then by the T1 relator we have
$t_{\alpha}t_{\beta}^{*}=0$ or $t_{\mu}t_{\nu}^{*}=0$ respectively, and in
either case $(t_{\alpha}t_{\beta}^{*})(t_{\mu}t_{\nu}^{*})=0$. If instead
$s(\alpha)=s(\beta)$ and $s(\mu)=s(\nu)$, then

$\displaystyle(t_{\alpha}t_{\beta}^{*})(t_{\mu}t_{\nu}^{*})$ $\displaystyle=$
$\displaystyle(t_{\alpha}t_{\beta}^{*})(t_{\beta}t_{\beta}^{*})(t_{\mu}t_{\mu}^{*})(t_{\mu}t_{\nu}^{*})\text{
by T3 and T2}$ $\displaystyle=$
$\displaystyle(t_{\alpha}t_{\beta}^{*})\left(\displaystyle\sum_{\lambda\in
MCE(\beta,\mu)}t_{\lambda}t_{\lambda}^{*}\right)(t_{\mu}t_{\nu}^{*})\text{ by
T4}$ $\displaystyle=$ $\displaystyle\displaystyle\sum_{\lambda\in
MCE(\beta,\mu)}t_{\alpha(\beta^{-1}\lambda)}t_{\nu(\mu^{-1}\lambda)}\text{ by
T2 and T3}$

where $\beta^{-1}\lambda$ denotes the unique path for which
$\beta(\beta^{-1}\lambda)=\lambda$, and similarly $\mu^{-1}\lambda$ denotes
the unique path for which $\mu(\mu^{-1}\lambda)=\lambda$.

This calculation shows that
$\operatorname{span}\\{t_{\alpha}t_{\beta}^{*}:\alpha,\beta\in\Lambda\\}$ is
closed under multiplication, so it is a $*$-subalgebra of $C^{*}(t)$, and it
contains each $t_{\alpha}$ since $t_{\alpha}=t_{\alpha}t_{s(\alpha)}^{*}$.
Thus
$\overline{\operatorname{span}}\\{t_{\alpha}t_{\beta}^{*}:\alpha,\beta\in\Lambda\\}$
is a $C^{*}$-subalgebra of $C^{*}(t)$ containing all of the generators, so
$C^{*}(t)=\overline{\operatorname{span}}\\{t_{\alpha}t_{\beta}^{*}:\alpha,\beta\in\Lambda\\}$.

∎

######  Example 2.43.

As we have already seen with groups and Fell bundles, a simple way of
representing an object often comes from its own action on itself, and such an
example arises for representations of $P$-graphs as well. To define the left-
regular representation of a $P$-graph $\Lambda$, let
$\mathcal{H}=\ell^{2}(\Lambda)$, with the typical orthonormal basis
$\\{e_{\alpha}\\}_{\alpha\in\Lambda}$. (Kribs and Power call this the Fock
space of $\Lambda$ in their papers [13, 14].) For $\mu\in\Lambda$ define a
“forward shift by $\mu$” operator $L_{\mu}$ by

$L_{\mu}e_{\alpha}=\begin{cases}e_{\mu\alpha}&\text{ if }s(\mu)=r(\alpha)\\\
0&\text{ if }s(\mu)\neq r(\alpha)\end{cases}$

and extending this linearly. A short computation on the inner product shows
that $L_{\mu}^{*}$ is the “backwards shift by $\mu$” operator, which is given
by

$L_{\mu}^{*}e_{\beta}=\begin{cases}e_{\alpha}&\text{ if there exists
}\alpha\text{ with }\mu\alpha=\beta\\\ 0&\text{ if no such }\alpha\text{
exists}\end{cases}$

(noting that if such an $\alpha$ does exist, it is unique by left
cancellation).

We will now show that $\\{L_{\mu}\\}_{\mu\in\Lambda}$ is a representation of
$\Lambda$. The T1 and T2 relations are immediate.

For the T3 relation, observe that

$L_{\mu}^{*}L_{\mu}e_{\alpha}=L_{\mu}^{*}\left(\begin{cases}e_{\mu\alpha}&\text{
if }s(\mu)=r(\alpha)\\\ 0&\text{ if }s(\mu)\neq
r(\alpha)\end{cases}\right)=\begin{cases}e_{\alpha}&\text{ if
}s(\mu)=r(\alpha)\\\ 0&\text{ if }s(\mu)\neq
r(\alpha)\end{cases}=L_{s(\mu)}e_{\alpha}$

as desired.

For the T4 relation, first note that

$L_{\mu}L_{\mu}^{*}e_{\alpha}=\begin{cases}e_{\alpha}&\text{ if
}\alpha\in\mu\Lambda\\\ 0&\text{ if }\alpha\not\in\mu\Lambda\end{cases}.$

Thus

$L_{\mu}L_{\mu}^{*}L_{\nu}L_{\nu}^{*}e_{\alpha}=\begin{cases}e_{\alpha}&\text{
if }\alpha\in\mu\Lambda\cap\nu\Lambda\\\ 0&\text{ if
}\alpha\not\in\mu\Lambda\cap\nu\Lambda\end{cases}.$

But since $\mu\Lambda\cap\nu\Lambda=\bigsqcup_{\lambda\in
MCE(\mu,\nu)}\lambda\Lambda$ by Lemma 2.38(4), then if
$\alpha\in\mu\Lambda\cap\nu\Lambda$ there exists a unique $\lambda_{0}\in
MCE(\mu,\nu)$ with $\alpha\in\lambda\Lambda$, so
$L_{\lambda_{0}}L_{\lambda_{0}}^{*}e_{\alpha}=e_{\alpha}$, and
$L_{\lambda}L_{\lambda}^{*}e_{\alpha}=0$ for $\lambda\in
MCE(\mu,\nu)\setminus\lambda_{0}$. Thus

$\displaystyle\sum_{\lambda\in
MCE(\mu,\nu)}L_{\lambda}L_{\lambda}^{*}e_{\alpha}=e_{\alpha}$

when $\alpha\in\mu\Lambda\cap\nu\Lambda$, so
$L_{\mu}L_{\mu}^{*}L_{\nu}L_{\nu}^{*}=\displaystyle\sum_{\lambda\in
MCE(\mu,\nu)}L_{\lambda}L_{\lambda}^{*}$ as desired.

Thus $\\{L_{\mu}\\}_{\mu\in\Lambda}$ is a representation, as desired.

There are several additional properties that we will sometimes ask of our
representation. In order, we will ask for our generators to be nonzero, that
our representation preserve its knowledge of the $P$-graded structure through
a certain coaction called a gauge coaction, and that our representation be
“tight” in the sense of [8]. More precisely:

######  Definition 2.44.

Let $(G,P)$ be a WQLO group and $\Lambda$ a finitely aligned $P$-graph. We say
that a representation $t$ of $\Lambda$…

  1. a.

is $\Lambda$-faithful if each $t_{\lambda}$ is nonzero.

  2. b.

has a gauge coaction if there is a $G$-coaction $\delta$ on $C^{*}(t)$ such
that

$\delta(t_{\lambda})=t_{\lambda}\otimes U_{d(\lambda)}$

for all $\lambda\in\Lambda$.

  3. c.

is tight if whenever $\mu\in\Lambda$ and $E\subset\mu\Lambda$ is finite and
exhaustive for $\mu\Lambda$, we have

$\displaystyle\prod_{\alpha\in
E}(t_{\mu}t_{\mu}^{*}-t_{\alpha}t_{\alpha}^{*})=0.$

(Recall that $E$ is exhaustive for $\mu\Lambda$ if for all $\nu\in\mu\Lambda$,
there is an $\alpha\in E$ such that $\alpha$ and $\nu$ have a common
extension.) This terminology is motivated by [8], which defines a tight
representation of a semilattice. We show in Appendix 1 that this notion of
tight is equivalent to the one in [8].

  4. d.

canonically covers another representation $s$ if there is a (necessarily
surjective) $*$-homomorphism from $C^{*}(t)$ to $C^{*}(s)$ given by
$t_{\lambda}\mapsto s_{\lambda}$. In such a case, we will write
$\pi_{s}^{t}:C^{*}(t)\rightarrow C^{*}(s)$ to denote the $*$-homomorphism,
which we call the canonical covering. The notation $\pi_{s}^{t}$ should
hopefully be suggestive of $t$ “covering” $s$.

  5. e.

is canonically isomorphic to another representation $s$ if there is an
isomorphism between $C^{*}(s)$ and $C^{*}(t)$ given by $t_{\lambda}\mapsto
s_{\lambda}$ (i.e. if $s$ and $t$ canonically cover each other).

It should be noted that the conditions (T1)-(T4) and tightness are all
polynomial relations of the form suitable for Lemma 2.9 (finite-alignment
being necessary in the case of (T4)). However, $\Lambda$-faithfulness and the
existence of a gauge coaction are not polynomial relators in an obvious way.

######  Remark 2.45.

The left-regular representation $L$ from Example 2.43 is $\Lambda$-faithful.

However, this representation is as far from tight as possible: given
$\mu\in\Lambda$ and $E\subset\mu\Lambda$ which is finite and exhaustive for
$\mu\Lambda$, $\displaystyle\prod_{\alpha\in
E}(L_{\mu}L_{\mu}^{*}-L_{\alpha}L_{\alpha}^{*})=0$ if and only if $\mu\in E$
(i.e. when one of the terms of the product is 0). To see this, note that if
$\alpha\in\mu\Lambda\setminus\mu$, then $L_{\alpha}L_{\alpha}^{*}e_{\mu}=0$,
so $(L_{\mu}L_{\mu}^{*}-L_{\alpha}L_{\alpha}^{*})e_{\mu}=e_{\mu}$, and thus
taking the product over $\alpha\in E\subset\mu\Lambda\setminus\mu$, we get
that $\displaystyle\prod_{\alpha\in
E}(L_{\mu}L_{\mu}^{*}-L_{\alpha}L_{\alpha}^{*})e_{\mu}=e_{\mu}\neq 0$.

We will show in Corollary 4.38 that $L$ has a gauge coaction if $(G,P)$ has a
strong reduction to an amenable group.

######  Lemma 2.46.

Let $(G,P)$ be a WQLO group, and let $\Lambda$ be a finitely aligned
$P$-graph. One may apply Lemma 2.9 to the relators (T1)-(T4), so there is a
representation of $\Lambda$ which is universal.

###### Proof.

Observe that (T1) and (T3) together imply that the generators are partial
isometries. Also, all of the relators are polynomials in the generators (in
the case of (T4), we must note that the expression is finite since the graph
is finitely aligned). Thus there is a universal representation of $\Lambda$. ∎

######  Definition 2.47.

We call the universal representation of a $P$-graph $\Lambda$ the
Toeplitz(-Cuntz-Krieger) representation of $\Lambda$. We will use
$\mathcal{T}$ to denote this representation, and write
$\mathcal{T}C^{*}(\Lambda)$ or $C^{*}(\mathcal{T})$ for the algebra generated
by it which we call the Toeplitz(-Cuntz-Krieger) algebra of the graph.

######  Lemma 2.48.

Let $(G,P)$ be a WQLO group, and let $\Lambda$ be a finitely aligned
$P$-graph.

  1. 1.

Let $Rep(\Lambda)$ denote the set of all representations of $\Lambda$ up to
canonical isomorphism. Then $Rep(\Lambda)$ can be given a partial ordering
$\geq_{rep}$ by saying that $s\geq_{rep}t$ if and only if $s$ canonically
covers $t$ (and $t\cong s$ if and only if they are canonically isomorphic).

  2. 2.

There is a bijection $\Psi$ from $Rep(\Lambda)$ to ideals in
$\mathcal{T}C^{*}(\Lambda)$ given by $t\mapsto\ker\pi_{t}^{\mathcal{T}}$,
which is order-reversing in the sense that
$t\leq_{rep}s\iff\Psi(t)\supseteq\Psi(s)$.

###### Proof.

For (1), we must check the partial order properties of reflexivity,
transitivity, and antisymmetry.

Certainly each representation canonically covers itself by the identity map,
so $\leq_{rep}$ is reflexive.

If $t\leq_{rep}s$ and $s\leq_{rep}r$, then there are canonical coverings
$\pi_{s}^{r}:C^{*}(r)\rightarrow C^{*}(s)$ and
$\pi_{t}^{s}:C^{*}(s)\rightarrow C^{*}(t)$. Then the composition
$\pi_{t}^{r}:=\pi_{t}^{s}\circ\pi_{s}^{r}$ is a canonical covering of $t$ by
$r$, so $t\leq_{rep}r$, giving transitivity.

If $t\leq_{rep}s$ and $s\leq_{rep}t$, then there are canonical coverings
$\pi_{t}^{s}:C^{*}(s)\rightarrow C^{*}(t)$ and
$\pi_{s}^{t}:C^{*}(t)\rightarrow C^{*}(s)$. Composing these maps in either
direction shows that they fix each generator, and therefore are the identity
maps. Therefore $\pi_{t}^{s}$ and $\pi_{s}^{t}$ are inverses and hence
canonical isomorphisms.

Thus $\leq_{rep}$ is indeed a partial order, as desired.

For (3), it is certainly true that $\Psi$ is a well-defined function from
$Rep(\Lambda)$ to $\\{\text{ideals in }\mathcal{T}C^{*}(\Lambda)\\}$.

We will first check that if $s,t\in Rep(\Lambda)$, then
$t\leq_{rep}s\iff\Psi(t)\supseteq\Psi(s)$. If $t\leq_{rep}s$, then there is a
canonical covering $\pi^{s}_{t}:C^{*}(s)\rightarrow C^{*}(t)$, and since
$\pi^{s}_{t}\circ\pi^{\mathcal{T}}_{s}=\pi^{\mathcal{T}}_{t}$, then
$\Psi(t)=\ker\pi^{\mathcal{T}}_{t}\supseteq\pi^{\mathcal{T}}_{s}=\Psi(s)$.
Conversely, if $\Psi(t)\supseteq\Psi(s)$, then by Lemma 2.8 applied to
$\pi^{\mathcal{T}}_{t}$ and $\pi^{\mathcal{T}}_{s}$, there is a
$*$-homomorphism $\pi_{t}^{s}:C^{*}(s)\rightarrow C^{*}(t)$ with
$\pi_{t}^{s}\circ\pi^{\mathcal{T}}_{s}=\pi^{\mathcal{T}}_{t}$so $\pi_{t}^{s}$
is a canonical covering and $t\leq_{rep}s$. This completes the proof that
$\Psi$ is order-reversing.

Now to show that $\Psi$ is a bijection, if $\Psi(s)=\Psi(t)$, then by the
above $s\leq_{rep}t$ and $t\leq_{rep}s$, so $s\cong t$ and hence $\Psi$ is
injective. For surjectivity, if $J\lhd\mathcal{T}C^{*}(\Lambda)$ is an ideal,
represent $\Lambda$ by $\lambda\mapsto\mathcal{T}_{\lambda}+J$. Then this
representation has kernel $J$, so $\Psi$ is surjective, as desired. ∎

As a consequence of this, representations have the order structure of the
family of ideals of an algebra, and in particular have a lattice order (where
meet and join are represented by intersection and addition of ideals).

######  Remark 2.49.

$\Lambda$-faithfulness is a “largeness” condition in that if $s\leq t$ and $s$
is $\Lambda$-faithful, then so is $t$.

Tightness is a “smallness” condition in that if $s\leq t$ and $t$ is tight,
then so is $s$.

####  2.5.1 Gauge coactions and Coactionization

It may not be obvious what role “has a gauge coaction” plays: is it saying a
representation is “large” or “small”?

The following proposition is a tangled web of results, but it shows that the
existence of a gauge coaction is properly considered a “largeness” condition:
any representation can be “lifted” (1) to have a gauge coaction (4), the
lifted version covers the original (2) and is the smallest gauge coacting
representation to do so (6), and they are isomorphic if and only if the
original had a gauge coaction (5). We also simplify the notion of a gauge
coaction by showing that any homomorphism satisfying $t_{\lambda}\mapsto
t_{\lambda}\otimes U_{d(\lambda)}$ is automatically a gauge coaction (3), and
show that a representation is $\Lambda$-faithful or tight if and only if its
coactionization is (7).

######  Proposition 2.50.

Let $(G,P)$ be a WQLO group, and let $\Lambda$ be a finitely aligned
$P$-graph. Let $t$ be a representation of $\Lambda$.

  1. 1.

There is a representation $t^{\prime}$ of $\Lambda$ given by
$t^{\prime}_{\lambda}=t_{\lambda}\otimes U_{d(\lambda)}$.

  2. 2.

There is a canonical covering $t^{\prime}\mapsto t$.

  3. 3.

If there is a $*$-homomorphism $\delta:C^{*}(t)\rightarrow C^{*}(t)\otimes
C^{*}(G)$ satisfying $\delta(t_{\lambda})=t^{\prime}_{\lambda}$, then $\delta$
is a coaction, and hence a gauge coaction.

  4. 4.

$t^{\prime}$ has a gauge coaction.

  5. 5.

$t$ is canonically isomorphic to $t^{\prime}$ if and only if $t$ has a gauge
coaction.

  6. 6.

If $s\leq t$, then $s^{\prime}\leq t^{\prime}$, and in particular for any
representation $s$, $s^{\prime}$ is the smallest gauge coacting representation
that covers $s$.

  7. 7.

$t^{\prime}$ is $\Lambda$-faithful (respectively, tight) if and only if $t$
is.

Due to these properties, we believe that $t^{\prime}$ deserves to be called
the coactionization of $t$.

###### Proof.

The proof of (1) is a routine confirmation of the T1-T4 relators, which we
include here. We have:

T1: | $t_{v}^{\prime}t_{w}^{\prime}$ | = | $(t_{v}\otimes U_{e})(t_{w}\otimes U_{e})=t_{v}t_{w}\otimes U_{e}=\delta_{v,w}t_{v}\otimes U_{e}=\delta_{v,w}t_{v}^{\prime}$
---|---|---|---
T2: | $t_{\mu}^{\prime}t_{\nu}^{\prime}$ | = | $(t_{\mu}\otimes U_{d(\mu)})(t_{\nu}\otimes U_{d(\nu)})=(t_{\mu}t_{\nu})\otimes U_{d(\mu)}U_{d(\nu)}$
|  | = | $t_{\mu\nu}\otimes U_{d(\mu\nu)}=t_{\mu\nu}^{\prime}$
T3: | ${t_{\mu}^{\prime}}^{*}t_{\mu}^{\prime}$ | = | $(t_{\mu}\otimes U_{d(\mu)})^{*}(t_{\mu}\otimes U_{d(\mu)})=t_{\mu}^{*}t_{\mu}\otimes U_{d(\mu)}^{*}U_{d(\mu)}$
|  | = | $t_{s(\mu)}\otimes U_{e}=t_{s(\mu)}^{\prime}$
T4: | $t_{\nu}^{\prime}{t_{\nu}^{\prime}}^{*}t_{\mu}^{\prime}t_{\mu}^{\prime*}$ | = | $(t_{\nu}\otimes U_{d(\nu)})(t_{\nu}\otimes U_{d(\nu)})^{*}(t_{\mu}\otimes U_{d(\mu)})(t_{\mu}\otimes U_{d(\mu)})^{*}$
|  | = | $(t_{\nu}t_{\nu}^{*}t_{\mu}t_{\mu}^{*})\otimes U_{e}=\displaystyle\sum_{\lambda\in MCE(\mu,\nu)}t_{\lambda}t_{\lambda}^{*}\otimes U_{e}$
|  | = | $\displaystyle\sum_{\lambda\in MCE(\mu,\nu)}t_{\lambda}^{\prime}{t_{\lambda}^{\prime}}^{*}$

all as desired.

For (2), let $V$ denote the trivial representation of $G$ (given by
$V_{g}=1\in\mathbb{C}$ for all $g\in G$). Then by the universal property of
$C^{*}(G)$, there is a covering map
$\pi^{U}_{V}:C^{*}(G)\rightarrow\mathbb{C}$ given by
$\pi^{U}_{V}(U_{g})=V_{g}=1$.

Now by Lemma 2.14, there is a map
$\operatorname{id}_{C^{*}(t)}\otimes\pi^{U}_{V}:C^{*}(t)\otimes
C^{*}(G)\rightarrow C^{*}(t)\otimes\mathbb{C}$ satisfying

$\displaystyle(\operatorname{id}_{C^{*}(t)}\otimes\pi^{U}_{V})(t^{\prime}_{\lambda})$
$\displaystyle=$
$\displaystyle(\operatorname{id}_{C^{*}(t)}\otimes\pi^{U}_{V})(t_{\lambda}\otimes
U_{d(\lambda)})$ $\displaystyle=$ $\displaystyle t_{\lambda}\otimes 1.$

By identifying $C^{*}(t)$ with $C^{*}(t)\otimes\mathbb{C}$ in the natural way,
this gives the desired canonical covering.

For (3), we must check that $\delta$ is injective, nondegenerate, and
satisfies the coaction identity of Definition 2.25. To this end, note that if
$\pi_{t}^{t^{\prime}}$ is the canonical covering from the previous part, then
$\pi_{t}^{t^{\prime}}\circ\delta=\operatorname{id}_{C^{*}(t)}$, so $\delta$
must be injective. To check nondegeneracy, we must check that

$\overline{\operatorname{span}}\left[\delta(C^{*}(t))(C^{*}(t)\otimes
C^{*}(G))\right]=C^{*}(t)\otimes C^{*}(G).$

To this end, note that for all $g\in G$ and $\mu,\nu\in\Lambda$,

$\delta(t_{s(\mu)})(t_{\mu}t_{\nu}^{*}\otimes U_{g})=(t_{s(\mu)}\otimes
U_{1})(t_{\mu}t_{\nu}^{*}\otimes U_{g})=t_{\mu}t_{\nu}^{*}\otimes U_{g}.$

That is, $\\{t_{\mu}t_{\nu}^{*}\otimes U_{g}:\mu,\nu\in\Lambda,g\in
G\\}\subseteq\delta(C^{*}(t))(C^{*}(t)\otimes C^{*}(G))$. But since
$C^{*}(t)=\overline{\operatorname{span}}\\{t_{\mu}t_{\nu}^{*}:\mu,\nu\in\Lambda\\}$
and $C^{*}(G)=\overline{\operatorname{span}}\\{U_{g}\\}_{g\in G}$, then these
simple tensors have dense span $C^{*}(t)\otimes C^{*}(G)$. Thus

$C^{*}(t)\otimes
C^{*}(G)=\overline{\operatorname{span}}\\{t_{\mu}t_{\nu}^{*}\otimes
U_{g}\\}_{\mu,\nu\in\Lambda,g\in
G}=\overline{\operatorname{span}}\left[\delta(C^{*}(t))(C^{*}(t)\otimes
C^{*}(G))\right]$

as desired.

Finally, it suffices to check the coaction identity on each generator
$t_{\mu}$. To this end,

$\displaystyle((\delta\otimes\operatorname{id}_{G})\circ\delta)(t_{\mu})$
$\displaystyle=$
$\displaystyle(\delta\otimes\operatorname{id}_{G})(t_{\mu}\otimes U_{d(\mu)})$
$\displaystyle=$ $\displaystyle t_{\mu}\otimes U_{d(\mu)}\otimes U_{d(\mu)}$
$\displaystyle=$ $\displaystyle t_{\mu}\otimes\delta_{G}(U_{d(\mu)})$
$\displaystyle=$
$\displaystyle((\operatorname{id}_{C^{*}(t)}\otimes\delta_{G})(t_{\mu}\otimes
U_{d(\mu)})$ $\displaystyle=$
$\displaystyle((\operatorname{id}_{C^{*}(t)}\otimes\delta_{G})\circ\delta)(t_{\mu})$

as desired.

For (4), recall that there is a coaction $\delta_{G}:C^{*}(G)\rightarrow
C^{*}(G)\otimes C^{*}(G)$ given by $U_{g}\mapsto U_{g}\otimes U_{g}$. By Lemma
2.14 we may define $\bar{\delta}:C^{*}(t)\otimes C^{*}(G)\rightarrow
C^{*}(t)\otimes C^{*}(G)\otimes C^{*}(G)$ by
$\bar{\delta}=\operatorname{id}_{C^{*}(t)}\otimes\delta_{G}$. Let
$\delta=\bar{\delta}\mid_{C^{*}(t^{\prime})}$. Then for $\lambda\in\Lambda$,

$\delta(t^{\prime})=\delta(t_{\lambda}\otimes
U_{d(\lambda)})=t_{\lambda}\otimes\delta_{G}(U_{d(\lambda)})=t_{\lambda}\otimes
U_{d(\lambda)}\otimes U_{d(\lambda)}=t_{\lambda}^{\prime}\otimes
U_{d(\lambda)}.$

Thus by (3), $\delta$ is a gauge coaction of $C^{*}(t^{\prime})$.

For (5), if $t$ is canonically isomorphic to $t^{\prime}$, then there is a
homomorphism between their $C^{*}$-algebras given by $t_{\lambda}\mapsto
t^{\prime}_{\lambda}=t_{\lambda}\otimes U_{d(\lambda)}$. By (3) this is
precisely a gauge coaction.

For (6), if $s\leq t$, recall that $\pi_{s}^{t}$ denotes the canonical
covering. Then by Lemma 2.14 there is a $*$-homomorphism
$\pi_{s^{\prime}}^{t^{\prime}}:C^{*}(t^{\prime})\rightarrow C^{*}(s^{\prime})$
given by
$\pi_{s^{\prime}}^{t^{\prime}}=\pi_{s}^{t}\otimes\operatorname{id}_{C^{*}(G)}$,
and it is immediate that this is a canonical covering. For the “in
particular”, if $s\leq t$ and $t$ has a gauge coaction, then $t\cong
t^{\prime}$ by (5), so $s^{\prime}\leq t^{\prime}\cong t$, and thus
$s^{\prime}\leq t$, as desired.

For (7), observe that for a fixed $\lambda\in\Lambda$,

$\left\|{t_{\lambda}^{\prime}}\right\|^{2}=\left\|{t_{\lambda}\otimes
U_{d(\lambda)}}\right\|^{2}=\left\|{t_{\lambda}^{*}t_{\lambda}\otimes
1}\right\|=\left\|{t_{\lambda}^{*}t_{\lambda}}\right\|=\left\|{t_{\lambda}}\right\|^{2}$

so any $t_{\lambda}^{\prime}$ is 0 if and only if $t_{\lambda}$ is 0. Thus $t$
is $\Lambda$-faithful if and only if $t^{\prime}$ is $\Lambda$-faithful.

For tightness, fix a $\mu\in\Lambda$ and finite $E\subset\mu\Lambda$ which is
exhaustive for $\mu\Lambda$. Then

$\displaystyle\left\|{\displaystyle\prod_{\alpha\in
E}t_{\mu}^{\prime}{t_{\mu}^{\prime}}^{*}-t_{\alpha}^{\prime}{t_{\alpha}^{\prime}}^{*}}\right\|$
$\displaystyle=$ $\displaystyle\left\|{\left(\displaystyle\prod_{\alpha\in
E}t_{\mu}{t_{\mu}}^{*}-t_{\alpha}{t_{\alpha}}^{*}\right)\otimes 1}\right\|$
$\displaystyle=$ $\displaystyle\left\|{\displaystyle\prod_{\alpha\in
E}t_{\mu}{t_{\mu}}^{*}-t_{\alpha}{t_{\alpha}}^{*}}\right\|$

and in particular $\displaystyle\prod_{\alpha\in
E}t_{\mu}^{\prime}{t_{\mu}^{\prime}}^{*}-t_{\alpha}^{\prime}{t_{\alpha}^{\prime}}^{*}=0$
if and only if $\displaystyle\prod_{\alpha\in
E}t_{\mu}{t_{\mu}}^{*}-t_{\alpha}{t_{\alpha}}^{*}=0$, as desired. ∎

Finally, note that property (3) says that the notion of coaction is
dramatically simplified when considering just gauge coactions. Indeed, the
necessary and sufficient condition to having a gauge coaction is that the map
$t_{\lambda}\mapsto t_{\lambda}\otimes U_{d(\lambda)}$ extends to a
$*$-homomorphism on $C^{*}(t)$.

###  2.6 The Classical Gauge-Invariant Uniqueness Theorem

In this section we will introduce the gauge-invariant uniqueness theorem for
directed graphs (what we would call $\mathbb{N}$-graphs), and explain how we
hope to generalize it.

The following definitions and theorems are adapted from Chapters 1 and 2 of
[18].

######  Definition 2.51.

A directed graph is a tuple $E=(E^{0},E^{1},r,s)$ where $E^{0}$ and $E^{1}$
are countable sets, and $r,s:E^{1}\rightarrow E^{0}$ are functions. We think
of $E^{0}$ as the set of vertices in our directed graph, $E^{1}$ as our set of
edges, and $r$ and $s$ as the range and source maps for $E^{1}$. We say a
directed graph is row-finite if for each $v\in V$, $|r^{-1}(v)|<\infty$.

A Cuntz-Krieger $E$-family of a row-finite directed graph $(E^{0},E^{1},r,s)$
is a collection of operators $\\{P_{v}\\}_{v\in E^{0}}\cup\\{S_{e}\\}_{e\in
E^{1}}$ in a $C^{*}$-algebra such that $\\{P_{v}\\}_{v\in E^{0}}$ are a family
of pairwise orthogonal projections, and

  1. (CK1) $P_{s(e)}=S_{e}^{*}S_{e}$ for all $e\in E^{1}$ and

  2. (CK2) $P_{v}=\displaystyle\sum_{\begin{subarray}{c}e\in E^{1}\\\ r(e)=v\end{subarray}}S_{e}S_{e}^{*}$ whenever $r^{-1}(v)$ is nonempty.

The reader may wish to compare this definition to the definition of a
representation of a $P$-graph (Definition 2.41). Having done so, there are a
few differences that we will wish to reconcile.

######  Remark 2.52.

Given a directed graph $(E^{0},E^{1},r,s)$, for $n>1$ we let
$E^{n}=\\{(e_{1},e_{2},...,e_{n}):s(e_{i})=r(e_{i+1})\text{ for }1\leq
i<n\\}$. Then $E^{*}=\bigcup_{n\geq 0}E^{n}$ forms an $\mathbb{N}$-graph in a
natural way, where the operation is concatenation of tuples, and
$d(e_{1},...,e_{n})=n$. This $\mathbb{N}$-graph is always finitely aligned,
since for any $\mathbb{N}$-graph,

$MCE(\alpha,\beta)=\begin{cases}\\{\alpha\\}&\text{ if }\beta\leq\alpha\\\
\\{\beta\\}&\text{ if }\alpha\leq\beta\\\ \emptyset&\text{
otherwise}\end{cases}$

so $|MCE(\alpha,\beta)|\leq 1<\infty$.

Any Cuntz-Krieger $E$-family extends to a representation of $E^{*}$ by
$S_{(e_{1},e_{2},...,e_{n})}=S_{e_{1}}S_{e_{2}}...S_{e_{n}}$. In proving this,
the (T1) and (T2) relators are immediate, the (T3) relator follows rapidly
from (CK1), and the (T4) relator follows from the expression above for
$MCE(\alpha,\beta)$.

One should note that the (CK2) condition is analogous to tightness, since for
any vertex $v$, $r^{-1}(v)$ is finite and exhaustive for $v\Lambda$, and since
$\\{S_{e}S_{e}^{*}\\}_{e\in r^{-1}(v)}$ are pairwise orthogonal projections,
then

$\displaystyle\prod_{e\in r^{-1}(v)}(P_{v}P_{v}^{*}-S_{e}S_{e}^{*})=0\text{ if
and only if }P_{v}=\displaystyle\sum_{\begin{subarray}{c}e\in E^{1}\\\
r(e)=v\end{subarray}}S_{e}S_{e}^{*}.$

In other words, the (CK2) condition is saying that the representation is tight
in the case that $\mu\in\Lambda^{0}$ and $E\subset\Lambda^{1}$. Since every
path in an $\mathbb{N}$-graph can be uniquely factored into paths of length 1,
tightness in all cases is equivalent to this.

Due to this last point, readers should be aware that in the literature
representations of directed graphs (and higher rank graphs) are what we would
call tight representations.

In the notation of [18], $C^{*}(E)$ denotes the universal tight representation
of $E$, and if $\\{T_{v}\\}_{v\in E^{0}}\cup\\{Q_{e}\\}_{e\in E^{1}}$ is a
Cuntz-Krieger $E$-family, then $\pi_{T,Q}:C^{*}(E)\rightarrow C^{*}(T,Q)$
denotes the canonical covering.

The foundational result we wish to generalize is this, as stated in [18,
Theorem 2.2]:

######  Theorem 2.53 (Gauge Invariant Uniqueness Theorem for Graphs).

Let $E=(E^{0},E^{1},r,s)$ be a row-finite directed graph, and suppose that
$\\{T_{v}\\}_{v\in E^{0}}\cup\\{Q_{e}\\}_{e\in E^{1}}$ is a Cuntz-Krieger
$E$-family in a $C^{*}$-algebra $B$ with each $Q_{v}\neq 0$. If there is a
continuous action $\beta:\mathbb{T}\rightarrow\operatorname{Aut}B$ such that
$\beta_{z}(T_{e})=zT_{e}$ for every $e\in E^{1}$ and $\beta_{z}(Q_{v})=Q_{v}$
for every $v\in E^{0}$, then $\pi_{T,Q}$ is an isomorphism of $C^{*}(E)$ onto
$C^{*}(T,Q)$.

######  Remark 2.54.

The reader may recognize many of the hypotheses of Theorem 2.53 from our
earlier list of terminology for representations (Definition 2.44). The
statement $Q_{v}\neq 0$ for $v\in E^{0}$ implies that $T_{\mu}\neq 0$ for all
$\mu\in E^{*}$, so it is equivalent to our $\Lambda$-faithfulness. The
continuous action $\beta$ is called a gauge action and is equivalent by Lemma
2.34 to a gauge coaction. Also recall that tightness is a built-in hypothesis
in the representations considered in [18].

Therefore, we may restate Theorem 2.53 like so: for any $\mathbb{N}$-graph,
there is exactly one $\Lambda$-faithful, tight, gauge coacting representation
up to canonical isomorphism.

What we mean by a gauge invariant uniqueness theorem for $P$-graphs is a
similar statement: for any $P$-graph, there is exactly one $\Lambda$-faithful,
tight, gauge coacting representation up to canonical isomorphism.

We have already seen that in generalizing from $\mathbb{N}$-graphs to $P$
graphs, we needed an additional hypothesis (finite alignment). We will see in
Lemma 4.24 that such a gauge invariant uniqueness theorem need not be true in
general, so another hypothesis is needed. We spend Chapter 3 developing this
hypothesis on $(G,P)$, and then in Chapter 4 prove a gauge invariant
uniqueness theorem for $P$-graphs that satisfy this new hypothesis.

##  3 Reductions of Ordered Groups

In this section we will develop the notion of a reduction of an ordered group.
Roughly speaking, a reduction occurs when a positive cone $P$ in a group $G$
does not carry “enough” information about the group, and one is able to
replace $(G,P)$ with some $(H,Q)$ while preserving the essential properties
that will be required for a $P$-graph (namely, the structure of intervals
$[1,p]$). A special case of a reduction (called a “strong reduction”) is when
one can embed the positive cone $P$ in another group $H$ such that this
embedding extends to an order homomorphism from $(G,P)$ to $(H,P)$.

###  3.1 Definition and Basic Properties

######  Definition 3.1.

An order homomorphism $\varphi:(G,P)\rightarrow(H,Q)$ is a reduction if for
all $p\in P$, $\varphi$ is a bijection between the interval $[1,p]:=\\{x\in
P:1\leq x\leq p\\}$ and the interval $[1,\varphi(p)]:=\\{y\in Q:1\leq
y\leq\varphi(p)\\}$.

We say that $\varphi$ is a strong reduction if $\varphi\mid_{P}:P\rightarrow
Q$ is a bijection between $P$ and $Q$.

We say $(G,P)$ has a (strong) reduction to an amenable group or (strongly)
reduces to an amenable group if there is an amenable group $H$, a positive
cone $Q\subset H$, and a (strong) reduction $\varphi:(G,P)\rightarrow(H,Q)$.

Intuitively, a reduction is a map that preserves the order structure of
$(G,P)$ “locally”, meaning on every interval. For strong reductions, the
reader should think of them as arising from embedding the positive cone $P$ in
two distinct groups $(G,P)$ and $(H,P)$ such that there is a homomorphism from
one to the other taking $P$ to itself bijectively.

######  Example 3.2.

Let $G=\langle a,b\rangle$, the free group on two generators. If
$P=\\{a,b\\}^{*}$, then define
$\varphi:(G,P)\rightarrow(\mathbb{Z},\mathbb{N})$ by
$\varphi(a)=\varphi(b)=1$, which is an order homomorphism.

Then $\varphi$ is a reduction since, for example, it maps the interval
$[e,abba]=\\{e,a,ab,abb,abba\\}$ bijectively onto the interval
$[\varphi(e),\varphi(abba)]=[0,4]=\\{0,1,2,3,4\\}$. Since $\mathbb{Z}$ is
amenable, then this shows that $(G,P)$ reduces to an amenable group.

However, $\varphi$ is not injective even when restricted to $P$, as
$\varphi(a)=1=\varphi(b)$, so $\varphi$ is not a strong reduction.

It may not be immediately obvious from the definition that a strong reduction
is a reduction, so we will prove this as a short lemma:

######  Lemma 3.3.

Let $\varphi:(G,P)\rightarrow(H,Q)$ be a strong reduction. Then it is a
reduction.

###### Proof.

We must show that if $\varphi$ is bijective as a map from $P$ to $Q$, then it
is bijective from $[1,p]$ to $[1,\varphi(p)]$ for each $p\in P$. Fixing a
$p\in P$, since $[1,p]\subset P$, then $\varphi$ is injective on $[1,p]$, so
it suffices to show it is surjective onto $[1,\varphi(p)]$.

To this end, fix some $q_{1}\in[1,\varphi(p)]$. Since $q_{1}\leq\varphi(p)$,
there is a $q_{2}\in Q$ such that $q_{1}q_{2}=\varphi(p)$. Now since
$q_{1},q_{2}\in Q$ and $\varphi$ is a bijection from $P$ to $Q$, there is a
$p_{1},p_{2}\in P$ such that $\varphi(p_{1})=q_{1}$ and
$\varphi(p_{2})=q_{2}$. Then
$\varphi(p)=q_{1}q_{2}=\varphi(p_{1})\varphi(p_{2})=\varphi(p_{1}p_{2})$.
Since $p_{1}p_{2}\in P$ and $\varphi$ is injective on $P$, then
$p_{1}p_{2}=p$, so $p_{1}\in[1,p]$, and in particular there is a
$p_{1}\in[1,p]$ such that $\varphi(p_{1})=q_{1}$. That is, $\varphi$ is a
bijection from $[1,p]$ onto $[1,\varphi(p)]$, as desired. ∎

We’ll now provide several alternative characterizations of being a reduction
and a strong reduction.

######  Proposition 3.4.

Let $\varphi:(G,P)\rightarrow(H,Q)$ be an order homomorphism. Then the
following are equivalent:

  1. 1.

$\varphi$ is a reduction in the sense of Definition 3.1.

  2. 2.

$d^{P}_{Q}:=\varphi\mid_{P}:P\rightarrow Q$ is a functor that makes $P$ into a
$Q$-graph.

  3. 3.

For all $q\in Q,p\in P$ such that $q\leq\varphi(p)$, there exist unique
$p_{1},p_{2}\in P$ such that $p_{1}p_{2}=p$ and $\varphi(p_{1})=q$.

  4. 4.

For each nonempty interval $[x,y]\subset G$, $\varphi$ is an order isomorphism
between $[x,y]$ and $[\varphi(x),\varphi(y)]$.

  5. 5.

The following two statements together:

    1. (a)

$S\cap\ker\varphi=\\{1\\}$ where $S=\bigcup_{p\in P}[1,p][1,p]^{-1}$.

    2. (b)

For all $p\in P$, $\varphi([1,p])=[1,\varphi(p)]$.

###### Proof.

Fix $\varphi:(G,P)\rightarrow(H,Q)$ an order homomorphism. We will prove that
$(2)\Rightarrow(3)\Rightarrow(1)\Rightarrow(4)\Rightarrow(2)$ and
$(1)\iff(5)$.

$(2\Rightarrow 3)$ Fix $q\in Q$, $p\in P$ with $q\leq\varphi(p)=d_{Q}^{P}(p)$.
Then since $P$ is a $Q$-graph, we can uniquely factorize $p$ as $p=p_{1}p_{2}$
where $q=d^{P}_{Q}(p_{1})=\varphi(p_{1})$, as desired.

$(3\Rightarrow 1)$ Fix $p\in P$. By Lemma 2.7,
$\varphi([1,p])\subset[1,\varphi(p)]$. It then suffices to show that if
$q\in[1,\varphi(p)]$, then there exists a unique $s\in[1,p]$ with
$\varphi(s)=q$. To this end, if $q\in[1,\varphi(p)]$, then by (3), there
exists $p_{1},p_{2}\in P$ such that $p_{1}p_{2}=p$ and with
$\varphi(p_{1})=q$. Then $p_{1}\in[1,p]$, so $p_{1}$ is such an $s$. If there
were some $p_{1}^{\prime}\in[1,p]$ with
$\varphi(p_{1})=\varphi(p_{1}^{\prime})$, then by the former, there exists
$p_{2}^{\prime}\in P$ with $p_{1}^{\prime}p_{2}^{\prime}=p$, so by the
uniqueness condition of (2), we have $p_{1}^{\prime}=p_{1}$ and
$p_{2}^{\prime}=p_{2}$, and thus our $p_{1}$ is unique, as desired.

$(1\Rightarrow 4)$ Fix $x,y\in G$, and suppose the interval $[x,y]=\\{z\in
G:x\leq z\leq y\\}$ is nonempty. Since this interval is nonempty, then $x\leq
y$, so $1\leq x^{-1}y$, and we will write $p=x^{-1}y\in P$. By left-
invariance, $x\leq z\leq y$ if and only if $1\leq x^{-1}z\leq x^{-1}y$, so
$[x,y]=x[1,x^{-1}y]=x[1,p]$. By left-invariance, $x[1,p]$ is order-isomorphic
to $[1,p]$, and $\varphi(x)[1,\varphi(p)]$ is order-isomorphic to
$[1,\varphi(p)]$, so it suffices to show that $[1,p]$ is order-isomorphic to
$[1,\varphi(p)]$. By (1), $\varphi$ is a bijection between $[1,p]$ and
$[1,\varphi(p)]$, so it suffices to show that $s\leq
t\iff\varphi(s)\leq\varphi(t)$ for $s,t\in[1,p]$. The $\Rightarrow$ direction
is immediate from Lemma 2.7. For the other direction, suppose $s,t\in[1,p]$
and $\varphi(s)\leq\varphi(t)$. Then $\varphi(s)\in[1,\varphi(t)]$, and by
(3), $\varphi$ maps $[1,t]$ surjectively onto $[1,\varphi(t)]$, so there
exists some $s^{\prime}\in[1,t]$ such that $\varphi(s^{\prime})=\varphi(s)$.
But $s^{\prime}\in[1,t]\subseteq[1,p]$, and $\varphi$ is injective on $[1,p]$,
so $s^{\prime}=s$. Thus $s\in[1,t]$, so $s\leq t$.

$(4\Rightarrow 2)$ Certainly $d^{P}_{Q}:=\varphi\mid_{P}$ is a functor from
$P$ to $Q$. It then suffices to check unique factorization. To this end,
suppose $p\in P$ and $q_{1},q_{2}\in Q$ with $d_{Q}^{P}(p)=q_{1}q_{2}$. Then
$q_{1}\in[1,d_{Q}^{P}(p)]=[1,\varphi(p)]$, so by (4) there is a unique
$p_{1}\in[1,p]$ with $q_{1}=\varphi(p_{1})=d_{Q}^{P}(p_{1})$. Since $p_{1}\leq
p$, then $p_{2}:=p_{1}^{-1}p\in P$. Thus $p$ is uniquely factorized as
$p=p_{1}p_{2}$ with $d_{Q}^{P}(p_{1})=q_{1}$.

$(1\Rightarrow 5a)$ Suppose that $s\in S\cap\ker\varphi$. Then $s=qr^{-1}$
where $q,r\in[1,p]$ for some $p\in P$. Then
$\varphi(r)=\varphi(s)\varphi(r)=\varphi(sr)=\varphi(q)$. By (1),
$\varphi\mid_{[1,p]}$ is injective, so $q=r$ and thus $s=qr^{-1}=1$, as
desired. .

$(1\Rightarrow 5b)$ is immediate from the fact that $\varphi$ is a bijection
between the intervals $[1,p]$ and $[1,\varphi(p)]$.

$(5\Rightarrow 1)$ Fix $p\in P$. By Lemma 2.7,
$\varphi([1,p])\subseteq[1,\varphi(p)]$. It then suffices to show that if
$q\in[1,\varphi(p)]$, then there exists a unique $s\in[1,p]$ with
$\varphi(s)=q$. By (5b), there exists at least one $s\in[1,p]$ with
$\varphi(s)=q$. Assume for the sake of contradiction that there were distinct
$s_{1},s_{2}\in[1,p]$ with $\varphi(s_{1})=q=\varphi(s_{2})$. Then
$s_{1}s_{2}^{-1}\in\ker\varphi$. But since
$s_{1}s_{2}^{-1}\in[1,p][1,p]^{-1}\subseteq S$, then by (5a) we have
$s_{1}s_{2}^{-1}=1$, so $s_{1}=s_{2}$, a contradiction of distinctness. Thus
$\varphi$ is bijective as a map from $[1,p]$ to $[1,\varphi(p)]$.

∎

######  Lemma 3.5.

Let $(G,P)$, $(H,Q)$ be ordered groups, and $\varphi:G\rightarrow H$ a group
homomorphism. The following are equivalent:

  1. 1.

$\varphi$ is a strong reduction in the sense of Definition 3.1.

  2. 2.

$\varphi\mid_{P}$ is an order isomorphism from $P$ to $Q$.

###### Proof.

$(2\Rightarrow 1)$ is immediate.

$(1\Rightarrow 2)$ Since $\varphi\mid_{P}$ is already a bijection, we must
show that for $p_{1},p_{2}\in P$, $p_{1}\leq
p_{2}\iff\varphi(p_{1})\leq\varphi(p_{2})$. The $\Rightarrow$ direction is
immediate from Lemma 2.7. For the other direction, suppose that
$\varphi(p_{1})\leq\varphi(p_{2})$. Then there is a $q\in Q$ such that
$\varphi(p_{1})q=\varphi(p_{2})$. Since $\varphi$ is a surjection from $P$ to
$Q$, there exists a $p\in P$ such that $\varphi(p)=q$, and hence
$\varphi(p_{1}p)=\varphi(p_{2})$. Since $\varphi$ is injective on $P$, then
$p_{1}p=p_{2}$, and thus $p_{1}\leq p_{2}$, as desired. ∎

While the theory of reductions of ordered groups can be entirely severed from
the amenability of the groups involved, it is perhaps unsurprising that a
reduction to an amenable group will allow us to do more interesting analysis
of the $C^{*}$-algebras of $P$-graphs that will appear in later sections.

The following example shows that the existence of a reduction to an amenable
group (or even a non-injective reduction!) depends on not just the group $G$,
but also the positive cone $P$.

######  Example 3.6.

Let $G=\langle a,b\rangle$, the free group on two generators. If
$P=\\{a,b\\}^{*}$, then we’ve seen that $(G,P)$ is an ordered group which
reduces to an amenable group via the map
$\varphi:(G,P)\rightarrow(\mathbb{Z},\mathbb{N})$ given by
$\varphi(a)=\varphi(b)=1$.

Suppose that $(G,R)$ is a total ordering on a group (meaning that $R\cup
R^{-1}=G$ in addition to $R\cap R^{-1}=\\{1\\}$). Then for any reduction
$\varphi$, by Proposition 3.4(5), since $S=\bigcup_{r\in R}[1,r][1,r]^{-1}=G$,
then $\ker\varphi=\ker\varphi\cap G=\\{1\\}$. That is, every reduction
$\varphi$ is injective.

In particular, taking $G=F_{2}$ and $R$ a total ordering on $G$ (such as the
ordering arising from the Magnus expansion given in [4, Section 3.2]),
$(F_{2},R)$ cannot reduce to an amenable group since the range group of a
reduction will always contain a copy of $F_{2}$.

######  Example 3.7.

For any group $(G,P)$, $\operatorname{id}_{G}$ is a reduction, so if $G$ is
amenable, $\operatorname{id}_{G}$ is a reduction to an amenable group.

######  Example 3.8.

Let $G=\langle a,b\rangle$, the free group on two generators and
$P=\\{a,b\\}^{*}$. One can give a strong reduction of $(G,P)$ onto the
amenable group $H=BS(1,2)=\langle c,t|tc=c^{2}t\rangle$. Let
$Q=\\{t,ct\\}^{*}$, which is a positive cone in $BS(1,2)$, and let
$\varphi:(G,P)\rightarrow(H,Q)$ by $\varphi(a)=ct,\varphi(b)=t$. Certainly
$\varphi$ is an order homomorphism. Then one may verify that $\varphi$ is
bijective from $P$ to $Q$ in this way: if $\varphi(p_{1})=\varphi(p_{2})$,
then write this word in $H$ as $c^{i}t^{j}$. It must be that $j$ is the length
of $p_{1}$ and the length of $p_{2}$, so $p_{1}$ and $p_{2}$ have equal
length. Then, $i$ will be the number that comes from substituting $1$ for $a$
and $0$ for $b$ in the original word and interpreting the result as a binary
number. Two binary numbers with the same number of digits are equal if and
only if their digits are equal, so it must be that $p_{1}=p_{2}$. Thus
$\varphi$ is injective on $P$, and hence bijective on $P$, meaning $\varphi$
is a strong reduction.

The following result shows that reductions preserve weakly quasi-lattice
order.

######  Lemma 3.9.

Let $\varphi:(G,P)\rightarrow(H,Q)$ be a reduction. Then:

  1. 1.

If $(H,Q)$ is WQLO, then so is $(G,P)$.

  2. 2.

If $\varphi(P)=Q$ and $(G,P)$ is WQLO, then so is $(H,Q)$.

###### Proof.

For (1), assume for the sake of contradiction that $(H,Q)$ is WQLO but $(G,P)$
is not. Then there are $x,y\in P$ with a common upper bound but no least
common upper bound. That is, there are upper bounds $b_{1}>b_{2}>b_{3}>...$
such that $x,y\leq b_{i}$ for all $i\in\mathbb{N}$.

For all $i\in\mathbb{N}$, since $1\leq x\leq b_{i}$, then $1\leq b_{i}$. In
particular, $b_{i}\in[1,b_{1}]$ for all $i\in\mathbb{N}$. Also note that
$x,y\in[1,b_{1}]$.

Now, since $\varphi$ is a reduction, $\varphi$ is an order isomorphism from
$[1,b_{1}]$ to $[1,\varphi(b_{1})]$, so in $Q$,
$\varphi(b_{1})>\varphi(b_{2})>...$ is a strictly decreasing sequence of
common upper bounds of $\varphi(x)$ and $\varphi(y)$. This contradicts the
fact that $(H,Q)$ is WQLO.

For (2), assume for the sake of contradiction that $\varphi(P)=Q$ and $(G,P)$
is WQLO but that $(H,Q)$ is not WQLO. Then there are $z,w\in Q$ with a common
upper bound but no least common upper bound. That is, there are upper bounds
$c_{1}>c_{2}>c_{3}>...$ such that $z,w\leq c_{i}$ for all $i\in\mathbb{N}$.

For all $i\in\mathbb{N}$, since $1\leq z\leq c_{i}$, then $1\leq c_{i}$. In
particular, $c_{i}\in[1,c_{1}]$ for all $i\in\mathbb{N}$. Also note that
$z,w\in[1,c_{1}]$.

Now, since $\varphi(P)=Q$, there is some $p\in P$ such that
$\varphi(p)=c_{1}$. Since $\varphi$ is a reduction, $\varphi$ is an order
isomorphism from $[1,p]$ to $[1,\varphi(p)]=[1,c_{1}]$. Let $x$, $y$, and
$b_{i}$ denote the unique elements in $[1,p]$ such that
$\varphi(x)=z,\varphi(y)=w$, and $\varphi(b_{i})=c_{i}$ for all
$i\in\mathbb{N}$. Then in $P$, $b_{1}>b_{2}>b_{3}...$ is a strictly decreasing
sequence of common upper bounds of $x$ and $y$. This contradicts the fact that
$(G,P)$ is WQLO.

∎

######  Remark 3.10.

There is a notion of amenability for semigroups, and $(G,P)$ may strongly
reduce to an amenable group even if neither $G$ nor $P$ is amenable as a group
or semigroup (respectively). For example, in $(F_{2},P_{2})$, the free group
on 2 generators with positive cone $P_{2}$ which is the free monoid on 2
generators, we will show that $(F_{2},P_{2})$ strongly reduces to
$(\mathbb{Z}\wr\mathbb{Z},\mathbb{N}\wr\mathbb{N})$, which is amenable, but
$F_{2}$ is famously not amenable, and $P_{2}$ is also not amenable as a
semigroup.

The fact that $P_{2}$ is not amenable but can be embedded in an amenable group
has been known for more than half a century. In [11], the author shows an
embedding of $P_{2}$ into $\mathbb{Z}\wr\mathbb{Z}^{2}$.

###  3.2 Constructions

In this section, we will show that the notion of “(strongly) reduces to an
(amenable) group” behaves well with the usual group theory constructions such
as composition, hereditary subgroups, direct products, and free products.

####  3.2.1 Composition

######  Lemma 3.11.

A composition of (strong) reductions is a (strong) reduction. That is, if
$\varphi:(G,P)\rightarrow(H,Q)$ and $\psi:(H,Q)\rightarrow(F,R)$ are two
(strong) reductions, then $\psi\circ\varphi$ is a (strong) reduction.

###### Proof.

Suppose $\varphi:(G,P)\rightarrow(H,Q)$ and $\psi:(H,Q)\rightarrow(F,R)$ are
two reductions. Then we will show that $\psi\circ\varphi$ is a reduction.

Certainly, $\psi\circ\varphi$ is an order homomorphism, so by Lemma 3.4 (3),
we must show that for all $p\in P$, $\psi\circ\varphi$ bijectively maps
$[1,p]$ onto $[1,\psi(\varphi(p))]$.

To this end, we know that $\varphi:[1,p]\rightarrow[1,\varphi(p)]$ is
bijective, and $\psi:[1,\varphi(p)]\rightarrow[1,\psi(\varphi(p))]$ are
bijections since $\varphi$ and $\psi$ are reductions, so therefore their
composition is a bijection, as desired.

For the composition of strong reductions, if $\varphi:(G,P)\rightarrow(H,Q)$
and $\psi:(H,Q)\rightarrow(F,R)$ are two strong reductions, then $\varphi$
maps $P$ bijectively onto $Q$ and $\psi$ maps $Q$ bijectively onto $R$, so
$\psi\circ\varphi$ maps $P$ bijectively onto $R$, as desired.

∎

####  3.2.2 Hereditary Subgroups

In the context of ordered groups, the “correct” notion of a subgroup is often
a hereditary subgroups in the sense of [3, Corollary 5.6]. In this section we
will remind the reader of the definition and some basic results about the
concept.

######  Definition 3.12.

Given an ordered group $(G,P)$, if $Q\subseteq P$ is a subsemigroup, we say
that $Q$ is hereditary in $P$ if $p_{1},p_{2}\in P$, $p_{1}p_{2}\in Q$ implies
that $p_{1},p_{2}\in Q$.

If $(G,P)$ is an ordered group, we will say that $(H,Q)$ is a hereditary
subgroup if $H$ is a subgroup of $G$, $Q$ is a subsemigroup of $P\cap H$, and
$Q$ is hereditary in $P$.

######  Lemma 3.13.

Let $(H,P)$ be an ordered group, and $Q\subseteq P$ a hereditary subsemigroup.
For the sake of clarity, let $\leq_{P}$ and $\leq_{Q}$ denote the orderings on
$G$ arising from $P$ and $Q$, respectively. Then for all $q\in Q$ and $p\in
P$, $1\leq_{P}p\leq_{P}q$ if and only if $1\leq_{Q}p\leq_{Q}q$.

In particular, for $q\in Q$, the intervals $[1,q]_{\leq_{P}}$ and
$[1,q]_{\leq_{Q}}$ are equal as sets.

###### Proof.

Fix some $q\in Q,p\in P$.

Suppose $1\leq_{P}p\leq_{P}q$. Then $p\in P$ and there exists an $s\in P$ such
that $ps=q$. Since $q\in Q$ and $Q$ is hereditary, then this implies that
$p,s\in Q$, so $1\leq_{Q}p\leq_{Q}q$, as desired.

Suppose $1\leq_{Q}p\leq_{Q}q$. Then $p\in Q$ and there exists an $s\in Q$ such
that $ps=q$. Since $Q\subseteq P$, then $1\leq_{P}p\leq_{P}q$, as desired.

The “in particular” part is immediate.

∎

######  Lemma 3.14.

Let $(G,P),(H,Q)$ be ordered groups, and $\varphi:G\rightarrow H$ a
homomorphism. Let $H^{\prime}=\varphi(G)$, $Q^{\prime}=\varphi(P)$, and let
$\varphi^{\prime}$ denote the codomain restriction of $\varphi$ to the
codomain $H^{\prime}$. Then the following are equivalent:

  1. 1.

$\varphi$ is a reduction.

  2. 2.

$(H^{\prime},Q^{\prime})$ is a hereditary subgroup of $(H,Q)$ and
$\varphi^{\prime}$ is a surjective reduction of $(G,P)$ onto
$(H^{\prime},Q^{\prime})$.

###### Proof.

$(1\Rightarrow 2)$ If $\varphi$ is a reduction, we will first show that
$Q^{\prime}=\varphi(P)$ is a hereditary subsemigroup of $Q$. If
$q=\varphi(p)\in\varphi(P)$, and $rr^{\prime}=q$ for some $r,r^{\prime}\in Q$,
then $r\in[1,\varphi(p)]$, so by Proposition 3.4(3), there exists a unique
$s\in[1,p]$ such that $\varphi(s)=r$. Then $\varphi(s)r^{\prime}=\varphi(p)$,
so $r^{\prime}=\varphi(s^{-1}p)$. Since $s\leq p$, then there exists
$s^{\prime}\in P$ such that $ss^{\prime}=p$, so
$r^{\prime}=\varphi(s^{\prime})$. Thus $r,r^{\prime}\in\varphi(P)$, so
$\varphi(P)$ is a hereditary subsemigroup of $Q$.

Since additionally $H^{\prime}<H$ and $Q^{\prime}<Q\cap\varphi(G)$, then
$(H^{\prime},Q^{\prime})$ is a hereditary subgroup of $(H,Q)$.

To show that $\varphi^{\prime}$ is a surjective reduction, it is immediately
surjective since its codomain was restricted to the image. Now fix a $p\in P$.
Since $\varphi$ is a reduction, then by Proposition 3.4 (3), $\varphi$ maps
$[1,p]$ bijectively onto $[1,\varphi(p)]$. By Lemma 3.13, we have
$[1,\varphi(p)]=[1,\varphi^{\prime}(p)]$ as sets, so $\varphi^{\prime}$ maps
$[1,p]$ bijectively onto $[1,\varphi^{\prime}(p)]$. Then by Proposition 3.4
(3), $\varphi^{\prime}$ is a reduction.

$(2\Rightarrow 1)$ Fix a $p\in P$. Since $\varphi^{\prime}$ is a reduction,
then by Proposition 3.4 (3), $\varphi^{\prime}$ maps $[1,p]$ bijectively onto
$[1,\varphi^{\prime}(p)]$. By Lemma 3.13, we have
$[1,\varphi^{\prime}(p)]_{\leq Q^{\prime}}=[1,\varphi(p)]_{\leq Q}$ as sets,
so $\varphi$ maps $[1,p]$ bijectively onto $[1,\varphi(p)]$. Then by
Proposition 3.4 (3), $\varphi$ is a reduction.

∎

######  Lemma 3.15.

Let $(G,P)$ be an ordered group and $(H,Q)$ a hereditary subgroup of $(G,P)$.
Then the inclusion map $i:(H,Q)\rightarrow(G,P)$ is a reduction.

###### Proof.

Observe that the inclusion map, restricted to its image, is the identity map,
and thus a surjective reduction. Then the inclusion map is a reduction by
Lemma 3.14.

∎

We can now conclude that “reduces to an amenable group” is closed under taking
hereditary subgroups.

######  Corollary 3.16.

Let $(G,P)$ be an ordered group and $(H,Q)$ a hereditary subgroup of $(G,P)$.
If $(G,P)$ reduces to an amenable group, then $(H,Q)$ reduces to an amenable
group.

###### Proof.

Let $i:H\rightarrow G$ denote the inclusion map, which by the previous lemma
is a reduction.

If $\varphi:(G,P)\rightarrow(F,R)$ is a reduction to an amenable group, then
$\varphi\circ i:(H,Q)\rightarrow(F,R)$ is a composition of reductions, and
hence a reduction by Lemma 3.11. It has amenable range, so it is a reduction
to an amenable group. ∎

####  3.2.3 Direct Products

Now, we’ll show that reductions respect direct products, which implies that
“reduces to an amenable group” is preserved under direct products.

First, let us check that direct products preserve WQLO:

######  Lemma 3.17.

Let $(G,P)$ and $(H,Q)$ be ordered groups. Let $P\times Q$ denote the
submonoid of $G\times H$ generated by (the canonical copies of) $P$ and $Q$.
Then $(G\times H,P\times Q)$ is an ordered group, and if $(G,P)$ and $(H,Q)$
are WQLO, then so is $(G\times H,P\times Q)$.

###### Proof.

By definition, $P\times Q$ is a submonoid of $G\times H$, and it is immediate
that $P\times Q\cap(P\times Q)^{-1}=(P\cap P^{-1})\times(Q\cap
Q^{-1})=\\{1_{G}\\}\times\\{1_{H}\\}$, so $(G\times H,P\times Q)$ is an
ordered group.

If $(G,P)$ and $(H,Q)$ are WQLO, then one may quickly confirm that given
$(p_{1},q_{1}),(p_{2},q_{2})\in P\times Q$, their least upper bound is
$(p_{1}\vee p_{2},q_{1}\vee q_{2})$.

∎

######  Lemma 3.18.

For $i=1,2$, let $(G_{i},P_{i})$ and $(H_{i},Q_{i})$ be ordered groups,
$\varphi_{i}:G_{i}\rightarrow H_{i}$ a homomorphism. Let $G=G_{1}\times
G_{2}$, and similarly define $P,H,Q$. Then there is a homomorphism
$\varphi:G\rightarrow H$ given by
$\varphi(g_{1},g_{2})=(\varphi_{1}(g_{1}),\varphi_{2}(g_{2}))$, and $\varphi$
is an order homomorphism (respectively, reduction or strong reduction) if
$\varphi_{1}$ and $\varphi_{2}$ both are.

###### Proof.

The existence of such a homomorphism is an elementary fact of group theory.

If $\varphi_{1},\varphi_{2}$ are both order homomorphisms, then
$\varphi_{1}(P_{1})\subseteq Q_{1}$ and $\varphi_{2}(P_{2})\subseteq Q_{2}$,
so $\varphi(P_{1}\times
P_{2})\subseteq\varphi_{1}(P_{1})\times\varphi_{2}(P_{2})\subseteq Q_{1}\times
Q_{2}$, as desired.

If $\varphi_{1},\varphi_{2}$ are both reductions, then we will use Proposition
3.4 (3) as our notion of a reduction, so it suffices to show that for all
$p_{1}\in P_{1},p_{2}\in P_{2}$, $[(1,1),(p_{1},p_{2})]$ is mapped bijectively
onto $[(1,1),(\varphi_{1}(p_{1}),\varphi_{2}(p_{2})]$. But in $(G_{1}\times
G_{2},P_{1}\times P_{2})$, we have that
$[(1,1),(x_{1},x_{2})]=[1,x_{1}]\times[1,x_{2}]$, so we know that
$\varphi=\varphi_{1}\times\varphi_{2}$ bijectively carries
$[(1,1),(p_{1},p_{2})]=[1,p_{1}]\times[1,p_{2}]$ onto
$[(1,1),(\varphi_{1}(p_{1}),\varphi_{2}(p_{2}))]=[1,\varphi_{1}(p_{1})]\times[1,\varphi_{2}(p_{2})]$,
as desired.

If $\varphi_{1},\varphi_{2}$ are both strong reductions, then $\varphi_{1}$
and $\varphi_{2}$ map $P_{1}$ and $P_{2}$ bijectively onto $Q_{1}$ and $Q_{2}$
respectively, so for all $q=(q_{1},q_{2})\in Q$ there is a unique $p_{1}\in
P_{1},p_{2}\in P_{2}$ such that
$\varphi_{1}(p_{1})=q_{1},\varphi_{2}(p_{2})=q_{2}$, and hence
$p=(p_{1},p_{2})$ is the unique element of $P$ with $\varphi(p)=q$, so
$\varphi$ is a bijection from $P$ to $Q$, as desired.

∎

######  Corollary 3.19.

If $(G_{1},P_{1})$, $(G_{2},P_{2})$ are ordered groups which reduce to
amenable ordered groups, then $(G_{1}\times G_{2},P_{1}\times P_{2})$ reduces
to an amenable group.

###### Proof.

For $i=1,2$, let $\varphi_{i}:(G_{i},P_{i})\rightarrow(H_{i},Q_{i})$ denote
the reduction of $(G_{i},P_{i})$ to an amenable ordered group.

By the previous Lemma, we know that $(G_{1}\times G_{2},P_{1}\times P_{2})$
reduces to $(H_{1}\times H_{2},Q_{1}\times Q_{2})$. Since $H_{1}$ and $H_{2}$
are amenable, then $H_{1}\times H_{2}$ is amenable, so this is a reduction to
an amenable group. ∎

####  3.2.4 Free Products

In this section we will show that the class of groups which have reductions to
amenable groups is closed under (finite) free products. That is, we will show
that if $(G,P)$ and $(H,Q)$ reduce to amenable groups, then $(G*H,P*Q)$ also
reduces to an amenable group.

This will consist of two steps, analogous to Lemma 3.18 and Corollary 3.19
from the direct product case. A free product analogue of Lemma 3.18 is
straightforward. However, in the free product case, an analogue of Corollary
3.19 is more difficult, since the free product of two amenable groups is
almost never amenable. Therefore, our second step is longer.

Given ordered groups $(G_{1},P_{1})$ and $(G_{2},P_{2})$, we denote by
$P_{1}*P_{2}$ the submonoid of $G_{1}*G_{2}$ generated by the naturally
embedded copies of $P_{1}$ and $P_{2}$. First, we will prove a lemma
characterizing the order on $(G_{1}*G_{2},P_{1}*P_{2})$:

######  Lemma 3.20.

Let $(G_{1},P_{1}),(G_{2},P_{2})$ be ordered groups, and let
$(G,P)=(G_{1}*G_{2},P_{1}*P_{2})$.

  1. 1.

For $p\in P$, we may write $p=p_{1}p_{2}...p_{2n}$ where $p_{i}\in P_{1}$ if
$i$ is odd, and $p_{i}\in P_{2}$ if $i$ even, and $p_{i}\neq 1$ for $1<i<2n$.

  2. 2.

This representation of $p$ is unique. That is, there is exactly one such
choice of $n$ and elements $p_{i}$ such that $p_{i}\in P_{1}$ if $i$ is odd,
and $p_{i}\in P_{2}$ if $i$ even, and $p_{i}\neq 1$ for $1<i<2n$.

  3. 3.

Given such a representation $p=p_{1}p_{2}...p_{2n}$, we have
$[1,p]=\bigcup_{i=1}^{2n}X_{i}$, where $X_{i}=p_{1}...p_{i-1}[1,p_{i}]$.

  4. 4.

Given such a decomposition $[1,p]=\bigcup_{i=1}^{2n}X_{i}$, if $a\in X_{i}$
and $b\in X_{j}$ for $i<j$, then $a\leq b$.

###### Proof.

For (1), by definition every element of $P$ can be written as a product of
elements of $P_{1}$ and $P_{2}$. If two consecutive elements are from the same
$P_{i}$, they can be combined, and if any intermediate term is a 1, it can be
removed and the now-consecutive terms combined. Finally, if necessary a term
of 1 can be put at the beginning or end of the expression to make the
alternating product begin with a term from $P_{1}$ and end with a term from
$P_{2}$.

For (2), this representation is unique because of the Normal Form Theorem for
Free Products ([16, Chapter IV, Theorem 1.2]).

For (3), fix a $p=p_{1}p_{2}...p_{2n}$. It is immediate that each $X_{i}$ is a
subset of $[1,p]$, so it suffices to show containment in the other direction.
To this end, suppose $q\in[1,p]$, so $q\in P$ and $p=qr$ for some $r\in P$.
Then writing $q=q_{1}...q_{2m}$ and $r=r_{1}...r_{2k}$, we have that
$p=p_{1}p_{2}...p_{2n}=q_{1}...q_{2m}r_{1}...r_{2k}$. We now have a few cases
depending on whether or not $q_{2m}$ and $r_{1}$ are the identity.

  * •

$q_{2m}=1,r_{1}=1$: Then
$p=p_{1}p_{2}...p_{2n}=q_{1}...q_{2m-1}r_{2}...r_{2k}$ is the unique
alternating presentation with nonidentity terms, so $p_{1}=q_{1},p_{2}=q_{2}$,
etc. In particular, $q=q_{1}...q_{2m-1}=p_{1}...p_{2m-1}\in
X_{2m}=p_{1}...p_{2m-1}[1,p_{2m}]$.

  * •

$q_{2m}\neq 1,r_{1}\neq 1$: Then
$p=p_{1}p_{2}...p_{2n}=q_{1}...q_{2m}r_{1}...r_{2k}$ is the unique alternating
presentation with nonidentity terms, so $p_{1}=q_{1},p_{2}=q_{2}$, etc. In
particular, $q=q_{1}...q_{2m}=p_{1}...p_{2m}\in
X_{2m}=p_{1}...p_{2m-1}[1,p_{2m}]$.

  * •

$q_{2m}\neq 1,r_{1}=1$: Then
$p=p_{1}p_{2}...p_{2n}=q_{1}...(q_{2m}r_{2})r_{3}...r_{2k}$ is the unique
alternating presentation with nonidentity terms, so $p_{1}=q_{1},p_{2}=q_{2}$,
etc, ending with $p_{2m}=q_{2m}r_{2}$. Thus $q_{2m}\in[1,p_{2m}]$, so
$q=q_{1}...q_{2m}=p_{1}p_{2}...p_{2m-1}q_{2m}\in
X_{2m}=p_{1}...p_{2m-1}[1,p_{2m}]$.

  * •

$q_{2m}=1,r_{1}\neq 1$: Then
$p=p_{1}p_{2}...p_{2n}=q_{1}...q_{2m-2}(q_{2m-1}r_{1})r_{2}...r_{2k}$ is the
unique alternating presentation with nonidentity terms, so
$p_{1}=q_{1},p_{2}=q_{2}$, etc, ending with $p_{2m-1}=q_{2m-1}r_{1}$. Thus
$q_{2m-1}\in[1,p_{2m-1}]$, so
$q=q_{1}...q_{2m-1}=p_{1}p_{2}...p_{2m-2}q_{2m-1}\in
X_{2m-1}=p_{1}...p_{2m-2}[1,p_{2m-1}]$.

In all four cases, $q$ is in some $X_{i}$, so $[1,p]=\bigcup_{i=1}^{2n}X_{i}$,
as desired.

For (4), note that for each $y\in X_{i}$, we have $p_{1}p_{2}...p_{i-1}\leq
y\leq p_{1}p_{2}...p_{i}$, so $a\leq p_{1}p_{2}...p_{i}\leq
p_{1}p_{2}...p_{j-1}\leq b$, as desired. ∎

We can now confirm that weak quasi-lattice order is preserved under free
products.

######  Lemma 3.21.

If $(G_{1},P_{1})$ and $(G_{2},P_{2})$ are ordered groups (respectively, WQLO
groups) then so is $(G_{1}*G_{2},P_{1}*P_{2})$.

###### Proof.

If $(G_{1},P_{1})$ and $(G_{2},P_{2})$ are ordered groups, then by definition
$P_{1}*P_{2}$ is a submonoid of $G*H$. We must now check that
$(P_{1}*P_{2})\cap(P_{1}*P_{2})^{-1}=\\{1\\}$, or equivalently that if $a,b\in
P_{1}*P_{2}$ with $ab=1$, then $a=b=1$. But if $ab=1$, then by writing
$a=p_{1}p_{2}...p_{2n}$ and $b=q_{1}q_{2}...q_{2m}$ as in the previous
theorem, then $1=ab=p_{1}p_{2}...p_{2n}q_{1}q_{2}...q_{2m}$. By the Normal
Form Theorem for Free Products ([16, Chapter IV, Theorem 1.2], a reduced
sequence in a free product can equal 1 if and only if its length is 1, so all
the $p_{i}$s and $q_{i}$s in the products must cancel with each other. But
since each term of the product is positive, they can only cancel if each term
is equal to 1. Thus $a=b=1$, as desired, so $(G_{1}*G_{2},P_{1}*P_{2})$ is an
ordered group.

If $(G_{1},P_{1})$ and $(G_{2},P_{2})$ are WQLO, we will check that
$(G_{1}*G_{2},P_{1}*P_{2})$ is WQLO. To this end, fix some $y_{1},y_{2}\in
P_{1}*P_{2}$ and suppose $y_{1},y_{2}$ have some common upper bound $y_{3}\in
P_{1}*P_{2}$. By part (1) of the previous lemma, we can write
$y_{3}=p_{1}p_{2}p_{3}...p_{2n-1}p_{2n}$, where each $p_{i}\in P_{1}$ for $i$
odd and $p_{i}\in P_{2}$ for $i$ even. By part (3) of that lemma,
$[1,y_{3}]=\bigcup_{i=1}^{2n}X_{i}$ where $X_{i}=p_{1}...p_{i-1}[1,p_{i}]$.
Since $y_{1},y_{2}\in[1,y_{3}]$, then $y_{1}\in X_{i},y_{2}\in X_{j}$ for some
$i,j\leq 2n$. Without loss of generality, suppose $i\leq j$.

If $i<j$, then by part (4) of the previous Lemma, we have $y_{1}\leq y_{2}$,
so $y_{1}\vee y_{2}=y_{2}$.

If $i=j$, then $y_{1},y_{2}\in X_{i}=p_{1}...p_{i-1}[1,p_{i}]$, so there are
$r_{1},r_{2}$ such that $y_{1}=p_{1}...p_{i-1}r_{1}$ and
$y_{2}=p_{1}...p_{i-1}r_{2}$ where $r_{1},r_{2}\in P_{1}$ if $i$ is odd, and
$r_{1},r_{2}\in P_{2}$ if $i$ is even. In either case,
$p_{1}...p_{i-1}(r_{1}\vee r_{2})$ will be the supremum of $y_{1}$ and
$y_{2}$.

∎

We will now show that if two groups have a reduction, their free product
reduces to the free product of the reductions.

######  Lemma 3.22.

For $i=1,2$, let $\varphi_{i}:(G_{i},P_{i})\rightarrow(H_{i},Q_{i})$ be a
reduction. Then let

$G=G_{1}*G_{2},P=P_{1}*P_{2},H=H_{1}*H_{2},\text{ and }Q=Q_{1}*Q_{2},$

and let $\varphi:G\rightarrow H$ be the homomorphism satisfying
$\varphi\mid_{G_{i}}=\varphi_{i}$ for $i=1,2$. Then
$\varphi:(G,P)\rightarrow(H,Q)$ is a reduction.

###### Proof.

Certainly, $\varphi$ is a homomorphism. By Proposition 3.4 (3), it suffices to
show that for all $p\in P$, the interval $[1,p]$ is mapped bijectively onto
$[1,\varphi(p)]$.

Fix some $p\in P$, and then by Lemma 3.20, we may write
$p=p_{1}p_{2}...p_{2n}$ where $p_{i}$ is in $P_{1}$ if $i$ is odd, and in
$P_{2}$ if $i$ is even. Then $[1,p]=\bigcup_{i=1}^{2n}X_{i}$, where
$X_{i}=p_{1}...p_{i-1}[1,p_{i}]$.

Now, for $1\leq i\leq 2n$, let $q_{i}=\varphi_{1}(p_{i})$ if $i$ is odd, and
$q_{i}=\varphi_{2}(p_{i})$ if $i$ is even. Thus $\varphi(p)=q_{1}...q_{2n}$.
Note that for $1<i<2n$, $q_{i}\neq 1$ since $p_{i}\neq 1$ and by Proposition
3.4 (4), both $\varphi_{1}$ and $\varphi_{2}$ send strictly positive elements
to strictly positive elements.

Thus $\varphi(p)=q_{1}...q_{2n}$ is the unique way to write $\varphi(p)\in Q$
as an alternating product of non-unit elements, so by Lemma 3.20,
$[1,\varphi(p)]=\bigcup_{i=1}^{2n}Y_{i}$ where
$Y_{i}=q_{1}...q_{i-1}[1,q_{i}]$. But since $\varphi_{1}$ and $\varphi_{2}$
are reductions, we know that each $X_{i}$ is carried bijectively to its
$Y_{i}$, so it suffices to explain why bijectivity is preserved when taking
the union of the $X_{i}$ and $Y_{i}$s. It is immediate that the map from
$\bigcup X_{i}$ to $\bigcup Y_{i}$ is surjective.

For injectivity, first note that for $1\leq i\leq j\leq n$,

$Y_{i}\cap Y_{j}=\begin{cases}\emptyset&j>i+1\\\ \\{q_{1}...q_{i}\\}&j=i+1\\\
Y_{i}&i=j\end{cases}$

and similarly,

$X_{i}\cap X_{j}=\begin{cases}\emptyset&j>i+1\\\ \\{p_{1}...p_{i}\\}&j=i+1\\\
X_{i}&i=j\end{cases}$

Then, if $\varphi(s_{1})\in Y_{i}$ and $\varphi(s_{2})\in Y_{j}$ for
$s_{1},s_{2}\in[1,p]$ such that $\varphi(s_{1})=\varphi(s_{2})$, either $i=j$
so $s_{1},s_{2}\in X_{i}$ and thus by injectivity on $X_{i}$ we have
$s_{1}=s_{2}$, or $j=i+1$, so $\varphi(s_{1})=\varphi(s_{2})=q_{1}...q_{i}$,
so $s_{1}=s_{2}=p_{1}...p_{i}$. In either case, $s_{1}=s_{2}$, so we have
injectivity of $\varphi$. Thus $\varphi$ is bijective on $[1,p]$ as desired,
so it is a reduction.

∎

This completes the first step, showing that if two groups have reductions to
amenable groups, their free product reduces to a free product of amenable
groups. Now comes the harder step: showing that a free product of amenable
groups reduces to an amenable group!

The key is a construction from group theory called a wreath product. Recall
the following definition of a wreath product:

######  Definition 3.23.

Let $G$ and $H$ be groups. Let

$G^{H}=\\{f:H\rightarrow G\text{ a function}|\operatorname{supp}(f)\text{ is
finite}\\}$

where $\operatorname{supp}(f)=\\{h\in H:f(h)\neq 1\\}$, and give $G^{H}$ a
group structure by pointwise multiplication. Note that $G^{H}$ is isomorphic
to a direct sum of $|H|$ copies of $G$, hence the notation.

Give $G^{H}$ an action $\alpha:H\rightarrow Aut(G^{H})$ by translation:
$\left[\alpha_{h}(f)\right](h^{\prime})=f(h^{-1}h^{\prime})$.

Then we define the (restricted) wreath product to be the semidirect product
$G\wr H:=G^{H}\rtimes_{\alpha}H$, so there is a (split) short exact sequence:

$1\rightarrow G^{H}\rightarrow G\wr H\rightarrow H\rightarrow 1.$

The following remark establishes the basic properties of wreath products, and
are all routine to verify.

######  Remark 3.24.

For each $g\in G$ and $h\in H$, let $g\delta_{h}$ denote the function in
$G^{H}$ given by $g\delta_{h}(h^{\prime})=\begin{cases}g&h=h^{\prime}\\\
1&h\neq h^{\prime}\end{cases}$. Then there is a natural embedding of $G$ into
$G\wr H$ by $g\mapsto(g\delta_{1_{H}},1_{H})$ and there is a natural embedding
of $H$ into $G\wr H$ by $h\mapsto(1_{G^{H}},h)$. In a slight abuse of
notation, we will write $g$ or $g\delta_{1}$ for the embedded element of $G$
in $G\wr H$, and write $h$ for the embedded element of $H$ in $G\wr H$.

Now note that

$\displaystyle\left[\alpha_{h_{2}}(g\delta_{h_{1}})\right](h_{3})$
$\displaystyle=$ $\displaystyle g\delta_{h_{1}}(h_{2}^{-1}h_{3})$
$\displaystyle=$ $\displaystyle\begin{cases}g&h_{1}=h_{2}^{-1}h_{3}\\\
1&h_{1}\neq h_{2}^{-1}h_{3}\end{cases}$ $\displaystyle=$
$\displaystyle\begin{cases}g&h_{2}h_{1}=h_{3}\\\ 1&h_{2}h_{1}\neq
h_{3}\end{cases}$ $\displaystyle=$ $\displaystyle g\delta_{h_{2}h_{1}}(h_{3})$

so unbinding the $h_{3}$, we have that
$\alpha_{h_{2}}(g\delta_{h_{1}})=g\delta_{h_{2}h_{1}}$. Now,

$\displaystyle h_{2}(g\delta_{h_{1}})h_{2}^{-1}$ $\displaystyle=$
$\displaystyle(1_{G},h_{2})(g\delta_{h_{1}},1_{H})(1_{G},h_{2}^{-1})$
$\displaystyle=$ $\displaystyle(1_{G},h_{2})(g\delta_{h_{1}},h_{2}^{-1})$
$\displaystyle=$ $\displaystyle(1_{G},h_{2})(g\delta_{h_{1}},h_{2}^{-1})$
$\displaystyle=$ $\displaystyle(\alpha_{h_{2}}(g\delta_{h_{1}}),1_{H})$
$\displaystyle=$ $\displaystyle g\delta_{h_{2}h_{1}}$

Since $G^{H}$ is generated by the $g\delta_{h}$, then together (the embedded
copies of) $G$ and $H$ generate $G\wr H$.

Given $P,Q$ positive cones in $G,H$, we let $P\wr Q$ denote the monoid
generated by the naturally embedded copies of $P$ and $Q$ within $G\wr H$.

With these natural embeddings of $G$ and $H$ into $G\wr H$, by the universal
property of free products of groups, we get a homomorphism
$\varphi:G*H\rightarrow G\wr H$ given by sending (the naturally embedded
copies of) $G$ and $H$ to (the naturally embedded copies of) $G$ and $H$. We
will call this the natural homomorphism of $G*H$ into $G\wr H$. Since $G\wr H$
is generated by $G$ and $H$, it is immediate that the natural homomorphism is
surjective. For the same reason, $P*Q$ will be mapped surjectively onto $P\wr
Q$. We will show later that $\varphi:(G*H,P*Q)\rightarrow(G\wr H,P\wr Q)$ is a
strong reduction.

Finally, note that every element of $G^{H}$ can be written as a (possibly
empty) product of a finite number of $g_{i}\delta_{h_{i}}$ where each $g_{i}$
is nonunital and each $h_{i}$ is distinct, and this presentation is unique up
to a permutation of the $g_{i}\delta_{h_{i}}$. Note that the element
$g_{i}\delta_{h_{i}}$ commutes with the element $g_{j}\delta_{h_{j}}$ as long
as $h_{i}\neq h_{j}$. When $f\in G^{H}$ is written as
$f=g_{1}\delta_{h_{1}}...g_{n}\delta_{h_{n}}$ where the $h_{i}$ are distinct,
then

$f(h)=\begin{cases}g_{1}&\text{ if }h=h_{1}\\\ g_{2}&\text{ if }h=h_{2}\\\
\vdots\\\ g_{n}&\text{ if }h=h_{n}\\\ 1&\text{ otherwise}\end{cases}.$

Finally, we will remind the reader of this fact about amenability of wreath
products:

######  Lemma 3.25.

Let $G$ and $H$ be amenable groups. Then $G\wr H$ is amenable.

###### Proof.

Recall that the class of amenable groups are closed under finite direct sums,
direct limits, and extensions.

From our definition of the wreath product, there is a short exact sequence

$1\rightarrow G^{H}\rightarrow G\wr H\rightarrow H\rightarrow 1.$

Recall also that $G^{H}$ is isomorphic to the direct sum of $|H|$ copies of
$G$. Thus $G^{H}$ is a direct limit of $\\{G^{n}\\}_{n\in\mathbb{N}}$. Since
amenable groups are closed under finite direct sum and $G$ is amenable by
hypothesis, then each $G^{n}$ is amenable, and since amenable groups are
closed under direct limit, then $G^{H}$ is amenable.

Also, $H$ is amenable by hypothesis, so $G\wr H$ is an extension of an
amenable group by an amenable group. Thus $G\wr H$ is amenable.

∎

Now, we will have two results that establish why
$\varphi:(G_{1}*G_{2},P_{1}*P_{2})\rightarrow(G_{1}\wr G_{2},P_{1}\wr P_{2})$
is a reduction.

######  Lemma 3.26.

Let $(G_{1},P_{1})$ and $(G_{2},P_{2})$ be ordered groups, and let
$p=p_{1}p_{2}...p_{2n}\in P_{1}\wr P_{2}$ be an element such that $p_{i}\in
P_{1}$ for $i$ odd and $p_{i}\in P_{2}$ for $i$ even and $p_{i}\neq 1$ for
$1<i<n$. Then, as an element of $P_{1}\wr P_{2}\subset G_{1}^{G_{2}}\rtimes
G_{2}$, $p=(f,p_{\leq 2n})$ where $p_{\leq 2n}=p_{2}p_{4}...p_{2n}$ and

$f(h)=\begin{cases}p_{1}&\text{ if }h=1_{G_{2}}\\\ p_{3}&\text{ if }h=p_{2}\\\
p_{5}&\text{ if }h=p_{2}p_{4}\\\ \vdots\\\ p_{2n-1}&\text{ if
}h=p_{2}...p_{2n-2}\\\ 1&\text{ otherwise}\end{cases}$

###### Proof.

By elementary algebra, we may rewrite the product $p=p_{1}p_{2}p_{3}...p_{2n}$
as:

$p=p_{1}p_{2}p_{3}...p_{2n}=p_{1}(p_{2}p_{3}p_{2}^{-1})...(p_{2}p_{4}...p_{2n-2}p_{2n-1}p_{2n-2}^{-1}...p_{4}^{-1}p_{2}^{-1})(p_{2}p_{4}...p_{2n})$

With the shorthand that $p_{\leq 0}=1_{H}$ and $p_{\leq
2i}=p_{2}p_{4}...p_{2i}$ for $1\leq i\leq n$, we then have that

$p=\left[(p_{\leq 0}p_{1}p_{\leq 0}^{-1})(p_{\leq 2}p_{3}p_{\leq
2}^{-1})(p_{\leq 4}p_{5}p_{\leq 4}^{-1})...(p_{\leq 2n-2}p_{2n-1}p_{\leq
2n-2}^{-1})\right]p_{\leq 2n}.$

Let $f=(p_{\leq 0}p_{1}p_{\leq 0}^{-1})(p_{\leq 2}p_{3}p_{\leq
2}^{-1})(p_{\leq 4}p_{5}p_{\leq 4}^{-1})...(p_{\leq 2n-2}p_{2n-1}p_{\leq
2n-2}^{-1})$ be the bracketed term, so then $p$ is represented in
$G_{1}^{G_{2}}\rtimes G_{2}$ by $(f,p_{\leq 2n})$. It then suffices to show
that $f$ assumes the values as claimed.

Recall that for $i$ odd, $p_{i}$ is embedded as $p_{i}\delta_{1_{H}}$, and for
$i$ even, $p_{i}$ is embedded as an element of $G_{2}$. By the calculation in
the above remark, we then have that $p_{\leq 2i}p_{2i+1}p_{\leq
2i}^{-1}=p_{\leq 2i}(p_{2i+1}\delta_{1_{H}})p_{\leq
2i}^{-1}=p_{2i+1}\delta_{p_{\leq 2i}}$, so $f=(p_{1}\delta_{p_{\leq
0}})(p_{3}\delta_{p_{\leq 2}})...(p_{2n-1}\delta_{p_{\leq 2n-2}})$.

Note that $p_{\leq 2i+2}=p_{\leq 2i}p_{2i+2}$, and since $p_{2i+2}>1$ for
$0\leq i<n-1$, we have that $p_{\leq 2i}<p_{\leq 2i+2}$ for $0\leq i<n-1$, and
in particular the $p_{\leq 2i}$ are distinct (except possibly that $p_{\leq
2n-2}=p_{\leq 2n}$).

Since the $\\{p_{\leq 0},...,p_{\leq 2n-2}\\}$ are distinct, then by the above
remark, we know that

$f=(p_{1}\delta_{p_{\leq 0}})(p_{3}\delta_{p_{\leq
2}})...(p_{2n-1}\delta_{p_{\leq 2n-2}})$

is the unique representation of $f$ as a product of $g_{i}\delta_{h_{i}}$
(except possibly that the $p_{1}\delta_{p_{\leq 0}}$ term is equal to 1), so

$f(h)=\begin{cases}p_{1}&\text{ if }h=p_{\leq 0}\\\ p_{3}&\text{ if }h=p_{\leq
2}\\\ \vdots\\\ p_{2n-1}&\text{ if }h=p_{\leq 2n-2}\\\ 1&\text{
otherwise}\end{cases}$

as desired.

∎

######  Proposition 3.27.

Let $(G_{1},P_{1})$, $(G_{2},P_{2})$ be two ordered groups. Let $\varphi$
denote the natural homomorphism from $G_{1}*G_{2}$ onto $G_{1}\wr G_{2}$. If
$p,q\in P:=P_{1}*P_{2}$ and $\varphi(p)=\varphi(q)$, then $p=q$.

In particular, $\varphi$ is a strong reduction of $G_{1}*G_{2}$ onto $G_{1}\wr
G_{2}$.

###### Proof.

Since $p,q\in P$, then by Lemma 3.20 we may write $p=p_{1}p_{2}...p_{2n}$
where $p_{i}$ is in $P_{1}$ if $i$ is odd, and in $P_{2}$ if $i$ is even, and
all $p_{i}\neq 1$ for $1<i<n$. Similarly, we may write $q=q_{1}q_{2}...q_{2m}$
where $q_{i}$ is in $P_{1}$ if $i$ is odd, and in $P_{2}$ if $i$ is even, and
all $p_{i}\neq 1$ for $1<i<m$.

Now, by the previous Lemma, we may write $\varphi(p)=(f_{p},p_{\leq 2n})$ and
$\varphi(q)=(f_{q},q_{\leq 2m})$ where $p_{\leq 2i}=p_{2}p_{4}...p_{2i}$,
$q_{\leq 2i}=q_{2}q_{4}...q_{2i}$,

$f_{p}(h)=\begin{cases}p_{1}&\text{ if }h=1_{G_{2}}\\\ p_{3}&\text{ if
}h=p_{2}\\\ p_{5}&\text{ if }h=p_{2}p_{4}\\\ \vdots\\\ p_{2n-1}&\text{ if
}h=p_{2}...p_{2n-2}\\\ 1&\text{ otherwise}\end{cases}\text{ and
}f_{q}(h)=\begin{cases}q_{1}&\text{ if }h=1_{G_{2}}\\\ q_{3}&\text{ if
}h=q_{2}\\\ q_{5}&\text{ if }h=q_{2}q_{4}\\\ \vdots\\\ q_{2m-1}&\text{ if
}h=q_{2}...q_{2m-2}\\\ 1&\text{ otherwise}\end{cases}$

Since $\varphi(p)=\varphi(q)$, then $f_{p}=f_{q}$, so $p_{1}=q_{1}$.
Furthermore,
$\\{p_{2},p_{2}p_{4},...,p_{2}...p_{2n-2}\\}=\operatorname{supp}(f_{p})\setminus\\{1\\}=\operatorname{supp}(f_{q})\setminus\\{1\\}=\\{q_{2},q_{2}q_{4},...,q_{2}...q_{2m-2}\\}$,
and these sets are strictly increasing sequences, so it must be that $n=m$ and
$p_{2}=q_{2}$, $p_{2}p_{4}=q_{2}q_{4}$, and so on. By cancelling common terms,
we get that $p_{2i}=q_{2i}$ for $1\leq i<n$. Returning to our expressions for
$f_{p}$ and $f_{q}$, we can now compare their values at
$p_{2}=q_{2},p_{2}p_{4}=q_{2}q_{4}$, etc to get that
$p_{3}=q_{3},p_{5}=q_{5}$, and so on. Finally, recall that $(f_{p},p_{\leq
2n})=\varphi(p)=\varphi(q)=(f_{q},q_{\leq 2n})$ so $p_{\leq 2n}=q_{\leq 2n}$
and by cancelling the common factor of $p_{2}...p_{2n-2}=q_{2}...q_{2n-2}$, we
have that $p_{2n}=q_{2n}$.

Thus for all $1\leq i\leq 2n$, we have that $p_{i}=q_{i}$, so $p=q$, as
desired.

For the “in particular”, by Corollary 3.5 it suffices to show that $\varphi$
is a bijection of $P_{1}*P_{2}$ onto $P_{1}\wr P_{2}$. It is immediate that
$\varphi$ is surjective, and by the previous part of the proposition it must
be injective. Thus $\varphi$ is a bijection of $P_{1}*P_{2}$ onto $P_{1}\wr
P_{2}$, so $\varphi$ is a strong reduction as desired.

∎

We have done the hard work already, so we may now reap our rewards:

######  Proposition 3.28.

Let $(G,P),(H,Q)$ be ordered groups, and suppose that $G$ and $H$ are
amenable. Then the natural homomorphism $\varphi:(G*H,P*Q)\rightarrow(G\wr
H,P\wr Q)$ is a strong reduction to an amenable group.

###### Proof.

By Propositon 3.27, $\varphi$ is a strong reduction. By Lemma 3.25, $G\wr H$
is amenable, so $\varphi$ is a strong reduction to an amenable group.

∎

And combining this with a previous result gives:

######  Corollary 3.29.

The class of ordered groups which reduce to an amenable group is closed under
finite free products.

###### Proof.

It suffices to check two-term free products. Suppose that $(G_{1},P_{1})$ and
$(G_{2},P_{2})$ are ordered groups with reductions $\varphi_{1},\varphi_{2}$
to amenable groups $(H_{1},Q_{1})$ and $(H_{2},Q_{2})$ respectively. Let
$G=G_{1}*G_{2}$, $P=P_{1}*P_{2}$, $H=H_{1}*H_{2}$, and $Q=Q_{1}*Q_{2}$.

Now by Lemma 3.22 there is a reduction $\varphi$ of $(G,P)$ onto $(H,Q)$. By
Proposition 3.28, since $(H,Q)$ is the free product of two amenable groups,
then there is a strong reduction
$\varphi^{\prime}:(H_{1}*H_{2},Q_{1}*Q_{2})\rightarrow(H_{1}\wr H_{2},Q_{1}\wr
Q_{2})$ where $H_{1}\wr H_{2}$ is amenable. Then
$\varphi^{\prime}\varphi:(G,P)\rightarrow(F,R)$ is a composition of
reductions, hence a reduction by Lemma 3.11, and since its range is amenable,
$(G,P)$ reduces to an amenable group. ∎

###  3.3 Summary

To summarize our results on reductions to amenable groups:

###### Proof.

If $(G,P)$ is an ordered group with $G$ amenable, then
$\operatorname{id}_{G}:(G,P)\rightarrow(G,P)$ is a reduction to an amenable
group, so every amenable ordered group contains a reduction to an amenable
group.

We’ve seen that this class is closed under hereditary subgroups in Corollary
3.16, direct products in Lemma 3.19, and free products in Corollary 3.29.

∎

######  Corollary 3.30.

Let $G=\mathbb{Z}^{2}*\mathbb{Z}$ and $P=\mathbb{N}^{2}*\mathbb{N}$. Then
$(G,P)$ strongly reduces to an amenable group.

###### Proof.

By Proposition 3.27, $(\mathbb{Z}^{2}*\mathbb{Z},\mathbb{N}^{2}*\mathbb{N})$
strongly reduces to
$(\mathbb{Z}^{2}\wr\mathbb{Z},\mathbb{N}^{2}\wr\mathbb{N})$, which is amenable
by Lemma 3.25. ∎

Although the theory we have developed gives a rich class of examples of groups
which reduce to amenable groups, there are still many open questions worthy of
exploration:

######  Question 3.31.

Are there any ordered groups that reduce to an amenable group, but do not
strongly reduce to an amenable group?

######  Question 3.32.

Is the class of ordered groups which reduce to amenable groups closed under
other group theoretic constructions (direct limits, HNN extensions,
amalgamated free products, graph products, etc)?

For the latter question, we would conjecture that the answer is yes, and that
this should follow from this class being closed under quotients by “positive
relators”:

######  Conjecture 3.33.

Let $(G,P)$ be an ordered group. Let $p_{1},p_{2}\in P$ be incomparable
(meaning $p_{1}\not\leq p_{2}$ and $p_{1}\not\geq p_{2}$), and let $N$ denote
the normal subgroup of $G$ generated by $p_{1}p_{2}^{-1}$. Then:

  1. a.

If $(G,P)$ reduces to an amenable group, then $(G/N,P/N)$ reduces to an
amenable group.

  2. b.

If $(G,P)$ strongly reduces to an amenable group, then $(G/N,P/N)$ strongly
reduces to an amenable group.

Of these two sub-conjectures, (b) seems more plausible: one could imagine
“carrying the quotient forward” into the reduction. However, attempting that
in the case of a non-strong reduction of (a) seems more difficult. For
instance, we’ve seen that $(F_{2},P_{2})$ reduces to
$(\mathbb{Z},\mathbb{N})$, and taking $p_{1}=ab,p_{2}=ba$ results in the
relator $ab=ba$, so $F_{2}/N\cong\mathbb{Z}^{2}$ and
$P_{2}/N\cong\mathbb{N}^{2}$, but $(\mathbb{Z}^{2},\mathbb{N}^{2})$ does not
reduce to $(\mathbb{Z},\mathbb{N})$ or any quotient thereof. While
$(\mathbb{Z}^{2},\mathbb{N}^{2})$ reduces to a different amenable group
(itself), it is not clear in general how one would avoid this type of issue.

##  4 Gauge-Invariant Uniqueness for $P$-graphs

In this section, our goal is to prove a gauge-invariant uniqueness theorem for
$P$-graphs: there is exactly one representation of a $P$-graph which is
$\Lambda$-faithful, tight, and has a gauge coaction. However, we will show in
Lemma 4.24 that the gauge-invariant uniqueness theorem can fail in general, so
we require an additional hypothesis. That additional hypothesis is that
$(G,P)$ can reduce to an amenable ordered group, and we will prove a gauge-
invariant uniqueness theorem for such $P$-graphs in Theorem 1.

###  4.1 A Co-Universal Algebra

In this section, we will prove a weaker form of a gauge invariant uniqueness
theorem which says that there is exactly one representation of a $P$-graph
which is $\Lambda$-faithful, tight, and has a _normal_ gauge coaction. This
result is a slight generalization of [3, Theorem 5.3] to the context of
_weakly_ quasi-lattice ordered groups.

To prove this, we define and study the balanced algebra of a representation,
culminating in Theorem 4.15, which classifies the balanced algebras of a
graph.

To give a brief summary of the argument:

  1. 1.

Use the structure of “infinite paths” in $\Lambda$ to construct a tight,
$\Lambda$-faithful representation called the ultrafilter representation $f$.

  2. 2.

For a representation $t$, we define the balanced algebra
$\mathcal{B}(t)=\overline{\operatorname{span}}\\{t_{\mu}t_{\nu}^{*}:d(\mu)=d(\nu)\\}$
(Lemma 4.7). If there is a gauge coaction on $t$, there is a conditional
expectation $\Phi_{t}:C^{*}(t)\rightarrow\mathcal{B}(t)$ which is given by

$\Phi_{t}(t_{\mu}t_{\nu}^{*})=\begin{cases}t_{\mu}t_{\nu}^{*}&\text{ if
}d(\mu)=d(\nu)\\\ 0&\text{ otherwise}\end{cases},$

and this conditional expectation is faithful if and only if the gauge coaction
is normal (Lemma 4.8). We define a balanced covering to be a map
$\psi^{t}_{s}:\mathcal{B}(t)\rightarrow\mathcal{B}(s)$ given by
$t_{\mu}t_{\nu}^{*}\mapsto s_{\mu}s_{\nu}^{*}$.

  3. 3.

Recalling that $\mathcal{T}$ denotes the Toeplitz representation of $\Lambda$
from Definition 2.47, in Theorem 4.15 we show that
$\ker\psi^{\mathcal{T}}_{t}$ is generated by the “bolts” and range projections
it contains. Using this, we show in Lemma 4.16 that any tight representation