(2.12) $\pi(a)q_{F}=0.$

Moreover, by [16, Proposition 3.2], if $(\pi,t)$ is injective then (under the
assumption of strong compact alignment) (2.11) implies that

(2.13) $\phi_{{\underline{n}}}(a)\in{\mathcal{K}}(X_{{\underline{n}}})\text{
for all }{\underline{0}}\leq{\underline{n}}\leq{\underline{1}}_{[d]}.$

Conversely, if (2.12) and (2.13) hold for some $F\subseteq[d]$ and $a\in A$,
then Proposition 2.5.10 gives that

(2.14)
$\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{{\underline{n}}}(\phi_{{\underline{n}}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=\pi(a)q_{F}=0,$

and so $\pi(a)\in B_{({\underline{0}},{\underline{m}}]}^{(\pi,t)}$ for any
${\underline{m}}\geq{\underline{1}}_{F}$. The following proposition justifies
the usage of $\\{{\mathcal{I}}_{F}\\}_{F\subseteq[d]}$.

######  Proposition 2.5.11.

[16, Proposition 3.4] Let $X$ be a strong compactly aligned product system
with coefficients in a C*-algebra $A$. Suppose that $(\pi,t)$ is an injective
Nica-covariant representation of $X$ and fix $a\in A$. If $\pi(a)\in
B_{({\underline{0}},{\underline{m}}]}^{(\pi,t)}$ for some
${\underline{m}}\in\mathbb{Z}_{+}^{d}$, then $a\in{\mathcal{I}}_{F}$ for
$F=\operatorname{supp}{\underline{m}}$.

We define the _ideal of the CNP-relations with respect to $(\pi,t)$_ as
follows:

$\mathfrak{J}_{\mathcal{I}}^{(\pi,t)}:=\left\langle\pi({\mathcal{I}}_{F})q_{F}\mid
F\subseteq[d]\right\rangle\subseteq\mathrm{C}^{*}(\pi,t).$

We will say that $(\pi,t)$ is a _CNP-representation (of $X$)_ if it satisfies

$\pi(a)q_{F}=\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{{\underline{n}}}(\phi_{{\underline{n}}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=0\text{
for all }a\in{\mathcal{I}}_{F},F\subseteq[d].$

It follows that $\mathfrak{J}_{\mathcal{I}}^{(\pi,t)}=\\{0\\}$ if and only if
$(\pi,t)$ is a CNP-representation. We write ${\mathcal{N}}{\mathcal{O}}_{X}$
for the universal C*-algebra with respect to the CNP-representations, and
refer to it as the _Cuntz-Nica-Pimsner algebra of $X$_, i.e.,

${\mathcal{N}}{\mathcal{O}}_{X}\equiv{\mathcal{N}}{\mathcal{T}}_{X}/\mathfrak{J}_{\mathcal{I}}^{(\overline{\pi}_{X},\overline{t}_{X})}.$

We write $(\pi_{X}^{\mathcal{I}},t_{X}^{\mathcal{I}})$ for the _universal CNP-
representation (of $X$)_, i.e.,
$(\pi_{X}^{\mathcal{I}},t_{X}^{\mathcal{I}})=(Q\circ\overline{\pi}_{X},Q\circ\overline{t}_{X})$,
where $Q\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{O}}_{X}$
is the canonical quotient map. Since ${\mathcal{N}}{\mathcal{O}}_{X}$ is an
equivariant quotient of ${\mathcal{N}}{\mathcal{T}}_{X}$, the representation
$(\pi_{X}^{\mathcal{I}},t_{X}^{\mathcal{I}})$ admits a gauge action.

In [16] it is shown that ${\mathcal{N}}{\mathcal{O}}_{X}$ coincides with the
Cuntz-Nica-Pimsner algebra of Sims and Yeend [51], and thus with the strong
covariance algebra of Sehnem [48]. In particular,
$(\pi_{X}^{\mathcal{I}},t_{X}^{\mathcal{I}})$ is injective by [51, Theorem
4.1], since $(\mathbb{Z}^{d},\mathbb{Z}_{+}^{d})$ satisfies [51, Condition
(3.5)]. Moreover, ${\mathcal{N}}{\mathcal{O}}_{X}$ is co-universal with
respect to the injective Nica-covariant representations of $X$ that admit a
gauge action [51]. The co-universal property of
${\mathcal{N}}{\mathcal{O}}_{X}$ has been verified in several works [10, 16,
17, 49] in more general contexts. Moreover, unitarily equivalent strong
compactly aligned product systems $X$ and $Y$ satisfy
${\mathcal{N}}{\mathcal{O}}_{X}\cong{\mathcal{N}}{\mathcal{O}}_{Y}$
canonically.

######  Proposition 2.5.12.

Let $X$ and $Y$ be unitarily equivalent strong compactly aligned product
systems with coefficients in C*-algebras $A$ and $B$, respectively. Then the
bijection of Proposition 2.3.4 preserves the CNP-representations.

######  Proof..

This is immediate by Propositions 2.4.3 and 2.5.6, and Remark 2.3.2. ∎

We finish the subsection with a proposition concerning canonical
$*$-epimorphisms arising from injective Nica-covariant representations. This
trick was used implicitly in [16, 31].

######  Proposition 2.5.13.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let $(\tilde{\pi},\tilde{t})$ and $(\pi,t)$ be Nica-covariant
representations of $X$ such that $(\pi,t)$ is injective, and suppose there
exists a canonical $*$-epimorphism
$\Phi\colon\mathrm{C}^{*}(\tilde{\pi},\tilde{t})\to\mathrm{C}^{*}(\pi,t)$.
Then the following are equivalent:

  1. (i)

$\ker\Phi\cap B_{[{\underline{0}},\infty]}^{(\tilde{\pi},\tilde{t})}=\\{0\\}$;

  2. (ii)

$\ker\Phi\cap
B_{[{\underline{0}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}=\\{0\\}$ for
all ${\underline{m}}\in\mathbb{Z}_{+}^{d}$;

  3. (iii)

$\ker\Phi\cap
B_{[{\underline{0}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})}=\\{0\\}$;

  4. (iv)

$\ker\Phi\cap
B_{[{\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}=\\{0\\}$
for all $F\subseteq[d]$.

######  Proof..

First note that $(\tilde{\pi},\tilde{t})$ is injective since $(\pi,t)$ is
injective. By definition we see that
$B_{[{\underline{0}},\infty]}^{(\tilde{\pi},\tilde{t})}$ is the inductive
limit of the cores
$B_{[{\underline{0}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}$ for
${\underline{m}}\in\mathbb{Z}_{+}^{d}$, and thus [13, Lemma III.4.1] yields
the equivalence of item (i) with item (ii).

If item (ii) holds then it implies item (iii) by applying for
${\underline{m}}={\underline{1}}_{[d]}$. If item (iii) holds then it implies
item (iv) since
$B_{[{\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}\subseteq
B_{[{\underline{0}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})}$ for all
$F\subseteq[d]$.

It now suffices to show that item (iv) implies item (ii), so fix
${\underline{m}}\in\mathbb{Z}_{+}^{d}$. Without loss of generality, assume
that ${\underline{m}}$ has at least one entry greater than $1$ (the assumption
deals with the case where ${\underline{m}}$ has no entries greater than $1$).
To reach contradiction, assume that $\ker\Phi\cap
B_{[{\underline{0}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}\neq\\{0\\}$.
Take $0\neq f\in\ker\Phi\cap
B_{[{\underline{0}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}$, so that we
may write

$f=\tilde{\pi}(a)+\sum\\{\tilde{\psi}_{\underline{n}}(k_{\underline{n}})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{m}}\\},$

for some $a\in A$ and $k_{\underline{n}}\in{\mathcal{K}}(X_{\underline{n}})$
for all ${\underline{0}}\neq{\underline{n}}\leq{\underline{m}}$. Note that we
can write $\tilde{\pi}(a)=\tilde{\psi}_{\underline{0}}(k_{\underline{0}})$ for
$k_{\underline{0}}:=\phi_{\underline{0}}(a)$. Without loss of generality, we
may assume that $f$ is written irreducibly, so that we may choose a minimal
${\underline{0}}\leq{\underline{r}}\leq{\underline{m}}$ such that
$k_{\underline{r}}\neq 0$, and
$\tilde{\psi}_{\underline{r}}(k_{\underline{r}})\notin
B_{({\underline{r}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}$. The
minimality of ${\underline{r}}$ means that if we have
${\underline{0}}\leq{\underline{n}}\leq{\underline{m}}$ such that
$k_{\underline{n}}\neq 0$ and ${\underline{n}}\leq{\underline{r}}$, then
${\underline{n}}={\underline{r}}$. If ${\underline{r}}={\underline{m}}$, then
$f=\tilde{\psi}_{\underline{m}}(k_{\underline{m}})$ and
$\Phi(f)=\psi_{\underline{m}}(k_{\underline{m}})=0$. Injectivity of $(\pi,t)$
then implies that $k_{\underline{m}}=0$ and hence $f=0$, a contradiction. So
without loss of generality assume that ${\underline{r}}<{\underline{m}}$.
Fixing $\xi_{\underline{r}},\eta_{\underline{r}}\in X_{\underline{r}}$, we
have that

$\tilde{t}_{\underline{r}}(\xi_{\underline{r}})^{*}f\tilde{t}_{\underline{r}}(\eta_{\underline{r}})=\tilde{\pi}(b)+\sum\\{\tilde{\psi}_{\underline{n}}(k_{\underline{n}}^{\prime})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{m}}-{\underline{r}}\\},$

where $k_{\underline{n}}^{\prime}$ is a suitably defined element of
${\mathcal{K}}(X_{\underline{n}})$ for all
${\underline{0}}\neq{\underline{n}}\leq{\underline{m}}-{\underline{r}}$ and
$b:=\left\langle\xi_{\underline{r}},k_{\underline{r}}\eta_{\underline{r}}\right\rangle$,
due to the minimality of ${\underline{r}}$. Applying $\Phi$, we obtain that

$\pi(b)+\sum\\{\psi_{\underline{n}}(k_{\underline{n}}^{\prime})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{m}}-{\underline{r}}\\}=0,$

which in turn implies that
$b\in\pi^{-1}(B_{({\underline{0}},{\underline{m}}-{\underline{r}}]}^{(\pi,t)})$.
Let $F:=\operatorname{supp}({\underline{m}}-{\underline{r}})$, which is non-
empty since ${\underline{m}}\neq{\underline{r}}$. An application of (2.12)
then yields that $\pi(b)q_{F}=0$. Note also that
$\phi_{{\underline{n}}}(b)\in{\mathcal{K}}(X_{\underline{n}})$ for all
${\underline{0}}\leq{\underline{n}}\leq{\underline{1}}_{[d]}$ using (2.13),
which applies since $(\pi,t)$ is injective. Therefore we obtain that

$\tilde{\pi}(b)\tilde{q}_{F}=\tilde{\pi}(b)+\sum\\{(-1)^{|{\underline{n}}|}\tilde{\psi}_{\underline{n}}(\phi_{\underline{n}}(b))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}\in
B_{[{\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}.$

It then follows that

$\displaystyle\Phi(\tilde{\pi}(b)\tilde{q}_{F})$
$\displaystyle=\Phi(\tilde{\pi}(b)+\sum\\{(-1)^{|{\underline{n}}|}\tilde{\psi}_{\underline{n}}(\phi_{\underline{n}}(b))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\})$
$\displaystyle=\pi(b)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{\underline{n}}(\phi_{\underline{n}}(b))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=\pi(b)q_{F}=0.$

Hence $\tilde{\pi}(b)\tilde{q}_{F}\in\ker\Phi\cap
B_{[{\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}$ and so
$\tilde{\pi}(b)\tilde{q}_{F}=0$ by assumption. So we have that

$\tilde{t}_{\underline{r}}(\xi_{\underline{r}})^{*}\tilde{\psi}_{\underline{r}}(k_{\underline{r}})\tilde{t}_{\underline{r}}(\eta_{\underline{r}})=\tilde{\pi}(b)\in
B_{({\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}$

for all $\xi_{\underline{r}},\eta_{\underline{r}}\in X_{\underline{r}}$.
Consequently, we have that

$\displaystyle\tilde{\psi}_{\underline{r}}({\mathcal{K}}(X_{\underline{r}}))\tilde{\psi}_{\underline{r}}(k_{\underline{r}})\tilde{\psi}_{\underline{r}}({\mathcal{K}}(X_{\underline{r}}))\subseteq[\tilde{t}_{\underline{r}}(X_{\underline{r}})B_{({\underline{0}},{\underline{1}}_{F}]}^{(\tilde{\pi},\tilde{t})}\tilde{t}_{\underline{r}}(X_{\underline{r}})^{*}]\subseteq[\tilde{t}_{\underline{r}}(X_{\underline{r}})B_{({\underline{0}},{\underline{m}}-{\underline{r}}]}^{(\tilde{\pi},\tilde{t})}\tilde{t}_{\underline{r}}(X_{\underline{r}})^{*}]\subseteq
B_{({\underline{r}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}.$

By using an approximate unit of
$\tilde{\psi}_{\underline{r}}({\mathcal{K}}(X_{\underline{r}}))$ and the fact
that $B_{({\underline{r}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}$ is
closed in $\mathrm{C}^{*}(\tilde{\pi},\tilde{t})$, we deduce that
$\tilde{\psi}_{\underline{r}}(k_{\underline{r}})\in
B_{({\underline{r}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}$, which
contradicts the irreducibility assumption. Therefore we obtain that
$\ker\Phi\cap
B_{[{\underline{0}},{\underline{m}}]}^{(\tilde{\pi},\tilde{t})}=\\{0\\}$ for
all ${\underline{m}}\in\mathbb{Z}_{+}^{d}$, as required. ∎

###  2.6. The $IXI$ product system

In this subsection we present a product system construction that will be
useful in Section 5, and which extends the results of [34, Section 9].

Let $X$ be a C*-correspondence over a C*-algebra $A$ and let $I\subseteq A$ be
an ideal. Consider $IXI\subseteq XI$ and recall that $XI$ is closed. An
application of the Hewitt-Cohen Factorisation Theorem gives that $IXI$ is a
closed linear subspace of $XI$, i.e., $IXI=[IXI]$. It is clear that $IXI$ is
also closed under the right action of $A$, and hence $IXI$ carries the
structure of a right Hilbert $A$-module. Next, we define a left action
$\phi_{IXI}$ of $A$ on $IXI$ by

$\phi_{IXI}\colon
A\to{\mathcal{L}}(IXI);\phi_{IXI}(a)=\phi_{X}(a)|_{IXI}\text{ for all }a\in
A.$

Hence $IXI$ inherits the structure of a C*-correspondence over $A$ from $X$.
By restricting $\phi_{IXI}$ to $I$, we may view $IXI$ as a C*-correspondence
over $I$.

As was the case with ${\mathcal{K}}(XI)$, we have a natural embedding of
${\mathcal{K}}(IXI)$ in ${\mathcal{K}}(X)$. The proof of the following result
follows an almost identical trajectory to that of [22, Lemma 2.6 (1)], and it
is left to the reader.

######  Lemma 2.6.1.

Let $X$ be a C*-correspondence over a C*-algebra $A$ and let $I\subseteq A$ be
an ideal. Then there exists an embedding
$\iota\colon{\mathcal{K}}(IXI)\to{\mathcal{K}}(X)$ such that

$\iota(\Theta_{\xi,\eta}^{IXI})=\Theta_{\xi,\eta}^{X}\text{ for all
}\xi,\eta\in IXI.$

Consequently ${\mathcal{K}}(IXI)$ is $*$-isomorphic to
$\overline{\operatorname{span}}\\{\Theta_{\xi,\eta}^{X}\mid\xi,\eta\in IXI\\}$
via the map $\iota$, with

$\iota^{-1}(k)=k|_{IXI}\text{ for all
}k\in\overline{\operatorname{span}}\\{\Theta_{\xi,\eta}^{X}\mid\xi,\eta\in
IXI\\}.$

Now let $X$ be a product system over $P$. Recall that if $I$ is positively
invariant for $X$, then $IX_{p}I=IX_{p}$ for all $p\in P$ by Lemma 2.2.6. In
this case we obtain a product system $IXI$, with a construction that is
compatible with the product system structure of $X$.

######  Proposition 2.6.2.

Let $P$ be a unital subsemigroup of a discrete group $G$. Let $X$ be a product
system over $P$ with coefficients in a C*-algebra $A$ and let $I\subseteq A$
be an ideal that is positively invariant for $X$. Then
$IXI:=\\{IX_{p}I\\}_{p\in P}$ inherits from $X$ a canonical structure as a
product system over $P$ with coefficients in $I$, identifying each $IX_{p}I$
as a sub-C*-correspondence of $X_{p}$.

######  Proof..

We will denote the multiplication maps of $X$ by $\\{u_{p,q}^{X}\\}_{p,q\in
P}$. The discussion preceding Lemma 2.6.1 gives that $IX_{p}I$ is a
C*-correspondence over $I$ for all $p\in P$. For $p,q\in P$, let the unitary
map $\iota_{p,q}$ be induced by the canonical identifications

$IX_{p}I\otimes_{I}IX_{q}I\cong IX_{p}I\otimes_{A}IX_{q}I\cong
IX_{p}\otimes_{A}X_{q}I,$

where we use Lemma 2.2.6 in the second identification and write (in an abuse
of notation) $IX_{p}I\otimes_{A}IX_{q}I$ for the closed linear space generated
by $IX_{p}I\odot_{A}IX_{q}I$ in $X_{p}\otimes_{A}X_{q}$. We then define the
multiplication map $u_{p,q}^{IXI}$ for $IXI$ by
$u_{p,q}^{IXI}:=u_{p,q}^{X}\circ\iota_{p,q}$. It is clear that $u_{p,q}^{IXI}$
is a linear isometry and an $I$-bimodule map onto
$u_{p,q}^{X}(IX_{p}\otimes_{A}X_{q}I)=IX_{pq}I$. It follows that $IXI$
together with the multiplication maps $\\{u_{p,q}^{IXI}\\}_{p,q\in P}$ becomes
a product system, using the product system axioms of $X$. ∎

######  Proposition 2.6.3.

Let $P$ be a unital subsemigroup of a discrete group $G$. Let $X$ be a product
system over $P$ with coefficients in a C*-algebra $A$ and let $I\subseteq A$
be an ideal that is positively invariant for $X$. Let
$\\{\iota_{p}^{pq}\\}_{p,q\in P}$ denote the connecting $*$-homomorphisms of
$X$ and let $\\{j_{p}^{pq}\\}_{p,q\in P}$ denote the connecting
$*$-homomorphisms of $IXI$. Then

$j_{p}^{pq}(k_{p}|_{IX_{p}I})=\iota_{p}^{pq}(k_{p})|_{IX_{pq}I}\text{ for all
}p,q\in
P\;\text{and}\;k_{p}\in\overline{\operatorname{span}}\\{\Theta_{\xi_{p},\eta_{p}}^{X_{p}}\mid\xi_{p},\eta_{p}\in
IX_{p}I\\}.$

######  Proof..

Without loss of generality we may assume that $p\neq e$ and $q\neq e$, as the
claim is straightforward otherwise. By Lemma 2.6.1 we have that
$k_{p}|_{IX_{p}I}\in{\mathcal{K}}(IX_{p}I)$. Recalling that $IX_{pq}I$ is a
closed linear subspace of $X_{pq}$, it suffices to show that

$j_{p}^{pq}(k_{p}|_{IX_{p}I})u_{p,q}^{IXI}(\xi_{p}\otimes\xi_{q})=\iota_{p}^{pq}(k_{p})u_{p,q}^{IXI}(\xi_{p}\otimes\xi_{q}),\text{
for all }\xi_{p}\in IX_{p}I,\xi_{q}\in IX_{q}I.$

Indeed, fixing $\xi_{p}\in IX_{p}I$ and $\xi_{q}\in IX_{q}I$, we have that

$\displaystyle
j_{p}^{pq}(k_{p}|_{IX_{p}I})u_{p,q}^{IXI}(\xi_{p}\otimes\xi_{q})$
$\displaystyle=u_{p,q}^{IXI}((k_{p}\xi_{p})\otimes\xi_{q})=u_{p,q}^{X}((k_{p}\xi_{p})\otimes\xi_{q})$
$\displaystyle=\iota_{p}^{pq}(k_{p})u_{p,q}^{X}(\xi_{p}\otimes\xi_{q})=\iota_{p}^{pq}(k_{p})u_{p,q}^{IXI}(\xi_{p}\otimes\xi_{q}),$

as required. ∎

Next we show that the properties of compact alignment and strong compact
alignment of $IXI$ are inherited from $X$.

######  Proposition 2.6.4.

Let $P$ be a unital right LCM subsemigroup of a discrete group $G$. Let $X$ be
a compactly aligned product system over $P$ with coefficients in a C*-algebra
$A$ and let $I\subseteq A$ be an ideal that is positively invariant for $X$.
Then the product system $IXI$ is compactly aligned.

######  Proof..

Let $\\{\iota_{p}^{pq}\\}_{p,q\in P}$ (resp. $\\{j_{p}^{pq}\\}_{p,q\in P}$)
denote the connecting $*$-homomorphisms of $X$ (resp. $IXI$). It suffices to
show that the compact alignment condition holds for rank-one operators. To
this end, let $p,q\in P\setminus\\{e\\}$ be such that $pP\cap qP=wP$ for some
$w\in P$, and fix $\xi_{p},\eta_{p}\in IX_{p}I$ and $\xi_{q},\eta_{q}\in
IX_{q}I$. We must show that

$j_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})j_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I})\in{\mathcal{K}}(IX_{w}I).$

Recall that

$\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I}=\Theta_{\xi_{p},\eta_{p}}^{X_{p}}|_{IX_{p}I}\quad\text{and}\quad\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I}=\Theta_{\xi_{q},\eta_{q}}^{X_{q}}|_{IX_{q}I},$

by Lemma 2.6.1. An application of Proposition 2.6.3 then gives that

(2.15)
$j_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})j_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I})=\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})|_{IX_{w}I}\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})|_{IX_{w}I}=[\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})]|_{IX_{w}I}.$

By Lemma 2.6.1, it suffices to show that

$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\in\overline{\operatorname{span}}\\{\Theta_{\xi_{w},\eta_{w}}^{X_{w}}\mid\xi_{w},\eta_{w}\in
IX_{w}I\\}.$

Compact alignment of $X$ gives that

$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\in{\mathcal{K}}(X_{w}).$

Next, let $(u_{\lambda})_{\lambda\in\Lambda}$ denote an approximate unit of
$I$. For each $\lambda\in\Lambda$, we have that

$\phi_{w}(u_{\lambda})\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})=\iota_{p}^{w}(\phi_{p}(u_{\lambda})\Theta_{\xi_{p},\eta_{p}}^{X_{p}})=\iota_{p}^{w}(\Theta_{\phi_{p}(u_{\lambda})\xi_{p},\eta_{p}}^{X_{p}}).$

Using this and the fact that
$\left\|\cdot\right\|\text{-}\lim_{\lambda}\phi_{p}(u_{\lambda})\xi_{p}=\xi_{p}$
(as $\xi_{p}\in IX_{p}I$), we obtain that

$\left\|\cdot\right\|\text{-}\lim_{\lambda}\phi_{w}(u_{\lambda})\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})=\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}}).$

By analogous reasoning, we have that

$\left\|\cdot\right\|\text{-}\lim_{\lambda}\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\phi_{w}(u_{\lambda})=\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}}).$

Therefore, we obtain that

$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})=\left\|\cdot\right\|\text{-}\lim_{\lambda}\phi_{w}(u_{\lambda})\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\phi_{w}(u_{\lambda}).$

Consequently, we have that
$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})$
is expressible as the norm-limit of a net that is contained in
$\phi_{w}(I){\mathcal{K}}(X_{w})\phi_{w}(I)$. We also have that

$\phi_{w}(I){\mathcal{K}}(X_{w})\phi_{w}(I)\subseteq\overline{\operatorname{span}}\\{\Theta_{\xi_{w},\eta_{w}}^{X_{w}}\mid\xi_{w},\eta_{w}\in
IX_{w}I\\}$

by Corollary 2.2.7. Thus

(2.16)
$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\in\overline{\operatorname{span}}\\{\Theta_{\xi_{w},\eta_{w}}^{X_{w}}\mid\xi_{w},\eta_{w}\in
IX_{w}I\\},$

as required. ∎

######  Proposition 2.6.5.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $I\subseteq A$ be an ideal that is positively invariant
for $X$. Then the product system $IXI$ is strong compactly aligned.

######  Proof..

Let
$\\{\iota_{\underline{n}}^{{\underline{n}}+{\underline{m}}}\\}_{{\underline{n}},{\underline{m}}\in\mathbb{Z}_{+}^{d}}$
(resp.
$\\{j_{\underline{n}}^{{\underline{n}}+{\underline{m}}}\\}_{{\underline{n}},{\underline{m}}\in\mathbb{Z}_{+}^{d}}$)
denote the connecting $*$-homomorphisms of $X$ (resp. $IXI$). Proposition
2.6.4 asserts that $IXI$ is compactly aligned, so it remains to check that
$IXI$ satisfies

$j_{\underline{n}}^{{\underline{n}}+{\underline{i}}}({\mathcal{K}}(IX_{\underline{n}}I))\subseteq{\mathcal{K}}(IX_{{\underline{n}}+{\underline{i}}}I)\textup{
whenever ${\underline{n}}\perp{\underline{i}}$, where
$i\in[d],{\underline{n}}\in\mathbb{Z}_{+}^{d}\setminus\\{{\underline{0}}\\}$}.$

It suffices to show that this holds for rank-one operators. To this end, fix
$\xi_{\underline{n}},\eta_{\underline{n}}\in IX_{\underline{n}}I$. An
application of Proposition 2.6.3 then yields that

$j_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{IX_{\underline{n}}I})=j_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}}|_{IX_{{\underline{n}}}I})=\iota_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}})|_{IX_{{\underline{n}}+{\underline{i}}}I}.$

By strong compact alignment of $X$, we have that
$\iota_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}})\in{\mathcal{K}}(X_{{\underline{n}}+{\underline{i}}})$.
By using an approximate unit $(u_{\lambda})_{\lambda\in\Lambda}$ of $I$ and
Corollary 2.2.7, we then obtain that

$\displaystyle\iota_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}})$
$\displaystyle=\left\|\cdot\right\|\text{-}\lim_{\lambda}\phi_{{\underline{n}}+{\underline{i}}}(u_{\lambda})\iota_{\underline{n}}^{{\underline{n}}+{\underline{i}}}(\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}})\phi_{{\underline{n}}+{\underline{i}}}(u_{\lambda})$
$\displaystyle\in[\phi_{{\underline{n}}+{\underline{i}}}(I){\mathcal{K}}(X_{{\underline{n}}+{\underline{i}}})\phi_{{\underline{n}}+{\underline{i}}}(I)]$
$\displaystyle\subseteq\overline{\operatorname{span}}\\{\Theta_{\xi_{{\underline{n}}+{\underline{i}}},\eta_{{\underline{n}}+{\underline{i}}}}^{X_{{\underline{n}}+{\underline{i}}}}\mid\xi_{{\underline{n}}+{\underline{i}}},\eta_{{\underline{n}}+{\underline{i}}}\in
IX_{{\underline{n}}+{\underline{i}}}I\\}.$

An application of Lemma 2.6.1 finishes the proof. ∎

Next we study the representations of $IXI$. Given a representation $(\pi,t)$
of $X$, we write $(\pi|_{I},t|_{IXI})$ for the family
$\\{(\pi|_{I},t_{p}|_{IX_{p}I})\\}_{p\in P}$. It is routine to check that
$(\pi|_{I},t|_{IXI})$ is a representation of $IXI$. For each $p\in P$, let
$\tilde{\psi}_{p}\colon{\mathcal{K}}(IX_{p}I)\to\mathrm{C}^{*}(\pi,t)$ be the
$*$-homomorphism induced by $(\pi|_{I},t_{p}|_{IX_{p}I})$. Then for all
$\xi_{p},\eta_{p}\in IX_{p}I$, we have that

$\tilde{\psi}_{p}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}}|_{IX_{p}I})=\tilde{\psi}_{p}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})=t_{p}|_{IX_{p}I}(\xi_{p})t_{p}|_{IX_{p}I}(\eta_{p})^{*}=t_{p}(\xi_{p})t_{p}(\eta_{p})^{*}=\psi_{p}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}}),$

from which it follows that

(2.17) $\tilde{\psi}_{p}(k_{p}|_{IX_{p}I})=\psi_{p}(k_{p})\text{ for all
}k_{p}\in\overline{\operatorname{span}}\\{\Theta_{\xi_{p},\eta_{p}}^{X_{p}}\mid\xi_{p},\eta_{p}\in
IX_{p}I\\}.$

When $P$ is a right LCM semigroup and $X$ is compactly aligned (and thus $IXI$
is compactly aligned by Proposition 2.6.4), Nica-covariance of $(\pi,t)$ is
inherited by $(\pi|_{I},t|_{IXI})$.

######  Proposition 2.6.6.

Let $P$ be a unital right LCM subsemigroup of a discrete group $G$. Let $X$ be
a compactly aligned product system over $P$ with coefficients in a C*-algebra
$A$ and let $I\subseteq A$ be an ideal that is positively invariant for $X$.
Let $(\pi,t)$ be a Nica-covariant representation of $X$. Then
$(\pi|_{I},t|_{IXI})$ is a Nica-covariant representation of $IXI$, and

(2.18)
$\mathrm{C}^{*}(\pi|_{I},t|_{IXI})=\overline{\operatorname{span}}\\{\pi(I)t_{p}(X_{p})\pi(I)t_{q}(X_{q})^{*}\pi(I)\mid
p,q\in P\\}.$

######  Proof..

Let $\\{\iota_{p}^{pq}\\}_{p,q\in P}$ (resp. $\\{j_{p}^{pq}\\}_{p,q\in P}$)
denote the connecting $*$-homomorphisms of $X$ (resp. $IXI$). For each $p\in
P$, let $\tilde{\psi}_{p}\colon{\mathcal{K}}(IX_{p}I)\to\mathrm{C}^{*}(\pi,t)$
be the $*$-homomorphism induced by $(\pi|_{I},t_{p}|_{IX_{p}I})$. Now fix
$p,q\in P\setminus\\{e\\},k_{p}\in{\mathcal{K}}(IX_{p}I)$ and
$k_{q}\in{\mathcal{K}}(IX_{q}I)$. We must show that

$\tilde{\psi}_{p}(k_{p})\tilde{\psi}_{q}(k_{q})=\begin{cases}\tilde{\psi}_{w}(j_{p}^{w}(k_{p})j_{q}^{w}(k_{q}))&\text{if}\;pP\cap
qP=wP\;\text{for some}\;w\in P,\\\ 0&\text{otherwise}.\end{cases}$

It suffices to show this for $k_{p}=\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I}$ and
$k_{q}=\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I}$ for some $\xi_{p},\eta_{p}\in
IX_{p}I$ and $\xi_{q},\eta_{q}\in IX_{q}I$. By (2.17) and Nica-covariance of
$(\pi,t)$, we have that

$\displaystyle\tilde{\psi}_{p}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})\tilde{\psi}_{q}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I})$
$\displaystyle=\psi_{p}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\psi_{q}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})$
$\displaystyle=\begin{cases}\psi_{w}(\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}}))&\text{if}\;pP\cap
qP=wP\;\text{for some}\;w\in P,\\\ 0&\text{otherwise}.\end{cases}$

Suppose $pP\cap qP=wP$ for some $w\in P$. Since the assumptions of Proposition
2.6.4 are satisfied, by (2.15) we have that

$[\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})]|_{IX_{w}I}=j_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})j_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I}),$

and by (2.16) we have that

$\iota_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{X_{p}})\iota_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{X_{q}})\in\overline{\operatorname{span}}\\{\Theta_{\xi_{w},\eta_{w}}^{X_{w}}\mid\xi_{w},\eta_{w}\in
IX_{w}I\\}.$

Another application of (2.17) then yields that

$\tilde{\psi}_{p}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})\tilde{\psi}_{q}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I})=\begin{cases}\tilde{\psi}_{w}(j_{p}^{w}(\Theta_{\xi_{p},\eta_{p}}^{IX_{p}I})j_{q}^{w}(\Theta_{\xi_{q},\eta_{q}}^{IX_{q}I}))&\text{if}\;pP\cap
qP=wP\;\text{for some}\;w\in P,\\\ 0&\text{otherwise},\end{cases}$

as required. The second claim is a direct consequence of Nica-covariance. ∎

The structure of $\mathrm{C}^{*}(\pi|_{I},t|_{IXI})$ is linked with that of
$\left\langle\pi(I)\right\rangle\subseteq\mathrm{C}^{*}(\pi,t)$.

######  Proposition 2.6.7.

Let $P$ be a unital right LCM subsemigroup of a discrete group $G$. Let $X$ be
a compactly aligned product system over $P$ with coefficients in a C*-algebra
$A$ and let $I\subseteq A$ be an ideal that is positively invariant for $X$.
Let $(\pi,t)$ be a Nica-covariant representation of $X$. Then the following
hold:

  1. (i)

$\left\langle\pi(I)\right\rangle=\overline{\operatorname{span}}\\{t_{p}(X_{p})\pi(I)t_{q}(X_{q})^{*}\mid
p,q\in P\\}$;

  2. (ii)

$\mathrm{C}^{*}(\pi|_{I},t|_{IXI})=[\pi(I)\left\langle\pi(I)\right\rangle\pi(I)]$,
and thus $\mathrm{C}^{*}(\pi|_{I},t|_{IXI})$ is a hereditary, full
C*-subalgebra of $\left\langle\pi(I)\right\rangle$.

######  Proof..

Item (i) follows by Nica-covariance of $(\pi,t)$. Item (ii) follows as a
standard C*-result. ∎

Suppose now that $P=\mathbb{Z}_{+}^{d}$ and $X$ is strong compactly aligned.
An application of the Gauge-Invariant Uniqueness Theorem permits the
identification of ${\mathcal{N}}{\mathcal{T}}_{IXI}$ and
${\mathcal{N}}{\mathcal{O}}_{IXI}$ with C*-subalgebras of
${\mathcal{N}}{\mathcal{T}}_{X}$ and ${\mathcal{N}}{\mathcal{O}}_{X}$,
respectively. We note that the Toeplitz-Nica-Pimsner case extends to right LCM
semigroups using [31, Theorem 6.4] (but we will not require it here).

######  Proposition 2.6.8.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $I\subseteq A$ be an ideal that is positively invariant
for $X$. Then the inclusion $IXI\hookrightarrow X$ induces a canonical
embedding

${\mathcal{N}}{\mathcal{T}}_{IXI}\cong\mathrm{C}^{*}(\overline{\pi}_{X}|_{I},\overline{t}_{X}|_{IXI})\subseteq{\mathcal{N}}{\mathcal{T}}_{X},$

and thus ${\mathcal{N}}{\mathcal{T}}_{IXI}$ is a hereditary, full
C*-subalgebra of $\left\langle\overline{\pi}_{X}(I)\right\rangle$.

######  Proof..

Note that $IXI$ is strong compactly aligned by Proposition 2.6.5, and
$(\overline{\pi}_{X}|_{I},\overline{t}_{X}|_{IXI})$ is a Nica-covariant
representation of $IXI$ by Proposition 2.6.6. Injectivity of
$(\overline{\pi}_{X}|_{I},\overline{t}_{X}|_{IXI})$ is inherited from
$(\overline{\pi}_{X},\overline{t}_{X})$, and
$(\overline{\pi}_{X}|_{I},\overline{t}_{X}|_{IXI})$ admits a gauge action by
restriction. Hence it suffices to show that
$\overline{\pi}_{X}|_{I}\times\overline{t}_{X}|_{IXI}\colon{\mathcal{N}}{\mathcal{T}}_{IXI}\to\mathrm{C}^{*}(\overline{\pi}_{X}|_{I},\overline{t}_{X}|_{IXI})$
is injective on the fixed point algebra; equivalently on the
$[{\underline{0}},{\underline{1}}_{[d]}]$-core by Proposition 2.5.13.

To this end, first note that we may take
$(\overline{\pi}_{X},\overline{t}_{X})$ to be the Fock representation without
loss of generality. Let
$\psi_{\underline{n}}\colon{\mathcal{K}}(IX_{\underline{n}}I)\to{\mathcal{N}}{\mathcal{T}}_{X}$
be the $*$-homomorphism induced by
$(\overline{\pi}_{X}|_{I},\overline{t}_{X,{\underline{n}}}|_{IX_{{\underline{n}}}I})$
for all ${\underline{n}}\in\mathbb{Z}_{+}^{d}$. Take
$f\in\ker\overline{\pi}_{X}|_{I}\times\overline{t}_{X}|_{IXI}\cap
B_{[{\underline{0}},{\underline{1}}_{[d]}]}^{(\overline{\pi}_{IXI},\overline{t}_{IXI})}$,
so that we may write

$f=\overline{\pi}_{IXI}(a)+\sum\\{\overline{\psi}_{IXI,{\underline{n}}}(k_{\underline{n}})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}\\},$

for some $a\in I$ and $k_{\underline{n}}\in{\mathcal{K}}(IX_{\underline{n}}I)$
for all ${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}$. Recall
that we write
$\overline{\pi}_{IXI}(a)=\overline{\psi}_{IXI,{\underline{0}}}(k_{\underline{0}})$
for $k_{\underline{0}}:=\phi_{IX_{\underline{0}}I}(a)$, and in turn
$\overline{\pi}_{X}|_{I}(a)=\psi_{\underline{0}}(k_{\underline{0}})$. Towards
contradiction, suppose that $f\neq 0$. Then we may choose
${\underline{0}}\leq{\underline{r}}\leq{\underline{1}}_{[d]}$ minimal such
that $k_{{\underline{r}}}\neq 0$. Let
$P_{{\underline{r}}}\colon{\mathcal{F}}X\to X_{{\underline{r}}}$ be the
canonical projection. Then

$\iota(k_{{\underline{r}}})=P_{{\underline{r}}}\left(\sum\\{\psi_{\underline{n}}(k_{\underline{n}})\mid{\underline{0}}\leq{\underline{n}}\leq{\underline{1}}_{[d]}\\}\right)P_{{\underline{r}}}=0,$

where
$\iota\colon{\mathcal{K}}(IX_{{\underline{r}}}I)\to{\mathcal{K}}(X_{{\underline{r}}})$
is the embedding guaranteed by Lemma 2.6.1. Hence $k_{{\underline{r}}}=0$,
which is a contradiction. We conclude that
$\overline{\pi}_{X}|_{I}\times\overline{t}_{X}|_{IXI}$ is injective on the
$[{\underline{0}},{\underline{1}}_{[d]}]$-core, as required. Proposition 2.6.7
then completes the proof. ∎

To establish the corresponding result for ${\mathcal{N}}{\mathcal{O}}_{IXI}$,
we first generalise [34, Proposition 9.2].

######  Proposition 2.6.9.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $I\subseteq A$ be an ideal that is positively invariant
for $X$. Then, for all $F\subseteq[d]$, we have that:

  1. (i)

${\mathcal{J}}_{F}(IXI)=I\cap{\mathcal{J}}_{F}(X)$;

  2. (ii)

${\mathcal{I}}_{F}(IXI)=I\cap{\mathcal{I}}_{F}(X)$.

######  Proof..

(i) The claim holds trivially for $F=\emptyset$, so assume that
$F\neq\emptyset$. Let $a\in{\mathcal{J}}_{F}(IXI)\subseteq I$. It suffices to
show that $a\in{\mathcal{J}}_{F}(X)$. To this end, fix $i\in[d]$. By
definition, we have that
$a\in(\phi_{IX_{\underline{i}}I})^{-1}({\mathcal{K}}(IX_{\underline{i}}I))$.
Since $I$ is positively invariant for $X_{\underline{i}}$, we may apply Lemma
2.2.6 to deduce that $IX_{\underline{i}}I=IX_{\underline{i}}$, and an
application of [34, Lemma 9.1] then yields that

$(\phi_{IX_{\underline{i}}I})^{-1}({\mathcal{K}}(IX_{\underline{i}}I))=I\cap\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}})).$

Hence $a\in\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))$ for
all $i\in[d]$, as required. Moreover, since $\ker\phi_{{\underline{i}}}\cap
I\subseteq\ker\phi_{IX_{\underline{i}}I}$ for every $i\in[d]$, we have that
$(\bigcap_{i\in
F}\ker\phi_{IX_{\underline{i}}I})^{\perp}\subseteq(\bigcap_{i\in
F}\ker\phi_{\underline{i}})^{\perp}\cap I$. Hence $a\in(\bigcap_{i\in
F}\ker\phi_{\underline{i}})^{\perp}$ and thus $a\in{\mathcal{J}}_{F}(X)$, as
required.

For the reverse inclusion, let $a\in I\cap{\mathcal{J}}_{F}(X)$. For each
$i\in[d]$, we have that

$a\in
I\cap\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))=(\phi_{IX_{\underline{i}}I})^{-1}({\mathcal{K}}(IX_{\underline{i}}I))$

by Lemma 2.2.6 and [34, Lemma 9.1]. Therefore, it suffices to show that
$a\in(\bigcap_{i\in F}\ker\phi_{IX_{\underline{i}}I})^{\perp}$. To this end,
fix $b\in\bigcap_{i\in F}\ker\phi_{IX_{\underline{i}}I}$. Another application
of [34, Lemma 9.1] gives that

$b\in I\cap(\bigcap_{i\in F}\ker\phi_{\underline{i}}).$

Since $a\in(\bigcap_{i\in F}\ker\phi_{\underline{i}})^{\perp}$, we therefore
have that $ab=0$, as required.

(ii) Once again, we may assume without loss of generality that
$F\neq\emptyset$. Take $a\in{\mathcal{I}}_{F}(IXI)\subseteq I$ and fix
${\underline{n}}\perp F$. We have that

$\displaystyle\left\langle X_{\underline{n}},aX_{\underline{n}}\right\rangle$
$\displaystyle\subseteq[\left\langle
X_{\underline{n}},IaIX_{\underline{n}}\right\rangle]\subseteq[\left\langle
IX_{\underline{n}},aIX_{\underline{n}}\right\rangle]$
$\displaystyle\subseteq[\left\langle
IX_{\underline{n}}I,\phi_{IX_{\underline{n}}I}(a)(IX_{\underline{n}}I)\right\rangle]\subseteq{\mathcal{J}}_{F}(IXI)\subseteq{\mathcal{J}}_{F}(X),$

by using Lemma 2.2.6 and item (i). Thus $a\in I\cap{\mathcal{I}}_{F}(X)$, as
required.

For the reverse inclusion, fix $a\in I\cap{\mathcal{I}}_{F}(X)$ and
${\underline{n}}\perp F$. We have that

$\left\langle
IX_{\underline{n}}I,\phi_{IX_{\underline{n}}I}(a)(IX_{\underline{n}}I)\right\rangle\subseteq\left\langle
X_{\underline{n}},aX_{\underline{n}}\right\rangle\subseteq
I\cap{\mathcal{J}}_{F}(X)={\mathcal{J}}_{F}(IXI),$

using positive invariance of $I$, the fact that $a\in{\mathcal{I}}_{F}(X)$ and
item (i). Thus $a\in{\mathcal{I}}_{F}(IXI)$, completing the proof. ∎

######  Proposition 2.6.10.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $I\subseteq A$ be an ideal that is positively invariant
for $X$. Then the inclusion $IXI\hookrightarrow X$ induces a canonical
embedding

${\mathcal{N}}{\mathcal{O}}_{IXI}\cong\mathrm{C}^{*}(\pi_{X}^{\mathcal{I}}|_{I},t_{X}^{\mathcal{I}}|_{IXI})\subseteq{\mathcal{N}}{\mathcal{O}}_{X},$

and thus ${\mathcal{N}}{\mathcal{O}}_{IXI}$ is a hereditary, full
C*-subalgebra of $\left\langle\pi_{X}^{\mathcal{I}}(I)\right\rangle$.

######  Proof..

Being the restriction of an injective Nica-covariant representation of $X$
that admits a gauge action,
$(\pi_{X}^{\mathcal{I}}|_{I},t_{X}^{\mathcal{I}}|_{IXI})$ is an injective
Nica-covariant representation of $IXI$ that admits a gauge action. Thus it
suffices to show that
$(\pi_{X}^{\mathcal{I}}|_{I},t_{X}^{\mathcal{I}}|_{IXI})$ is a CNP-
representation of $IXI$ by [16, Theorem 4.2].

To this end, fix $\emptyset\neq F\subseteq[d]$ and
$a\in{\mathcal{I}}_{F}(IXI)$. For each ${\underline{n}}\in\mathbb{Z}_{+}^{d}$,
let
$\tilde{\psi}_{\underline{n}}\colon{\mathcal{K}}(IX_{\underline{n}}I)\to{\mathcal{N}}{\mathcal{O}}_{X}$
be the $*$-homomorphism induced by
$(\pi_{X}^{\mathcal{I}}|_{I},t_{X,{\underline{n}}}^{\mathcal{I}}|_{IX_{\underline{n}}I})$.
We must show that

$\pi_{X}^{\mathcal{I}}|_{I}(a)+\sum\\{(-1)^{|{\underline{n}}|}\tilde{\psi}_{\underline{n}}(\phi_{IX_{\underline{n}}I}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=0.$

Fix ${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}$. We claim that

$\phi_{\underline{n}}(a)\in\overline{\operatorname{span}}\\{\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}}\mid\xi_{\underline{n}},\eta_{\underline{n}}\in
IX_{\underline{n}}I\\}.$

To see this, first note that $a\in I\cap{\mathcal{I}}_{F}(X)$ by item (ii) of
Proposition 2.6.9, and so
$\phi_{\underline{n}}(a)\in{\mathcal{K}}(X_{\underline{n}})$. Let
$(u_{\lambda})_{\lambda\in\Lambda}$ be an approximate unit of $I$. Then an
application of Corollary 2.2.7 yields that

$\phi_{\underline{n}}(u_{\lambda})\phi_{\underline{n}}(a)\phi_{\underline{n}}(u_{\lambda})\in\overline{\operatorname{span}}\\{\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}}\mid\xi_{\underline{n}},\eta_{\underline{n}}\in
IX_{\underline{n}}I\\}\text{ for all }\lambda\in\Lambda.$

Since $a\in I$, we also have that

$\left\|\cdot\right\|\text{-}\lim_{\lambda}\phi_{\underline{n}}(u_{\lambda})\phi_{\underline{n}}(a)\phi_{\underline{n}}(u_{\lambda})=\phi_{\underline{n}}(a),$

and consequently

$\phi_{\underline{n}}(a)\in\overline{\operatorname{span}}\\{\Theta_{\xi_{\underline{n}},\eta_{\underline{n}}}^{X_{\underline{n}}}\mid\xi_{\underline{n}},\eta_{\underline{n}}\in
IX_{\underline{n}}I\\},$

as claimed. Note also that
$\phi_{\underline{n}}(a)|_{IX_{\underline{n}}I}=\phi_{IX_{\underline{n}}I}(a)$
by definition. An application of (2.17) then gives that

$\tilde{\psi}_{\underline{n}}(\phi_{IX_{\underline{n}}I}(a))=\psi_{X,{\underline{n}}}^{\mathcal{I}}(\phi_{\underline{n}}(a))\text{
for all }{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}.$

From this we obtain that

$\displaystyle\pi_{X}^{\mathcal{I}}|_{I}(a)+\sum$
$\displaystyle\\{(-1)^{|{\underline{n}}|}\tilde{\psi}_{\underline{n}}(\phi_{IX_{\underline{n}}I}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=$
$\displaystyle=\pi_{X}^{\mathcal{I}}(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{X,{\underline{n}}}^{\mathcal{I}}(\phi_{\underline{n}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}=0,$

where the final equality follows from the fact that
$a\in{\mathcal{I}}_{F}(X)$, together with the fact that
$(\pi_{X}^{\mathcal{I}},t_{X}^{\mathcal{I}})$ is a CNP-representation of $X$.
Thus $(\pi_{X}^{\mathcal{I}}|_{I},t_{X}^{\mathcal{I}}|_{IXI})$ is a CNP-
representation of $IXI$, as required. Proposition 2.6.7 then completes the
proof. ∎

##  3\. Relative $2^{d}$-tuples and relative Cuntz-Nica-Pimsner algebras

Parametrising the gauge-invariant ideals of the Toeplitz-Pimsner algebra of a
C*-correspondence is facilitated using relative Cuntz-Pimsner algebras. Hence
our first goal is to suitably adapt the relative Cuntz-Pimsner algebra
construction for a strong compactly aligned product system $X$. We do this by
mimicking the construction of ${\mathcal{N}}{\mathcal{O}}_{X}$, and exploiting
strong compact alignment in order to define our relative Cuntz-Nica-Pimsner
algebras in terms of simple algebraic relations induced by finitely many
subsets of the coefficient algebra. As an intermediate step towards the
parametrisation, we study further quotients in-between
${\mathcal{N}}{\mathcal{T}}_{X}$ and ${\mathcal{N}}{\mathcal{O}}_{X}$.

###  3.1. Relative $2^{d}$-tuples and induced ideals

We start by paring down the properties of the family
${\mathcal{I}}:=\\{{\mathcal{I}}_{F}\\}_{F\subseteq[d]}$ and the CNP-relations
of [16] covered in Subsection 2.5.

######  Definition 3.1.1.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. A _$2^{d}$ -tuple (of $X$)_ is a family
${\mathcal{L}}:=\\{{\mathcal{L}}_{F}\\}_{F\subseteq[d]}$ such that
${\mathcal{L}}_{F}$ is a non-empty subset of $A$ for all $F\subseteq[d]$. A
$2^{d}$-tuple ${\mathcal{L}}$ of $X$ is called _relative_ if

${\mathcal{L}}_{F}\subseteq\bigcap\\{\phi_{{\underline{i}}}^{-1}({\mathcal{K}}(X_{{\underline{i}}}))\mid
i\in F\\}\text{ for all }\emptyset\neq F\subseteq[d].$

######  Remark 3.1.2.

We stipulate that the sets ${\mathcal{L}}_{F}$ are non-empty for convenience.
More specifically, the sets ${\mathcal{L}}_{F}$ are designed to generate
certain “relative CNP-relations”, and so ${\mathcal{L}}_{F}=\\{0\\}$ plays the
same role as ${\mathcal{L}}_{F}=\emptyset$. In the former case the relative
CNP-relations are satisfied trivially, and in the latter case they are
satisfied vacuously. Functionally there is no difference, and it is more
convenient to take ${\mathcal{L}}_{F}=\\{0\\}$ as in this case
${\mathcal{L}}_{F}$ is an ideal.

We write ${\mathcal{L}}\subseteq{\mathcal{L}}^{\prime}$ for $2^{d}$-tuples
${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ if and only if
${\mathcal{L}}_{F}\subseteq{\mathcal{L}}_{F}^{\prime}$ for all
$F\subseteq[d]$. This defines a partial order on the set of $2^{d}$-tuples of
$X$. We say that ${\mathcal{L}}={\mathcal{L}}^{\prime}$ if and only if
${\mathcal{L}}\subseteq{\mathcal{L}}^{\prime}$ and
${\mathcal{L}}^{\prime}\subseteq{\mathcal{L}}$.

Let $(\pi,t)$ be a Nica-covariant representation of $X$. The crucial property
of a relative $2^{d}$-tuple ${\mathcal{L}}$ is that

$\pi(a)q_{F}=\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{\underline{n}}(\phi_{\underline{n}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}\in\mathrm{C}^{*}(\pi,t)\text{
for all }a\in{\mathcal{L}}_{F},F\subseteq[d],$

using Proposition 2.5.10. This allows us to extend the ideas of [16] in a
natural way.

######  Definition 3.1.3.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. We define the _ideal of
the ${\mathcal{L}}$-relative CNP-relations with respect to $(\pi,t)$_ to be

${\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}:=\sum\\{{\mathfrak{J}}_{{\mathcal{L}},F}^{(\pi,t)}\mid
F\subseteq[d]\\}\subseteq\mathrm{C}^{*}(\pi,t),\;\text{where}\;\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}:=\left\langle\pi({\mathcal{L}}_{F})q_{F}\right\rangle\text{
for all }F\subseteq[d].$

We say that _${\mathcal{L}}$ induces
${\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}$_.

Being an algebraic sum of ideals in $\mathrm{C}^{*}(\pi,t)$, the space
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$ is itself an ideal in
$\mathrm{C}^{*}(\pi,t)$. When $(\pi,t)$ admits a gauge action, the ideal
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$ is gauge-invariant. By setting
${\mathcal{L}}={\mathcal{I}}$, we recover
$\mathfrak{J}_{\mathcal{I}}^{(\pi,t)}$ as defined in [16]. In many cases, we
may assume that ${\mathcal{L}}$ is a family of ideals without loss of
generality.

######  Proposition 3.1.4.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. Then
$\left\langle{\mathcal{L}}\right\rangle:=\\{\left\langle{\mathcal{L}}_{F}\right\rangle\\}_{F\subseteq[d]}$
is a relative $2^{d}$-tuple of $X$ such that
${\mathcal{L}}\subseteq\left\langle{\mathcal{L}}\right\rangle$ and
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}=\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle,F}^{(\pi,t)}$
for all $F\subseteq[d]$. In particular, we have that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}=\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle}^{(\pi,t)}$.

######  Proof..

It is immediate that
${\mathcal{L}}\subseteq\left\langle{\mathcal{L}}\right\rangle$, and that
$\left\langle{\mathcal{L}}\right\rangle$ is a relative $2^{d}$-tuple of $X$.
It remains to see that
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}=\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle,F}^{(\pi,t)}$
for all $F\subseteq[d]$, as the final claim follows as an immediate
consequence. To this end, fix $F\subseteq[d]$. Since
${\mathcal{L}}_{F}\subseteq\left\langle{\mathcal{L}}_{F}\right\rangle$, we
have that
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}\subseteq\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle,F}^{(\pi,t)}$.
For the reverse inclusion, for $b,c\in A$ and $a\in{\mathcal{L}}_{F}$, we have
that

$\pi(bac)q_{F}=\pi(b)\pi(a)\pi(c)q_{F}=\pi(b)(\pi(a)q_{F})\pi(c)\in\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)},$

using Proposition 2.5.9 in the second equality. It follows that
$\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle,F}^{(\pi,t)}\subseteq\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$,
as required. ∎

Let ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$ be relative $2^{d}$-tuples of
$X$. It is straightforward to see that their sum
${\mathcal{L}}_{1}+{\mathcal{L}}_{2}$, defined by
$({\mathcal{L}}_{1}+{\mathcal{L}}_{2})_{F}:={\mathcal{L}}_{1,F}+{\mathcal{L}}_{2,F}$
for all $F\subseteq[d]$, is also a relative $2^{d}$-tuple. When
${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$ consist of ideals, summing
respects the induced ideals.

######  Proposition 3.1.5.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$ be relative
$2^{d}$-tuples of $X$ that consist of ideals and let $(\pi,t)$ be a Nica-
covariant representation of $X$. Then

${\mathfrak{J}}_{{\mathcal{L}}_{1}+{\mathcal{L}}_{2},F}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{L}}_{1},F}^{(\pi,t)}+{\mathfrak{J}}_{{\mathcal{L}}_{2},F}^{(\pi,t)}\text{
for all }F\subseteq[d],$

and thus
${\mathfrak{J}}_{{\mathcal{L}}_{1}+{\mathcal{L}}_{2}}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{L}}_{1}}^{(\pi,t)}+{\mathfrak{J}}_{{\mathcal{L}}_{2}}^{(\pi,t)}$.

######  Proof..

This is immediate since $0\in{\mathcal{L}}_{1,F},{\mathcal{L}}_{2,F}$ for all
$F\subseteq[d]$. ∎

A first approach towards the parametrisation of the gauge-invariant ideals of
${\mathcal{N}}{\mathcal{T}}_{X}$ would be to establish a correspondence
between the relative $2^{d}$-tuples of $X$ and the gauge-invariant ideals of
${\mathcal{N}}{\mathcal{T}}_{X}$ that they induce. However, this is
insufficient as different relative $2^{d}$-tuples may induce the same gauge-
invariant ideal of ${\mathcal{N}}{\mathcal{T}}_{X}$. We provide an example to
this effect.

######  Remark 3.1.6.

Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ that consists of ideals
and $(\pi,t)$ be a Nica-covariant representation. Then the relative
$2^{d}$-tuple ${\mathcal{L}}^{\prime}$ defined by

${\mathcal{L}}^{\prime}_{F}:=\begin{cases}{\mathcal{L}}_{\emptyset}&\text{if
}F=\emptyset,\\\
{\mathcal{L}}_{F}+{\mathcal{L}}_{\emptyset}\cap(\bigcap\\{\phi_{{\underline{i}}}^{-1}({\mathcal{K}}(X_{{\underline{i}}}))\mid
i\in F\\})&\text{if }\emptyset\neq F\subseteq[d],\end{cases}$

satisfies
${\mathfrak{J}}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}$.
Indeed, on one hand we have that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}\subseteq\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}$
since ${\mathcal{L}}\subseteq{\mathcal{L}}^{\prime}$. On the other hand, fix
$\emptyset\neq F\subseteq[d]$ and $a\in{\mathcal{L}}_{\emptyset}$ satisfying
$\phi_{{\underline{i}}}(a)\in{\mathcal{K}}(X_{{\underline{i}}})$ for all $i\in
F$. We claim that

$\psi_{{\underline{n}}}(\phi_{{\underline{n}}}(a))\in\left\langle\pi({\mathcal{L}}_{\emptyset})\right\rangle={\mathfrak{J}}_{{\mathcal{L}},\emptyset}^{(\pi,t)}\text{
for all }{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}.$

We will prove this by induction on $|{\underline{n}}|$. When
$|{\underline{n}}|=1$, we must have that ${\underline{n}}={\underline{i}}$ for
some $i\in F$. In turn, for an approximate unit
$(k_{{\underline{i}},\lambda})_{\lambda\in\Lambda}\subseteq{\mathcal{K}}(X_{{\underline{i}}})$,
we have that

$\psi_{\underline{i}}(\phi_{\underline{i}}(a))=\left\|\cdot\right\|\text{-}\lim_{\lambda}\psi_{\underline{i}}(\phi_{\underline{i}}(a)k_{{\underline{i}},\lambda})=\left\|\cdot\right\|\text{-}\lim_{\lambda}\pi(a)\psi_{\underline{i}}(k_{{\underline{i}},\lambda})\in\left\langle\pi({\mathcal{L}}_{\emptyset})\right\rangle,$

as required. Now suppose that the claim holds for all ${\underline{n}}$ such
that ${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}$ and
$|{\underline{n}}|=N$ for some $1\leq N<|F|$. Fix ${\underline{m}}$ such that
${\underline{0}}\neq{\underline{m}}\leq{\underline{1}}_{F}$ and
$|{\underline{m}}|=N+1$. We may express ${\underline{m}}$ in the form
${\underline{m}}={\underline{n}}+{\underline{i}}$ for some
${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}$ and $i\in F$ such
that ${\underline{n}}\perp{\underline{i}}$ and $|{\underline{n}}|=N$. We have
that

$\displaystyle\psi_{\underline{m}}(\phi_{\underline{m}}(a))$
$\displaystyle=\left\|\cdot\right\|\text{-}\lim_{\lambda}\psi_{\underline{m}}(\iota_{\underline{n}}^{\underline{m}}(\phi_{\underline{n}}(a))\iota_{\underline{i}}^{\underline{m}}(k_{{\underline{i}},\lambda}))=\left\|\cdot\right\|\text{-}\lim_{\lambda}\psi_{\underline{n}}(\phi_{\underline{n}}(a))\psi_{\underline{i}}(k_{{\underline{i}},\lambda})\in\left\langle\pi({\mathcal{L}}_{\emptyset})\right\rangle,$

by the inductive hypothesis, Proposition 2.5.7, and Nica-covariance (noting
that
${\underline{n}}\vee{\underline{i}}={\underline{n}}+{\underline{i}}={\underline{m}}$
since ${\underline{n}}\perp{\underline{i}}$). This finishes the proof of the
claim. Therefore, we have that

$\pi(a)q_{F}=\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{{\underline{n}}}(\phi_{{\underline{n}}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}\in\left\langle\pi({\mathcal{L}}_{\emptyset})\right\rangle\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}.$

Hence

$\pi\left({\mathcal{L}}_{\emptyset}\cap(\bigcap\\{\phi_{{\underline{i}}}^{-1}({\mathcal{K}}(X_{{\underline{i}}}))\mid
i\in F\\})\right)q_{F}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)},$

from which it follows that
$\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$,
as required.

However, notice that ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ may differ
in general. For example, assume that the left actions of the fibres of $X$ are
by compact operators. Then any $2^{d}$-tuple of $X$ is automatically relative.
Let ${\mathcal{L}}_{\emptyset}$ be a non-zero ideal and set
${\mathcal{L}}_{F}=\\{0\\}$ for all $\emptyset\neq F\subseteq[d]$. Then
${\mathcal{L}}\neq{\mathcal{L}}^{\prime}$, as claimed.

To remedy this issue, we will instead look for the largest relative
$2^{d}$-tuple inducing a fixed gauge-invariant ideal of
${\mathcal{N}}{\mathcal{T}}_{X}$.

######  Definition 3.1.7.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{M}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. We say that
${\mathcal{M}}$ is a _maximal $2^{d}$-tuple (of $X$) with respect to
$(\pi,t)$_ if whenever ${\mathcal{L}}$ is a relative $2^{d}$-tuple of $X$ such
that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}=\mathfrak{J}_{\mathcal{M}}^{(\pi,t)}$
and ${\mathcal{M}}\subseteq{\mathcal{L}}$, we have that
${\mathcal{L}}={\mathcal{M}}$. When we replace $(\pi,t)$ by
$(\overline{\pi}_{X},\overline{t}_{X})$, we will refer to ${\mathcal{M}}$
simply as a _maximal $2^{d}$-tuple (of $X$)_.

The following proposition establishes existence and uniqueness of maximal
$2^{d}$-tuples.

######  Proposition 3.1.8.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. Then there exists a
unique relative $2^{d}$-tuple ${\mathcal{M}}$ of $X$ such that:

  1. (i)

${\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{M}}}^{(\pi,t)}$,
and

  2. (ii)

${\mathcal{L}}^{\prime}\subseteq{\mathcal{M}}$ for every relative
$2^{d}$-tuple ${\mathcal{L}}^{\prime}$ of $X$ satisfying
${\mathfrak{J}}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}$.

In particular, ${\mathcal{M}}$ is a maximal $2^{d}$-tuple of $X$ with respect
to $(\pi,t)$ which consists of ideals.

######  Proof..

For each $F\subseteq[d]$, define

${\mathcal{M}}_{F}:=\bigcup\\{{\mathcal{L}}_{F}^{\prime}\mid{\mathcal{L}}^{\prime}\;\text{is
a relative $2^{d}$-tuple of $X$ such
that}\;\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}=\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}\\}.$

The union is well-defined, since it includes ${\mathcal{L}}_{F}$ by
assumption, and the index takes values in the set $\mathcal{P}(A)^{2^{d}}$.
Each ${\mathcal{M}}_{F}$ is a non-empty subset of $A$ by construction, so
${\mathcal{M}}$ is a $2^{d}$-tuple of $X$. Fix $\emptyset\neq F\subseteq[d]$
and take $a\in{\mathcal{M}}_{F}$. Then $a\in{\mathcal{L}}_{F}^{\prime}$ for
some relative $2^{d}$-tuple ${\mathcal{L}}^{\prime}$, and hence
$\phi_{\underline{i}}(a)\in{\mathcal{K}}(X_{\underline{i}})$ for all $i\in F$.
Therefore, setting ${\mathcal{M}}:=\\{{\mathcal{M}}_{F}\\}_{F\subseteq[d]}$,
we have that ${\mathcal{M}}$ is a relative $2^{d}$-tuple of $X$.

Because ${\mathcal{L}}\subseteq{\mathcal{M}}$, we have that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}\subseteq\mathfrak{J}_{\mathcal{M}}^{(\pi,t)}$
trivially. For the other inclusion, fix $F\subseteq[d]$ and
$a\in{\mathcal{M}}_{F}$. It suffices to show that
$\pi(a)q_{F}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$. To this end, we have
that $a\in{\mathcal{L}}_{F}^{\prime}$ for some relative $2^{d}$-tuple
${\mathcal{L}}^{\prime}$ with the property that
$\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}=\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$.
By definition, we have that
$\pi(a)q_{F}\in\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}=\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$,
as required.

By construction, we have that ${\mathcal{L}}^{\prime}\subseteq{\mathcal{M}}$
for every relative $2^{d}$-tuple ${\mathcal{L}}^{\prime}$ satisfying
$\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}=\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$.
It is immediate that ${\mathcal{M}}$ is unique and maximal with respect to
$(\pi,t)$, being the largest relative $2^{d}$-tuple inducing
${\mathfrak{J}}_{\mathcal{L}}^{(\pi,t)}$.

Finally, by Proposition 3.1.4 we have that
$\left\langle{\mathcal{M}}\right\rangle:=\\{\left\langle{\mathcal{M}}_{F}\right\rangle\\}_{F\subseteq[d]}$
is a relative $2^{d}$-tuple such that
${\mathcal{M}}\subseteq\left\langle{\mathcal{M}}\right\rangle$ and
$\mathfrak{J}_{\mathcal{M}}^{(\pi,t)}=\mathfrak{J}_{\left\langle{\mathcal{M}}\right\rangle}^{(\pi,t)}$.
Applying maximality of ${\mathcal{M}}$, we have that
${\mathcal{M}}=\left\langle{\mathcal{M}}\right\rangle$. This shows that
${\mathcal{M}}$ consists of ideals, finishing the proof. ∎

######  Remark 3.1.9.

The motivating example of a maximal relative $2^{d}$-tuple is the family
${\mathcal{I}}$. To see this, first note that ${\mathcal{I}}$ is a relative
$2^{d}$-tuple by definition. Let ${\mathcal{M}}$ be the maximal $2^{d}$-tuple
inducing
${\mathfrak{J}}_{\mathcal{I}}^{(\overline{\pi}_{X},\overline{t}_{X})}$, so
that ${\mathcal{I}}\subseteq{\mathcal{M}}$. If $a\in{\mathcal{M}}_{F}$ for
$F\subseteq[d]$, then

$\overline{\pi}_{X}(a)\overline{q}_{X,F}\in{\mathfrak{J}}_{\mathcal{M}}^{(\overline{\pi}_{X},\overline{t}_{X})}={\mathfrak{J}}_{\mathcal{I}}^{(\overline{\pi}_{X},\overline{t}_{X})},$

and thus we have that $a\in{\mathcal{I}}_{F}$ by [16, Proposition 3.4].
Therefore ${\mathcal{M}}\subseteq{\mathcal{I}}$, showing that
${\mathcal{M}}={\mathcal{I}}$.

We wish to ascertain the conditions under which a relative $2^{d}$-tuple is
promoted to a maximal one. Using ${\mathcal{I}}$ as a prototype, we abstract
two of its properties.

######  Definition 3.1.10.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a $2^{d}$-tuple of $X$.

  1. (i)

We say that ${\mathcal{L}}$ is $X$-_invariant_ if $\left[\left\langle
X_{\underline{n}},{\mathcal{L}}_{F}X_{\underline{n}}\right\rangle\right]\subseteq{\mathcal{L}}_{F}$,
for all ${\underline{n}}\perp F$, $F\subseteq[d]$.

  2. (ii)

We say that ${\mathcal{L}}$ is _partially ordered_ if
${\mathcal{L}}_{F_{1}}\subseteq{\mathcal{L}}_{F_{2}}$ whenever $F_{1}\subseteq
F_{2}\subseteq[d]$.

When the underlying product system $X$ is clear from the context, we will
abbreviate “$X$-invariant” as simply “invariant”. Notice that when we take
$F=\emptyset$, condition (i) implies that ${\mathcal{L}}_{\emptyset}$ is
positively invariant for $X$ (provided that ${\mathcal{L}}_{\emptyset}$ is an
ideal). If ${\mathcal{L}}_{F}$ is an ideal, then we may drop the closed linear
span in condition (i). If ${\mathcal{L}}$ is a partially ordered relative
$2^{d}$-tuple, then

${\mathcal{L}}_{F}\subseteq{\mathcal{L}}_{[d]}\subseteq\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in[d]\\}\text{ for all }F\subseteq[d].$

In particular, we have that $\pi(a)q_{F}\in\mathrm{C}^{*}(\pi,t)$ for all
$a\in{\mathcal{L}}_{D}$ and $D,F\subseteq[d]$, where $(\pi,t)$ is a Nica-
covariant representation.

The $2^{d}$-tuple ${\mathcal{I}}$ is invariant, partially ordered and consists
of ideals. Since ${\mathcal{I}}$ is maximal by Remark 3.1.9, one may be
tempted to assert that invariance and partial ordering suffice to capture
maximality. However, this is not true in general, as the following
counterexample shows.

######  Example 3.1.11.

Let $B$ be a non-zero C*-algebra and let $A=B\oplus\mathbb{C}$ be its
unitisation. Consider the semigroup action
$\alpha\colon\mathbb{Z}_{+}^{2}\to\text{End}(A)$ such that

$\alpha_{(1,0)}=\text{id}_{A}\quad\text{and}\quad\alpha_{(0,1)}(b,\lambda)=(0,\lambda)\text{
for all }b\in B,\lambda\in\mathbb{C}.$

Applying the construction of Subsection 5.3 to the C*-dynamical system
$(A,\alpha,\mathbb{Z}_{+}^{2})$, we obtain a strong compactly aligned product
system $X_{\alpha}$ over $\mathbb{Z}_{+}^{2}$ with coefficients in $A$. In
particular, the left action of every fibre is by compacts, and so any
$2^{2}$-tuple of $X_{\alpha}$ is automatically a relative $2^{2}$-tuple.

Next we define the relative $2^{2}$-tuples ${\mathcal{L}}$ and
${\mathcal{L}}^{\prime}$ of $X_{\alpha}$ by

${{\mathcal{L}}_{\\{2\\}}=\\{0\\}}$${{\mathcal{L}}_{\\{1,2\\}}=B\oplus\\{0\\}}$${{\mathcal{L}}_{\emptyset}=\\{0\\}}$${{\mathcal{L}}_{\\{1\\}}=\\{0\\}}$

and
${{\mathcal{L}}_{\\{2\\}}^{\prime}=\\{0\\}}$${{\mathcal{L}}_{\\{1,2\\}}^{\prime}=B\oplus\\{0\\}}$${{\mathcal{L}}_{\emptyset}^{\prime}=\\{0\\}}$${{\mathcal{L}}_{\\{1\\}}^{\prime}=B\oplus\\{0\\}.}$

Notice that ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ are invariant,
partially ordered, consist of ideals and satisfy
${\mathcal{L}}\subsetneq{\mathcal{L}}^{\prime}$. Let $(\pi,t)$ be a Nica-
covariant representation. Since
${\mathcal{L}}\subseteq{\mathcal{L}}^{\prime}$, it is clear that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}\subseteq\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}$.
It is immediate that
$\pi({\mathcal{L}}_{F}^{\prime})q_{F}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$
for all $F\in\\{\emptyset,\\{2\\},\\{1,2\\}\\}$. Fix
$(b,0)\in{\mathcal{L}}_{\\{1\\}}^{\prime}$, and observe that

$\displaystyle\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}\ni\pi(b,0)q_{\\{1,2\\}}$
$\displaystyle=\pi(b,0)-\psi_{(1,0)}(\phi_{(1,0)}(b,0))-\psi_{(0,1)}(\phi_{(0,1)}(b,0))+\psi_{(1,1)}(\phi_{(1,1)}(b,0))$
$\displaystyle=\pi(b,0)-\psi_{(1,0)}(\phi_{(1,0)}(b,0))=\pi(b,0)q_{\\{1\\}},$

using the fact that $\phi_{(0,1)}(b,0)=\phi_{(1,1)}(b,0)=0$ by definition.
Therefore we conclude that
$\pi({\mathcal{L}}_{\\{1\\}}^{\prime})q_{\\{1\\}}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$,
and thus
$\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\pi,t)}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$.
Hence ${\mathcal{L}}$ is not maximal with respect to $(\pi,t)$.

To better understand maximality, we need to explore $2^{d}$-tuples induced by
Nica-covariant representations, demonstrating the conditions of Definition
3.1.10 in action.

######  Definition 3.1.12.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $(\pi,t)$ be a Nica-covariant representation of $X$. We
define ${\mathcal{L}}^{(\pi,t)}$ to be the $2^{d}$-tuple of $X$ given by

${\mathcal{L}}_{\emptyset}^{(\pi,t)}:=\ker\pi\quad\text{and}\quad{\mathcal{L}}_{F}^{(\pi,t)}:=\pi^{-1}(B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)})\text{
for all }\emptyset\neq F\subseteq[d].$

######  Proposition 3.1.13.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $(\pi,t)$ be a Nica-covariant representation of $X$.
Then ${\mathcal{L}}^{(\pi,t)}$ is invariant, partially ordered and consists of
ideals.

If $(\pi,t)$ is in addition injective, then
${\mathcal{L}}^{(\pi,t)}\subseteq{\mathcal{I}}$, and thus
${\mathcal{L}}^{(\pi,t)}$ is relative.

######  Proof..

It is clear that ${\mathcal{L}}^{(\pi,t)}$ consists of ideals by definition.
For invariance, fix $F\subseteq[d],a\in{\mathcal{L}}_{F}^{(\pi,t)}$, and
${\underline{m}}\perp F$. Due to Nica-covariance, we have that

$\pi(\left\langle
X_{{\underline{m}}},aX_{{\underline{m}}}\right\rangle)=t_{{\underline{m}}}(X_{{\underline{m}}})^{*}\pi(a)t_{{\underline{m}}}(X_{{\underline{m}}})\subseteq
t_{{\underline{m}}}(X_{{\underline{m}}})^{*}B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)}t_{{\underline{m}}}(X_{{\underline{m}}})\subseteq
B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)}.$

Hence ${\mathcal{L}}^{(\pi,t)}$ is invariant, as required. To see that
${\mathcal{L}}^{(\pi,t)}$ is partially ordered, first note that the inclusion
${\mathcal{L}}_{\emptyset}^{(\pi,t)}\subseteq{\mathcal{L}}_{F}^{(\pi,t)}$ is
immediate for all $F\subseteq[d]$. Likewise, whenever $\emptyset\neq
F\subseteq D\subseteq[d]$, we have that
$B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)}\subseteq
B_{({\underline{0}},{\underline{1}}_{D}]}^{(\pi,t)}$ and therefore
${\mathcal{L}}_{F}^{(\pi,t)}\subseteq{\mathcal{L}}_{D}^{(\pi,t)}$, as
required. The final claim follows by [16, Proposition 3.4] (see Proposition
2.5.11). ∎

In order to make further progress with maximality, we use relative
$2^{d}$-tuples to define relative Cuntz-Nica-Pimsner algebras.

######  Definition 3.1.14.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. We say that $(\pi,t)$ is
an _${\mathcal{L}}$ -relative CNP-representation (of $X$)_ if it satisfies

$\pi({\mathcal{L}}_{F})q_{F}=\\{0\\}\text{ for all }F\subseteq[d].$

The universal C*-algebra with respect to the ${\mathcal{L}}$-relative CNP-
representations of $X$ is denoted by
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$, and we refer to it as the
_${\mathcal{L}}$ -relative Cuntz-Nica-Pimsner algebra (of $X$)_.

We have that $(\pi,t)$ is an ${\mathcal{L}}$-relative CNP-representation if
and only if $\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}=\\{0\\}$; equivalently if
$\pi\times t$ factors through ${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$.
The following proposition is the analogue of Proposition 2.3.3 for relative
Cuntz-Nica-Pimsner algebras, and shows that
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$ exists and admits a universal
property.

######  Proposition 3.1.15.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$.
Then there exists a C*-algebra ${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$
and an ${\mathcal{L}}$-relative CNP-representation
$(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ of $X$ on
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$ such that:

  1. (i)

${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)=\mathrm{C}^{*}(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$;

  2. (ii)

if $(\pi,t)$ is an ${\mathcal{L}}$-relative CNP-representation of $X$, then
there exists a (unique) canonical $*$-epimorphism
$\Phi\colon{\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\to\mathrm{C}^{*}(\pi,t)$.

The pair
$({\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X),(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}}))$
is unique up to canonical $*$-isomorphism.

######  Proof..

This follows immediately by considering the quotient of
${\mathcal{N}}{\mathcal{T}}_{X}$ by
$\mathfrak{J}_{{\mathcal{L}}}^{(\overline{\pi}_{X},\overline{t}_{X})}$. ∎

We will refer to the representation
$(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ as the _universal
${\mathcal{L}}$-relative CNP-representation (of $X$)_. Since
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$ is an equivariant quotient of
${\mathcal{N}}{\mathcal{T}}_{X}$, the representation
$(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ admits a gauge action. Notice
that $\\{\\{0\\}\\}_{F\subseteq[d]}$ and ${\mathcal{I}}$ both constitute
relative $2^{d}$-tuples of $X$ and in particular

${\mathcal{N}}{\mathcal{O}}(\\{\\{0\\}\\}_{F\subseteq[d]},X)={\mathcal{N}}{\mathcal{T}}_{X}\quad\text{and}\quad{\mathcal{N}}{\mathcal{O}}({\mathcal{I}},X)={\mathcal{N}}{\mathcal{O}}_{X}.$

Likewise, when $X$ is a C*-correspondence we recover the relative Cuntz-
Pimsner algebras. Note that
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)={\mathcal{N}}{\mathcal{O}}(\left\langle{\mathcal{L}}\right\rangle,X)$
since the covariance is described C*-algebraically.

If $(\pi,t)$ is an injective Nica-covariant representation, then
${\mathcal{L}}^{(\pi,t)}$ is a relative $2^{d}$-tuple by Proposition 3.1.13,
and thus we can make sense of
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$.
The next proposition shows that
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$
arises as a kernel when $(\pi,t)$ admits a gauge action.

######  Proposition 3.1.16.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $(\pi,t)$ be an injective Nica-covariant representation
of $X$. Then $(\pi,t)$ is an ${\mathcal{L}}^{(\pi,t)}$-relative CNP-
representation of $X$, and consequently there exists a (unique) canonical
$*$-epimorphism

$\Phi\colon{\mathcal{N}}{\mathcal{O}}({\mathcal{L}}^{(\pi,t)},X)\to\mathrm{C}^{*}(\pi,t).$

Moreover, $\Phi$ is injective if and only if $(\pi,t)$ admits a gauge action,
in which case

$\ker\pi\times
t=\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}.$

######  Proof..

Fix $F\subseteq[d]$ and
$a\in{\mathcal{L}}_{F}^{(\pi,t)}=\pi^{-1}(B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)})$.
Then $\pi(a)q_{F}=0$ by (2.12). This shows that $(\pi,t)$ is an
${\mathcal{L}}^{(\pi,t)}$-relative CNP-representation. Thus universality of
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}}^{(\pi,t)},X)$ guarantees the
existence of $\Phi$.

For the second claim, we write $(\tilde{\pi},\tilde{t})$ for
$(\pi_{X}^{{\mathcal{L}}^{(\pi,t)}},t_{X}^{{\mathcal{L}}^{(\pi,t)}})$, for
notational convenience. If $\Phi$ is injective, then $(\pi,t)$ inherits the
gauge action $\beta$ of $(\tilde{\pi},\tilde{t})$. Conversely, suppose that
$(\pi,t)$ admits a gauge action $\gamma$, and note that $\Phi$ intertwines the
gauge actions (i.e., $\Phi$ is equivariant). Thus it suffices to show that
$\Phi$ is injective on the fixed point algebra, and by Proposition 2.5.13 this
amounts to showing that $\Phi$ is injective on the
$[{\underline{0}},{\underline{1}}_{[d]}]$-core. To reach contradiction,
suppose there exists $0\neq f\in\ker\Phi\cap
B_{[{\underline{0}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})}$, so
that we may write

$f=\tilde{\pi}(a)+\sum\\{\tilde{\psi}_{{\underline{n}}}(k_{{\underline{n}}})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}\\},$

for some $a\in A$ and $k_{\underline{n}}\in{\mathcal{K}}(X_{\underline{n}})$
for all ${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}$. Recall
that we write $\tilde{\pi}(a)=\tilde{\psi}_{\underline{0}}(k_{\underline{0}})$
for $k_{\underline{0}}:=\phi_{\underline{0}}(a)$. Without loss of generality,
we may assume that $f$ is written irreducibly, so that we may choose
${\underline{0}}\leq{\underline{r}}\leq{\underline{1}}_{[d]}$ minimal such
that $k_{\underline{r}}\neq 0$, and
$\tilde{\psi}_{{\underline{r}}}(k_{{\underline{r}}})\notin
B_{({\underline{r}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})}$. If
${\underline{r}}={\underline{1}}_{[d]}$, then
$f=\tilde{\psi}_{{\underline{1}}_{[d]}}(k_{{\underline{1}}_{[d]}})$ and
$\Phi(f)=\psi_{{\underline{1}}_{[d]}}(k_{{\underline{1}}_{[d]}})=0$.
Injectivity of $(\pi,t)$ then implies that $k_{{\underline{1}}_{[d]}}=0$ and
hence $f=0$, a contradiction. So without loss of generality assume that
${\underline{r}}<{\underline{1}}_{[d]}$. Then for every
$\xi_{{\underline{r}}},\eta_{{\underline{r}}}\in X_{{\underline{r}}}$, we have
that

$0=t_{{\underline{r}}}(\xi_{{\underline{r}}})^{*}\Phi(f)t_{{\underline{r}}}(\eta_{{\underline{r}}})=\pi(\left\langle\xi_{{\underline{r}}},k_{{\underline{r}}}\eta_{{\underline{r}}}\right\rangle)+\sum\\{\psi_{{\underline{n}}}(k_{{\underline{n}}}^{\prime})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}-{\underline{r}}\\},$

where each $k_{\underline{n}}^{\prime}$ is a suitably defined element of
${\mathcal{K}}(X_{\underline{n}})$ for all
${\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{[d]}-{\underline{r}}$.
Hence we have that $\left\langle
X_{{\underline{r}}},k_{{\underline{r}}}X_{{\underline{r}}}\right\rangle\subseteq{\mathcal{L}}_{F}^{(\pi,t)}$
for $F:=\operatorname{supp}({\underline{1}}_{[d]}-{\underline{r}})$ (which is
non-empty since ${\underline{1}}_{[d]}\neq{\underline{r}}$), and thus we
obtain that $\tilde{\pi}(\left\langle
X_{{\underline{r}}},k_{{\underline{r}}}X_{{\underline{r}}}\right\rangle)\subseteq
B_{({\underline{0}},{\underline{1}}_{[d]}-{\underline{r}}]}^{(\tilde{\pi},\tilde{t})}$
since $(\tilde{\pi},\tilde{t})$ is an ${\mathcal{L}}^{(\pi,t)}$-relative CNP-
representation. In particular, we have that

$\displaystyle\tilde{\psi}_{{\underline{r}}}({\mathcal{K}}(X_{{\underline{r}}}))\tilde{\psi}_{{\underline{r}}}(k_{{\underline{r}}})\tilde{\psi}_{{\underline{r}}}({\mathcal{K}}(X_{{\underline{r}}}))$
$\displaystyle\subseteq[\tilde{t}_{{\underline{r}}}(X_{{\underline{r}}})\tilde{\pi}(\left\langle
X_{{\underline{r}}},k_{{\underline{r}}}X_{{\underline{r}}}\right\rangle)\tilde{t}_{{\underline{r}}}(X_{{\underline{r}}})^{*}]$
$\displaystyle\subseteq[\tilde{t}_{{\underline{r}}}(X_{{\underline{r}}})B_{({\underline{0}},{\underline{1}}_{[d]}-{\underline{r}}]}^{(\tilde{\pi},\tilde{t})}\tilde{t}_{{\underline{r}}}(X_{{\underline{r}}})^{*}]\subseteq
B_{({\underline{r}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})},$

and by taking an approximate unit of
$\tilde{\psi}_{{\underline{r}}}({\mathcal{K}}(X_{{\underline{r}}}))$ we derive
that $\tilde{\psi}_{{\underline{r}}}(k_{{\underline{r}}})\in
B_{({\underline{r}},{\underline{1}}_{[d]}]}^{(\tilde{\pi},\tilde{t})}$, which
is a contradiction. Thus $\Phi$ is injective on the
$[{\underline{0}},{\underline{1}}_{[d]}]$-core, as required.

For the last assertion, we have that $\pi\times t=\Phi\circ Q$ and thus in
particular

$\ker\pi\times t=\ker\Phi\circ Q=\ker
Q=\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$

for the quotient map
$Q\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{O}}({\mathcal{L}}^{(\pi,t)},X)$,
and the proof is complete. ∎

Injective Nica-covariant representations admitting a gauge action provide an
ample supply of maximal relative $2^{d}$-tuples.

######  Proposition 3.1.17.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let $(\pi,t)$ be an injective Nica-covariant representation of
$X$ that admits a gauge action. Then ${\mathcal{L}}^{(\pi,t)}$ is a maximal
$2^{d}$-tuple of $X$ that is contained in ${\mathcal{I}}$.

######  Proof..

By Proposition 3.1.13, we have that
${\mathcal{L}}^{(\pi,t)}\subseteq{\mathcal{I}}$. For maximality, let
${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ such that
$\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$
and ${\mathcal{L}}^{(\pi,t)}\subseteq{\mathcal{L}}$. It suffices to show that
${\mathcal{L}}\subseteq{\mathcal{L}}^{(\pi,t)}$. Fix $F\subseteq[d]$ and
$a\in{\mathcal{L}}_{F}$. Then

$\overline{\pi}_{X}(a)\overline{q}_{X,F}\in\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$

by definition. Since
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\ker\pi\times
t$ by Proposition 3.1.16, we obtain that

$\displaystyle\pi(a)q_{F}$
$\displaystyle=\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{{\underline{n}}}(\phi_{{\underline{n}}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\}$
$\displaystyle=(\pi\times
t)(\overline{\pi}_{X}(a)+\sum\\{(-1)^{|{\underline{n}}|}\overline{\psi}_{X,{\underline{n}}}(\phi_{{\underline{n}}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{F}\\})$
$\displaystyle=(\pi\times t)(\overline{\pi}_{X}(a)\overline{q}_{X,F})=0,$

using Proposition 2.5.10 in the first and third equalities. Thus
$a\in\pi^{-1}(B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)})={\mathcal{L}}_{F}^{(\pi,t)}$
and hence ${\mathcal{L}}\subseteq{\mathcal{L}}^{(\pi,t)}$, as required. ∎

###  3.2. (E)-$2^{d}$-tuples

We are interested in a special class of $2^{d}$-tuples that describes the
relative Cuntz-Nica-Pimsner algebras
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$ in-between
${\mathcal{N}}{\mathcal{T}}_{X}$ and ${\mathcal{N}}{\mathcal{O}}_{X}$.

######  Proposition 3.2.1.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be an ${\mathcal{L}}$-relative CNP-representation of $X$. If
$(\pi,t)$ is injective, then ${\mathcal{L}}\subseteq{\mathcal{I}}$.
Conversely, if ${\mathcal{L}}\subseteq{\mathcal{I}}$ then the universal
${\mathcal{L}}$-relative CNP-representation
$(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ is injective.

######  Proof..

First assume that $(\pi,t)$ is injective. Fix $F\subseteq[d]$ and
$a\in{\mathcal{L}}_{F}$. Since $(\pi,t)$ is an ${\mathcal{L}}$-relative CNP-
representation, we have that $\pi(a)q_{F}=0$ and hence $\pi(a)\in
B_{({\underline{0}},{\underline{1}}_{F}]}^{(\pi,t)}$. Since $(\pi,t)$ is
assumed to be injective, an application of [16, Proposition 3.4] gives that
$a\in{\mathcal{I}}_{F}$. Hence ${\mathcal{L}}\subseteq{\mathcal{I}}$.

Conversely, assume that ${\mathcal{L}}\subseteq{\mathcal{I}}$. Then the
canonical quotient map
${\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{O}}_{X}$ (which is
injective on $X$) factors through the canonical $*$-epimorphism
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\to{\mathcal{N}}{\mathcal{O}}_{X}$.
Hence $(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ is injective, finishing
the proof. ∎

We see that the $2^{d}$-tuples ${\mathcal{L}}\subseteq{\mathcal{I}}$ are
special, and we give them their own name to reflect this.

######  Definition 3.2.2.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. We say that a $2^{d}$-tuple ${\mathcal{L}}$ of $X$ is an _(E)-
$2^{d}$-tuple (of $X$)_ if ${\mathcal{L}}\subseteq{\mathcal{I}}$.

Every (E)-$2^{d}$-tuple is automatically a relative $2^{d}$-tuple with
${\mathcal{L}}_{\emptyset}=\\{0\\}={\mathcal{I}}_{\emptyset}$. The “E” of
“(E)-$2^{d}$-tuple” stands for “Embedding”, since
$X\hookrightarrow{\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$ isometrically
for a relative $2^{d}$-tuple ${\mathcal{L}}$ if and only if
${\mathcal{L}}\subseteq{\mathcal{I}}$, by Proposition 3.2.1. By Proposition
3.1.13, injective Nica-covariant representations provide the quintessential
supply of (E)-$2^{d}$-tuples. It is straightforward to check that the sum of
two (E)-$2^{d}$-tuples is an (E)-$2^{d}$-tuple. We also make the following
observation.

######  Proposition 3.2.3.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a $2^{d}$-tuple of $X$. If
${\mathcal{L}}$ is invariant and satisfies
${\mathcal{L}}\subseteq{\mathcal{J}}$, then ${\mathcal{L}}$ is an
(E)-$2^{d}$-tuple of $X$.

######  Proof..

Fix $F\subseteq[d]$ and $a\in{\mathcal{L}}_{F}$. By assumption we have that

$\left\langle
X_{\underline{n}},aX_{\underline{n}}\right\rangle\subseteq\left[\left\langle
X_{\underline{n}},{\mathcal{L}}_{F}X_{\underline{n}}\right\rangle\right]\subseteq{\mathcal{L}}_{F}\subseteq{\mathcal{J}}_{F}\text{
for all }{\underline{n}}\perp F.$

By definition this means that $a\in{\mathcal{I}}_{F}$ and hence
${\mathcal{L}}\subseteq{\mathcal{I}}$, as required. ∎

Let $(\pi,t)$ be a Nica-covariant representation and
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$ be the ideal induced by an
(E)-$2^{d}$-tuple ${\mathcal{L}}$. Our goal in the next propositions is to
show that we can choose ${\mathcal{L}}$ to be in addition invariant and
partially ordered, and to consist of ideals.

######  Proposition 3.2.4.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be an (E)-$2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. Then the $2^{d}$-tuple
${\operatorname{Inv}}({\mathcal{L}})$ of $X$ defined by

${\operatorname{Inv}}({\mathcal{L}})_{F}:=\overline{\operatorname{span}}\\{X_{\underline{n}}({\mathcal{L}}_{F})\mid{\underline{n}}\perp
F\\}\text{ for all }F\subseteq[d]$

is an invariant (E)-$2^{d}$-tuple consisting of ideals, and is such that

${\mathcal{L}}\subseteq{\operatorname{Inv}}({\mathcal{L}})\quad\text{and}\quad\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}=\mathfrak{J}_{{\operatorname{Inv}}({\mathcal{L}}),F}^{(\pi,t)}\text{
for all }F\subseteq[d].$

In particular, we have that
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}=\mathfrak{J}_{{\operatorname{Inv}}({\mathcal{L}})}^{(\pi,t)}$.

######  Proof..

Recall that $X_{\underline{n}}({\mathcal{L}}_{F})\equiv\left[\left\langle
X_{\underline{n}},{\mathcal{L}}_{F}X_{\underline{n}}\right\rangle\right]$ for
all ${\underline{n}}\in\mathbb{Z}_{+}^{d}$ and $F\subseteq[d]$. By considering
${\underline{n}}={\underline{0}}$, we see that
${\mathcal{L}}\subseteq{\operatorname{Inv}}({\mathcal{L}})$.

Fix $F\subseteq[d]$. Then $X_{\underline{n}}({\mathcal{L}}_{F})$ is an ideal
of $A$ for all ${\underline{n}}\perp F$, and hence
${\operatorname{Inv}}({\mathcal{L}})_{F}$ is itself an ideal of $A$. Since
${\mathcal{L}}$ is an (E)-$2^{d}$-tuple, we have that
${\mathcal{L}}_{F}\subseteq{\mathcal{I}}_{F}$ and thus
$X_{\underline{n}}({\mathcal{L}}_{F})\subseteq{\mathcal{I}}_{F}$ for all
${\underline{n}}\perp F$ because ${\mathcal{I}}$ is invariant [16, Proposition
2.7]. Thus
${\operatorname{Inv}}({\mathcal{L}})_{F}\subseteq{\mathcal{I}}_{F}$, and so
${\operatorname{Inv}}({\mathcal{L}})$ is an (E)-$2^{d}$-tuple that consists of
ideals.

To see that ${\operatorname{Inv}}({\mathcal{L}})$ is invariant, fix
$F\subseteq[d],{\underline{m}}\perp F$ and
$a\in{\operatorname{Inv}}({\mathcal{L}})_{F}$. Since
${\operatorname{Inv}}({\mathcal{L}})_{F}$ is an ideal, it suffices to show
that $\left\langle
X_{\underline{m}},aX_{\underline{m}}\right\rangle\subseteq{\operatorname{Inv}}({\mathcal{L}})_{F}$.
Assume that
$a=\left\langle\xi_{\underline{n}},b\eta_{\underline{n}}\right\rangle$ for
some ${\underline{n}}\perp F,\xi_{\underline{n}},\eta_{\underline{n}}\in
X_{\underline{n}}$ and $b\in{\mathcal{L}}_{F}$. Then we have that

$\displaystyle\left\langle X_{\underline{m}},aX_{\underline{m}}\right\rangle$
$\displaystyle=\left\langle
X_{\underline{m}},\left\langle\xi_{\underline{n}},b\eta_{\underline{n}}\right\rangle
X_{\underline{m}}\right\rangle\subseteq\left\langle
X_{{\underline{n}}+{\underline{m}}},bX_{{\underline{n}}+{\underline{m}}}\right\rangle\subseteq
X_{{\underline{n}}+{\underline{m}}}({\mathcal{L}}_{F})\subseteq{\operatorname{Inv}}({\mathcal{L}})_{F},$

using that ${\underline{n}}+{\underline{m}}\perp F$ in the final inclusion. It
follows that $\left\langle
X_{\underline{m}},X_{\underline{n}}({\mathcal{L}}_{F})X_{\underline{m}}\right\rangle\subseteq{\operatorname{Inv}}({\mathcal{L}})_{F}$
for all ${\underline{n}}\perp F$. We obtain that $\left\langle
X_{\underline{m}},{\operatorname{Inv}}({\mathcal{L}})_{F}X_{\underline{m}}\right\rangle\subseteq{\operatorname{Inv}}({\mathcal{L}})_{F}$
by linearity and continuity of the inner product, as required.

Since ${\mathcal{L}}\subseteq{\operatorname{Inv}}({\mathcal{L}})$, we have
that
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}\subseteq\mathfrak{J}_{{\operatorname{Inv}}({\mathcal{L}}),F}^{(\pi,t)}$
for all $F\subseteq[d]$. For the reverse inclusion, fix $F\subseteq[d]$ and
$a\in{\operatorname{Inv}}({\mathcal{L}})_{F}$. Assume that
$a=\left\langle\xi_{\underline{n}},b\eta_{\underline{n}}\right\rangle$ for
some ${\underline{n}}\perp F,\xi_{\underline{n}},\eta_{\underline{n}}\in
X_{\underline{n}}$ and $b\in{\mathcal{L}}_{F}$. We have that

$\pi(a)q_{F}=t_{\underline{n}}(\xi_{\underline{n}})^{*}\pi(b)t_{\underline{n}}(\eta_{\underline{n}})q_{F}=t_{\underline{n}}(\xi_{\underline{n}})^{*}(\pi(b)q_{F})t_{\underline{n}}(\eta_{\underline{n}})\in\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)},$

where we have used Proposition 2.5.9 in the second equality. Therefore
$\pi(X_{\underline{n}}({\mathcal{L}}_{F}))q_{F}\subseteq\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$
for all ${\underline{n}}\perp F$, from which it follows that
$\pi({\operatorname{Inv}}({\mathcal{L}})_{F})q_{F}\subseteq\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$.
In total we have that
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}=\mathfrak{J}_{{\operatorname{Inv}}({\mathcal{L}}),F}^{(\pi,t)}$
for all $F\subseteq[d]$, and thus in particular
$\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}=\mathfrak{J}_{{\operatorname{Inv}}({\mathcal{L}})}^{(\pi,t)}$,
finishing the proof. ∎

In order to choose ${\mathcal{L}}$ to be partially ordered, we need the
following auxiliary proposition.

######  Proposition 3.2.5.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. Fix $D\subseteq
F\subseteq[d]$. If
$a\in\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in F\\}$ and $\pi(a)q_{D}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$, then
$\pi(a)q_{F}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$.

######  Proof..

Without loss of generality, we may assume that $D=[k]$ (with the convention
that if $k=0$ then $D=\emptyset$) and $F=[\ell]$ for some $0\leq k\leq\ell\leq
d$. Note that we have that

$\pi(a)q_{D}=\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{\underline{n}}(\phi_{\underline{n}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{D}\\}$

because
$a\in\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in F\\}$, and hence in particular
$a\in\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in D\\}$ since $D\subseteq F$. If $k=\ell$, then there is nothing to show.
If $k<\ell$, then we have that

$\displaystyle\pi(a)q_{D}p_{\underline{k+1}}$
$\displaystyle=\pi(a)p_{\underline{k+1}}+\sum\\{(-1)^{|{\underline{n}}|}\psi_{\underline{n}}(\phi_{\underline{n}}(a))p_{{\underline{k+1}}}\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{D}\\}$
$\displaystyle=\left\|\cdot\right\|\text{-}\lim_{\lambda}\pi(a)p_{{\underline{k+1}},\lambda}+\sum\\{(-1)^{|{\underline{n}}|}(\left\|\cdot\right\|\text{-}\lim_{\lambda}\psi_{\underline{n}}(\phi_{\underline{n}}(a))p_{{\underline{k+1}},\lambda})\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{D}\\}$
$\displaystyle=\left\|\cdot\right\|\text{-}\lim_{\lambda}(\pi(a)+\sum\\{(-1)^{|{\underline{n}}|}\psi_{\underline{n}}(\phi_{\underline{n}}(a))\mid{\underline{0}}\neq{\underline{n}}\leq{\underline{1}}_{D}\\})p_{{\underline{k+1}},\lambda}$
$\displaystyle=\left\|\cdot\right\|\text{-}\lim_{\lambda}\pi(a)q_{D}p_{{\underline{k+1}},\lambda}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)},$

where in the second line we have used Proposition 2.5.10 and in the last line
we have used that $\pi(a)q_{D}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$ by
assumption. To express $\pi(a)p_{{\underline{k+1}}}$ as a norm-limit, we use
that $a\in\phi_{\underline{k+1}}^{-1}({\mathcal{K}}(X_{{\underline{k+1}}}))$,
and so

$\pi(a)p_{{\underline{k+1}}}=\psi_{{\underline{k+1}}}(\phi_{{\underline{k+1}}}(a))=\left\|\cdot\right\|\text{-}\lim_{\lambda}\pi(a)p_{{\underline{k+1}},\lambda},$

by the comments succeeding Proposition 2.5.10. Hence we have that

$\pi(a)q_{D\cup\\{k+1\\}}=\pi(a)q_{D}-\pi(a)q_{D}p_{{\underline{k+1}}}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}.$

Since
$a\in\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in D\cup\\{k+1\\}\\}$, we can express $\pi(a)q_{D\cup\\{k+1\\}}$ as an
alternating sum by Proposition 2.5.10. Consequently, we may apply the
preceding argument for $k+2$. The assumption that
$a\in\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in F\\}$ ensures that we may argue in this way until $\\{k+1,\dots,\ell\\}$
has been exhausted, and we deduce that
$\pi(a)q_{F}\in\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}$, as required. ∎

######  Proposition 3.2.6.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be an $(E)$-$2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. Then the $2^{d}$-tuple
$\operatorname{PO}({\mathcal{L}})$ of $X$ defined by

$\operatorname{PO}({\mathcal{L}})_{F}:=\sum\\{\left\langle{\mathcal{L}}_{D}\right\rangle\mid
D\subseteq F\\}\text{ for all }F\subseteq[d]$

is a partially ordered (E)-$2^{d}$-tuple consisting of ideals, and is such
that

${\mathcal{L}}\subseteq\operatorname{PO}({\mathcal{L}})\quad\text{and}\quad{\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}={\mathfrak{J}}_{\operatorname{PO}({\mathcal{L}})}^{(\pi,t)}.$

If ${\mathcal{L}}$ is in addition invariant, then so is
$\operatorname{PO}({\mathcal{L}})$.

######  Proof..

Every $\operatorname{PO}({\mathcal{L}})_{F}$ is an ideal, being a finite sum
of ideals in a C*-algebra. Fix $F\subseteq[d]$. For each $D\subseteq F$, we
have that
$\left\langle{\mathcal{L}}_{D}\right\rangle\subseteq{\mathcal{I}}_{D}\subseteq{\mathcal{I}}_{F}$,
using that ${\mathcal{L}}$ is an (E)-$2^{d}$-tuple (and that
${\mathcal{I}}_{D}$ is an ideal) in the first inclusion, and the fact that
${\mathcal{I}}$ is partially ordered in the second. Hence
$\operatorname{PO}({\mathcal{L}})_{F}\subseteq{\mathcal{I}}_{F}$ for all
$F\subseteq[d]$, and thus $\operatorname{PO}({\mathcal{L}})$ is an
(E)-$2^{d}$-tuple. It is clear that $\operatorname{PO}({\mathcal{L}})$ is
partially ordered, and
${\mathcal{L}}\subseteq\operatorname{PO}({\mathcal{L}})$ by construction.

Next we check that
$\mathfrak{J}_{{\mathcal{L}}}^{(\pi,t)}=\mathfrak{J}_{\operatorname{PO}({\mathcal{L}})}^{(\pi,t)}$.
Since ${\mathcal{L}}\subseteq\operatorname{PO}({\mathcal{L}})$, we have that
$\mathfrak{J}_{{\mathcal{L}}}^{(\pi,t)}\subseteq\mathfrak{J}_{\operatorname{PO}({\mathcal{L}})}^{(\pi,t)}$.
For the reverse inclusion, fix $F\subseteq[d]$ and
$a\in\operatorname{PO}({\mathcal{L}})_{F}$. It suffices to show that
$\pi(a)q_{F}\in\mathfrak{J}_{{\mathcal{L}}}^{(\pi,t)}$. Let
$a=\sum\\{a_{D}\mid D\subseteq F\\}$, where each
$a_{D}\in\left\langle{\mathcal{L}}_{D}\right\rangle$, so that
$\pi(a_{D})q_{D}\in\mathfrak{J}_{\left\langle{\mathcal{L}}\right\rangle}^{(\pi,t)}={\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}$
for all $D\subseteq F$ by Proposition 3.1.4. For each $D\subseteq F$, an
application of Proposition 3.2.5 gives that
$\pi(a_{D})q_{F}\in\mathfrak{J}_{{\mathcal{L}}}^{(\pi,t)}$ and hence
$\pi(a)q_{F}\in{\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}$, as required.

Now suppose that ${\mathcal{L}}$ is in addition invariant. To establish
invariance of the family $\operatorname{PO}({\mathcal{L}})$, fix
$F\subseteq[d],{\underline{n}}\perp F$ and
$a\in\operatorname{PO}({\mathcal{L}})_{F}$. By definition, we can write
$a=\sum\\{a_{D}\mid D\subseteq F\\}$ for some
$a_{D}\in\left\langle{\mathcal{L}}_{D}\right\rangle$, for each $D\subseteq F$.
Since ${\mathcal{L}}$ is invariant, it follows that $\left\langle
X_{\underline{n}},\left\langle{\mathcal{L}}_{D}\right\rangle
X_{\underline{n}}\right\rangle\subseteq\left\langle{\mathcal{L}}_{D}\right\rangle$
and hence in particular $\left\langle
X_{\underline{n}},a_{D}X_{\underline{n}}\right\rangle\subseteq\left\langle{\mathcal{L}}_{D}\right\rangle$
for each $D\subseteq F$. Consequently, we have that

$\left\langle
X_{\underline{n}},aX_{\underline{n}}\right\rangle\subseteq\sum\\{\left\langle
X_{\underline{n}},a_{D}X_{\underline{n}}\right\rangle\mid D\subseteq
F\\}\subseteq\sum\\{\left\langle{\mathcal{L}}_{D}\right\rangle\mid D\subseteq
F\\}=\operatorname{PO}({\mathcal{L}})_{F},$

and therefore $\operatorname{PO}({\mathcal{L}})$ is invariant, finishing the
proof. ∎

Letting ${\mathcal{L}}$ be an (E)-$2^{d}$-tuple and $(\pi,t)$ be a Nica-
covariant representation, Propositions 3.2.4 and 3.2.6 combine to give that
$\operatorname{PO}({\operatorname{Inv}}({\mathcal{L}}))$ is an
(E)-$2^{d}$-tuple that is invariant, partially ordered, consists of ideals and
satisfies
${\mathcal{L}}\subseteq\operatorname{PO}({\operatorname{Inv}}({\mathcal{L}}))$
and
${\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}={\mathfrak{J}}_{\operatorname{PO}({\operatorname{Inv}}({\mathcal{L}}))}^{(\pi,t)}$.

Next we focus on the maximal (E)-$2^{d}$-tuples.

######  Definition 3.2.7.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $(\pi,t)$ be a Nica-covariant representation of $X$. An
_(M)- $2^{d}$-tuple ${\mathcal{M}}$ (of $X$) with respect to $(\pi,t)$_ is a
maximal $2^{d}$-tuple of $X$ with respect to $(\pi,t)$ that is also an
(E)-$2^{d}$-tuple of $X$. When we replace $(\pi,t)$ by
$(\overline{\pi}_{X},\overline{t}_{X})$, we will refer to ${\mathcal{M}}$
simply as an _(M)- $2^{d}$-tuple (of $X$)_.

The “M” of “(M)-$2^{d}$-tuple” stands for “Maximal”. Note that
(M)-$2^{d}$-tuples (with respect to $(\pi,t)$) maximalise collections of
(E)-$2^{d}$-tuples, instead of just relative $2^{d}$-tuples. Indeed, if
${\mathcal{M}}$ is an (M)-$2^{d}$-tuple with respect to $(\pi,t)$, then it
contains every relative $2^{d}$-tuple ${\mathcal{L}}$ of $X$ satisfying
${\mathfrak{J}}_{\mathcal{L}}^{(\pi,t)}={\mathfrak{J}}_{\mathcal{M}}^{(\pi,t)}$
by Proposition 3.1.8. Each such ${\mathcal{L}}$ satisfies
${\mathcal{L}}\subseteq{\mathcal{M}}\subseteq{\mathcal{I}}$ and is therefore
an (E)-$2^{d}$-tuple, as claimed.

######  Remark 3.2.8.

Both ${\mathcal{L}}:=\\{\\{0\\}\\}_{F\subseteq[d]}$ and ${\mathcal{I}}$ are
(M)-$2^{d}$-tuples. For ${\mathcal{I}}$ this is shown in Remark 3.1.9. On the
other hand, ${\mathcal{L}}$ is clearly an (E)-$2^{d}$-tuple. To see that
${\mathcal{L}}$ is maximal, let ${\mathcal{L}}^{\prime}$ be a relative
$2^{d}$-tuple such that
$\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\\{0\\}$
and ${\mathcal{L}}\subseteq{\mathcal{L}}^{\prime}$. We may replace
$(\overline{\pi}_{X},\overline{t}_{X})$ by the Fock representation
$(\overline{\pi},\overline{t})$. For the projection
$P_{{\underline{0}}}\colon{\mathcal{F}}X\to X_{{\underline{0}}}$, we have that

$\phi_{{\underline{0}}}({\mathcal{L}}^{\prime}_{F})=P_{{\underline{0}}}\big{(}\overline{\pi}({\mathcal{L}}^{\prime}_{F})\overline{q}_{F}\big{)}P_{{\underline{0}}}\subseteq
P_{{\underline{0}}}\big{(}\mathfrak{J}_{{\mathcal{L}}^{\prime}}^{(\overline{\pi},\overline{t})}\big{)}P_{{\underline{0}}}=\\{0\\},$

for all $F\subseteq[d]$, and thus ${\mathcal{L}}={\mathcal{L}}^{\prime}$ as
required.

Moreover, in Proposition 3.1.17 we have shown that ${\mathcal{L}}^{(\pi,t)}$
is an (M)-$2^{d}$-tuple whenever $(\pi,t)$ is injective and admits a gauge
action.

The (M)-$2^{d}$-tuples of $X$ parametrise the gauge-invariant ideals
$\mathfrak{J}$ of ${\mathcal{N}}{\mathcal{T}}_{X}$ for which
${\mathcal{N}}{\mathcal{T}}_{X}/\mathfrak{J}$ lies between
${\mathcal{N}}{\mathcal{T}}_{X}$ and ${\mathcal{N}}{\mathcal{O}}_{X}$. To
prove this, we will require the following proposition.

######  Proposition 3.2.9.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let $\mathfrak{J}\subseteq{\mathcal{N}}{\mathcal{T}}_{X}$
be a gauge-invariant ideal. Then the following are equivalent:

  1. (i)

$\overline{\pi}_{X}^{-1}(\mathfrak{J})=\\{0\\}$;

  2. (ii)

there exists a unique (M)-$2^{d}$-tuple of $X$ inducing $\mathfrak{J}$;

  3. (iii)

there exists an (E)-$2^{d}$-tuple of $X$ inducing $\mathfrak{J}$.

######  Proof..

First assume that $\overline{\pi}_{X}^{-1}(\mathfrak{J})=\\{0\\}$. Let
$Q_{\mathfrak{J}}\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{T}}_{X}/\mathfrak{J}$
denote the quotient map. Then the Nica-covariant representation
$(Q_{\mathfrak{J}}\circ\overline{\pi}_{X},Q_{\mathfrak{J}}\circ\overline{t}_{X})$
is injective and admits a gauge action. By Proposition 3.1.16, we have that

${\mathfrak{J}}=\ker
Q_{{\mathfrak{J}}}=\ker(Q_{\mathfrak{J}}\circ\overline{\pi}_{X})\times(Q_{\mathfrak{J}}\circ\overline{t}_{X})=\mathfrak{J}_{{\mathcal{L}}^{(Q_{\mathfrak{J}}\circ\overline{\pi}_{X},Q_{\mathfrak{J}}\circ\overline{t}_{X})}}^{(\overline{\pi}_{X},\overline{t}_{X})},$

since
$(Q_{\mathfrak{J}}\circ\overline{\pi}_{X})\times(Q_{\mathfrak{J}}\circ\overline{t}_{X})=Q_{\mathfrak{J}}$.
Thus
${\mathcal{L}}^{(Q_{\mathfrak{J}}\circ\overline{\pi}_{X},Q_{\mathfrak{J}}\circ\overline{t}_{X})}$
induces $\mathfrak{J}$, and by Proposition 3.1.17 it is an (M)-$2^{d}$-tuple
of $X$. By Proposition 3.1.8 it is also unique, proving that (i) implies (ii).

Item (ii) implies item (iii) trivially. So assume that there exists an
(E)-$2^{d}$-tuple ${\mathcal{L}}$ such that
$\mathfrak{J}_{{\mathcal{L}}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}$.
In particular $(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ is injective by
Proposition 3.2.1. Hence, letting

$Q_{\mathfrak{J}}\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)={\mathcal{N}}{\mathcal{T}}_{X}/{\mathfrak{J}}$

denote the quotient map, we have that

$\displaystyle\\{0\\}=\ker\pi_{X}^{\mathcal{L}}$ $\displaystyle=\ker
Q_{\mathfrak{J}}\circ\overline{\pi}_{X}=\\{a\in
A\mid\overline{\pi}_{X}(a)\in\mathfrak{J}\\}=\overline{\pi}_{X}^{-1}(\mathfrak{J}),$

showing that item (iii) implies item (i), finishing the proof. ∎

######  Remark 3.2.10.

Proposition 3.2.9 implies that the mapping

$\displaystyle\\{{\mathcal{M}}\mid{\mathcal{M}}\;\text{is an (M)-$2^{d}$-tuple
of}\;X\\}$
$\displaystyle\to\\{\mathfrak{J}\subseteq{\mathcal{N}}{\mathcal{T}}_{X}\mid\mathfrak{J}\;\text{is
a gauge-invariant ideal,}\;\overline{\pi}_{X}^{-1}(\mathfrak{J})=\\{0\\}\\}$
$\displaystyle{\mathcal{M}}$
$\displaystyle\mapsto\mathfrak{J}_{\mathcal{M}}^{(\overline{\pi}_{X},\overline{t}_{X})}$

is a bijection. More specifically, the mappings

$\displaystyle{\mathcal{M}}\mapsto\mathfrak{J}_{\mathcal{M}}^{(\overline{\pi}_{X},\overline{t}_{X})}$
$\displaystyle\;\text{for all (M)-$2^{d}$-tuples ${\mathcal{M}}$ of $X$},$
$\displaystyle\mathfrak{J}\mapsto{\mathcal{L}}^{(Q_{\mathfrak{J}}\circ\overline{\pi}_{X},Q_{\mathfrak{J}}\circ\overline{t}_{X})}$
$\displaystyle\;\text{for all gauge-invariant ideals
$\mathfrak{J}\subseteq{\mathcal{N}}{\mathcal{T}}_{X}$ satisfying
$\overline{\pi}_{X}^{-1}(\mathfrak{J})=\\{0\\}$},$

are mutually inverse, where
$Q_{\mathfrak{J}}\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{T}}_{X}/\mathfrak{J}$
is the quotient map.

The following Gauge-Invariant Uniqueness Theorem recovers [34, Corollary 11.8]
when $d=1$, and [16, Theorem 4.2] when ${\mathcal{L}}={\mathcal{I}}$.

######  Theorem 3.2.11 ($\mathbb{Z}_{+}^{d}$-GIUT for (M)-$2^{d}$-tuples).

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be an (M)-$2^{d}$-tuple of $X$ and
$(\pi,t)$ be a Nica-covariant representation of $X$. Then
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\cong\mathrm{C}^{*}(\pi,t)$ via a
(unique) canonical $*$-isomorphism if and only if $(\pi,t)$ admits a gauge
action and ${\mathcal{L}}^{(\pi,t)}={\mathcal{L}}$.

######  Proof..

We start with the following remark. If
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\cong\mathrm{C}^{*}(\pi,t)$
canonically, then $(\pi,t)$ is injective as $X$ embeds isometrically in
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$. On the other hand, if
${\mathcal{L}}^{(\pi,t)}={\mathcal{L}}$ then
$\ker\pi\equiv{\mathcal{L}}_{\emptyset}^{(\pi,t)}={\mathcal{L}}_{\emptyset}=\\{0\\}$,
as ${\mathcal{L}}$ is in particular an (E)-$2^{d}$-tuple. Hence in both
directions we may use that $(\pi,t)$ is injective.

Assume that
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\cong\mathrm{C}^{*}(\pi,t)$ via a
canonical $*$-isomorphism
$\Phi\colon{\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)\to\mathrm{C}^{*}(\pi,t)$.
For notational convenience, we will abbreviate
$(\pi_{X}^{\mathcal{L}},t_{X}^{\mathcal{L}})$ as $(\tilde{\pi},\tilde{t})$.
Let $\beta$ denote the gauge action of $(\tilde{\pi},\tilde{t})$. By defining
$\gamma_{{\underline{z}}}:=\Phi\circ\beta_{{\underline{z}}}\circ\Phi^{-1}$ for
each ${\underline{z}}\in\mathbb{T}^{d}$, we obtain a gauge action $\gamma$ of
$(\pi,t)$.

Next we show that ${\mathcal{L}}^{(\pi,t)}={\mathcal{L}}$. To this end, it
suffices to show that
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}$.
Then we can apply maximality of ${\mathcal{L}}^{(\pi,t)}$ (which holds by
Proposition 3.1.17) and of ${\mathcal{L}}$ (which holds by assumption),
together with uniqueness from item (ii) of Proposition 3.2.9, to obtain the
result. To this end, fix $F\subseteq[d]$ and $a\in{\mathcal{L}}_{F}$. Then
$\tilde{\pi}(a)\tilde{q}_{F}=0$ since $(\tilde{\pi},\tilde{t})$ is an
${\mathcal{L}}$-relative CNP-representation. Applying $\Phi$, we obtain that
$\pi(a)q_{F}=0$, and therefore $a\in{\mathcal{L}}_{F}^{(\pi,t)}$ by
definition. Hence
$\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}\subseteq\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}$.
Next, fix $F\subseteq[d]$ and $a\in{\mathcal{L}}_{F}^{(\pi,t)}$. An
application of (2.12) yields that $\pi(a)q_{F}=0$. Recall that
$\tilde{\pi}=Q\circ\overline{\pi}_{X}$ and
$\tilde{t}_{\underline{n}}=Q\circ\overline{t}_{X,{\underline{n}}}$ for all
${\underline{n}}\in\mathbb{Z}_{+}^{d}\setminus\\{{\underline{0}}\\}$, where
$Q\colon{\mathcal{N}}{\mathcal{T}}_{X}\to{\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$
is the quotient map. Hence, applying $\Phi^{-1}$ to $\pi(a)q_{F}=0$, we obtain
that $Q(\overline{\pi}_{X}(a)\overline{q}_{X,F})=0$, and therefore

$\overline{\pi}_{X}(a)\overline{q}_{X,F}\in\ker
Q=\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}.$

Hence we have that
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}\subseteq\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}$,
and thus
$\mathfrak{J}_{{\mathcal{L}}^{(\pi,t)}}^{(\overline{\pi}_{X},\overline{t}_{X})}=\mathfrak{J}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}$,
as required.

The converse is an immediate application of Proposition 3.1.16. Indeed, since
$(\pi,t)$ is injective and admits a gauge action, we obtain that
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}}^{(\pi,t)},X)\cong\mathrm{C}^{*}(\pi,t)$
canonically, and the fact that ${\mathcal{L}}^{(\pi,t)}={\mathcal{L}}$ then
gives that
${\mathcal{N}}{\mathcal{O}}({\mathcal{L}}^{(\pi,t)},X)={\mathcal{N}}{\mathcal{O}}({\mathcal{L}},X)$,
finishing the proof. ∎

###  3.3. Properties of ideals induced by relative $2^{d}$-tuples

The definition of (M)-$2^{d}$-tuples itself depends extensively on the
associated gauge-invariant ideals of ${\mathcal{N}}{\mathcal{T}}_{X}$. Hence,
to obtain a genuinely useful parametrisation, we must characterise
(M)-$2^{d}$-tuples in a way that does not (explicitly) depend on any gauge-
invariant ideal of ${\mathcal{N}}{\mathcal{T}}_{X}$. To this end, we need to
study the properties of the gauge-invariant ideals arising from relative
$2^{d}$-tuples.

When a relative $2^{d}$-tuple ${\mathcal{L}}$ consists of ideals and is
invariant, the ideals $\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$ take on a
particularly simple form akin to the one obtained in [16, Proposition 4.6].

######  Proposition 3.3.1.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ and let
$(\pi,t)$ be a Nica-covariant representation of $X$. If ${\mathcal{L}}$ is
invariant and consists of ideals, then

(3.1)
${\mathfrak{J}}_{{\mathcal{L}},F}^{(\pi,t)}=\overline{\operatorname{span}}\\{t_{{\underline{n}}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{{\underline{m}}}(X_{\underline{m}})^{*}\mid{\underline{n}},{\underline{m}}\in\mathbb{Z}_{+}^{d}\\}\text{
for all }F\subseteq[d].$

######  Proof..

The proof follows the arguments of [16, Proposition 4.6]. In short, fix
$F\subseteq[d]$ and denote the right hand side of (3.1) by $\mathfrak{J}_{F}$.
The fact that
$\mathfrak{J}_{F}\subseteq\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$ is
immediate by definition. To prove that
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}\subseteq\mathfrak{J}_{F}$, it
suffices to show that $\mathfrak{J}_{F}$ is an ideal of
$\mathrm{C}^{*}(\pi,t)$, as ${\mathfrak{J}}_{F}$ contains the generators of
$\mathfrak{J}_{{\mathcal{L}},F}^{(\pi,t)}$. Since $\mathfrak{J}_{F}$ is
selfadjoint, it suffices to show that
$\mathfrak{J}_{F}\mathrm{C}^{*}(\pi,t)\subseteq\mathfrak{J}_{F}$.

It is clear that
$\mathfrak{J}_{F}t_{\underline{r}}(X_{\underline{r}})^{*}\subseteq\mathfrak{J}_{F}$
for all ${\underline{r}}\in\mathbb{Z}_{+}^{d}$. Hence it remains to show that
$\mathfrak{J}_{F}t_{\underline{r}}(X_{\underline{r}})\subseteq\mathfrak{J}_{F}$
for all ${\underline{r}}\in\mathbb{Z}_{+}^{d}$. For
${\underline{n}},{\underline{m}},{\underline{r}}\in\mathbb{Z}_{+}^{d}$, we
have that

$t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{\underline{m}}(X_{\underline{m}})^{*}t_{\underline{r}}(X_{\underline{r}})\subseteq\left[t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})t_{{\underline{r}}^{\prime}}(X_{{\underline{r}}^{\prime}})^{*}\right],$

where
${\underline{m}}^{\prime}=-{\underline{m}}+{\underline{m}}\vee{\underline{r}}$
and
${\underline{r}}^{\prime}=-{\underline{r}}+{\underline{m}}\vee{\underline{r}}$,
due to Nica-covariance. If ${\underline{m}}^{\prime}\not\perp F$, then

$t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})t_{{\underline{r}}^{\prime}}(X_{{\underline{r}}^{\prime}})^{*}=\\{0\\}\subseteq\mathfrak{J}_{F}$

by Proposition 2.5.9. If ${\underline{m}}^{\prime}\perp F$, then positive
invariance of ${\mathcal{L}}_{F}$ for $X_{{\underline{m}}^{\prime}}$ and Lemma
2.2.6 together yield that

$\pi({\mathcal{L}}_{F})t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})=t_{{\underline{m}}^{\prime}}({\mathcal{L}}_{F}X_{{\underline{m}}^{\prime}})\subseteq
t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}}{\mathcal{L}}_{F})=t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})\pi({\mathcal{L}}_{F}).$

Then we have that

$\displaystyle\left[t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})t_{{\underline{r}}^{\prime}}(X_{{\underline{r}}^{\prime}})^{*}\right]$
$\displaystyle\subseteq\left[t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})t_{{\underline{m}}^{\prime}}(X_{{\underline{m}}^{\prime}})q_{F}t_{{\underline{r}}^{\prime}}(X_{{\underline{r}}^{\prime}})^{*}\right]$
$\displaystyle\subseteq\left[t_{{\underline{n}}+{\underline{m}}^{\prime}}(X_{{\underline{n}}+{\underline{m}}^{\prime}})\pi({\mathcal{L}}_{F})q_{F}t_{{\underline{r}}^{\prime}}(X_{{\underline{r}}^{\prime}})^{*}\right]\subseteq\mathfrak{J}_{F},$

where we have used Proposition 2.5.9 in the first inclusion. Hence in all
cases we have that

$t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F})q_{F}t_{\underline{m}}(X_{\underline{m}})^{*}t_{\underline{r}}(X_{\underline{r}})\subseteq\mathfrak{J}_{F},$

and thus
$\mathfrak{J}_{F}t_{\underline{r}}(X_{\underline{r}})\subseteq\mathfrak{J}_{F}$
for all ${\underline{r}}\in\mathbb{Z}_{+}^{d}$, as required. ∎

By requiring that ${\mathcal{L}}$ is partially ordered, we can say more.

######  Proposition 3.3.2.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ that is
invariant, partially ordered, and consists of ideals. Let $(\pi,t)$ be a Nica-
covariant representation of $X$. Then

$\displaystyle
q_{F}\mathfrak{J}_{{\mathcal{L}},D}^{(\pi,t)}q_{F}\subseteq\overline{\operatorname{span}}\\{t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F\cup
D})q_{F\cup
D}t_{\underline{m}}(X_{\underline{m}})^{*}\mid{\underline{n}},{\underline{m}}\perp
F\\}\subseteq\mathfrak{J}_{{\mathcal{L}},F\cup D}^{(\pi,t)}\text{ for all
}F,D\subseteq[d].$

Moreover, we have that

$q_{F}{\mathfrak{J}}_{{\mathcal{L}}}^{(\pi,t)}q_{F}\subseteq\sum\\{{\mathfrak{J}}_{{\mathcal{L}},D}^{(\pi,t)}\mid
F\subseteq D\subseteq[d]\\}\text{ for all }F\subseteq[d].$

######  Proof..

For the first part, fix $F,D\subseteq[d]$. Then we have that

$\displaystyle q_{F}\mathfrak{J}_{{\mathcal{L}},D}^{(\pi,t)}q_{F}$
$\displaystyle\subseteq\overline{\operatorname{span}}\\{q_{F}t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{D})q_{D}t_{\underline{m}}(X_{\underline{m}})^{*}q_{F}\mid{\underline{n}},{\underline{m}}\in\mathbb{Z}_{+}^{d}\\}$
$\displaystyle=\overline{\operatorname{span}}\\{t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{D})q_{F\cup
D}t_{\underline{m}}(X_{\underline{m}})^{*}\mid{\underline{n}},{\underline{m}}\perp
F\\}$
$\displaystyle\subseteq\overline{\operatorname{span}}\\{t_{\underline{n}}(X_{\underline{n}})\pi({\mathcal{L}}_{F\cup
D})q_{F\cup
D}t_{\underline{m}}(X_{\underline{m}})^{*}\mid{\underline{n}},{\underline{m}}\perp
F\\}\subseteq\mathfrak{J}_{{\mathcal{L}},F\cup D}^{(\pi,t)},$

where we have used Propositions 2.5.9 and 3.3.1, that $q_{F}$ and $q_{D}$ are
commuting projections, and that ${\mathcal{L}}$ is partially ordered (so
${\mathcal{L}}_{D}\subseteq{\mathcal{L}}_{F\cup D}$).

For the second part, fix $F\subseteq[d]$. Then applying the first part for all
$D\subseteq[d]$ yields that

$\displaystyle q_{F}\mathfrak{J}_{\mathcal{L}}^{(\pi,t)}q_{F}$
$\displaystyle=\sum\\{q_{F}\mathfrak{J}_{{\mathcal{L}},D}^{(\pi,t)}q_{F}\mid
D\subseteq[d]\\}\subseteq\sum\\{\mathfrak{J}_{{\mathcal{L}},F\cup
D}^{(\pi,t)}\mid
D\subseteq[d]\\}=\sum\\{\mathfrak{J}_{{\mathcal{L}},D}^{(\pi,t)}\mid
F\subseteq D\subseteq[d]\\},$

and the proof is complete. ∎

Let $I$ be an ideal of $A$ and fix $F\subseteq[d]$. For $a\in A$, we write

$\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}I)\|=0$

if for any $\varepsilon>0$ there exists ${\underline{m}}\perp F$ such that

$\|\phi_{\underline{n}}(a)+{\mathcal{K}}(X_{\underline{n}}I)\|<\varepsilon\text{
for all
}{\underline{n}}\geq{\underline{m}}\;\text{satisfying}\;{\underline{n}}\perp
F.$

When working with a $2^{d}$-tuple ${\mathcal{L}}$ satisfying certain
compatibility relations, the limit condition can be recast in a simpler form.

######  Proposition 3.3.3.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be an invariant $2^{d}$-tuple of $X$ which
consists of ideals and satisfies

${\mathcal{L}}_{F}\subseteq\bigcap\\{\phi_{\underline{i}}^{-1}({\mathcal{K}}(X_{\underline{i}}))\mid
i\in[d]\\}\text{ for all }F\subseteq[d].$

Then, for each $F\subseteq[d]$ and $a\in A$, we have that
$\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\|=0$
if and only if for any $\varepsilon>0$, there exists ${\underline{m}}\perp F$
and $k_{\underline{m}}\in{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})$
such that $\|\phi_{\underline{m}}(a)+k_{\underline{m}}\|<\varepsilon$.

######  Proof..

The forward implication is immediate by definition. So assume that for any
$\varepsilon>0$ there exists ${\underline{m}}\perp F$ and
$k_{\underline{m}}\in{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})$ such
that $\|\phi_{\underline{m}}(a)+k_{\underline{m}}\|<\varepsilon$. Fix
$\varepsilon>0$ and a corresponding ${\underline{m}}\perp F$ and
$k_{\underline{m}}\in{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})$. If
$F=[d]$, then ${\underline{m}}\perp F$ implies that
${\underline{m}}={\underline{0}}$, and there is nothing to show. So assume
that $F\subsetneq[d]$. Without loss of generality, we may assume that
${\underline{m}}\neq{\underline{0}}$. Indeed, if
${\underline{m}}={\underline{0}}$ then
$k_{\underline{0}}\in{\mathcal{K}}(A{\mathcal{L}}_{F})$ and hence
$k_{\underline{0}}=\phi_{\underline{0}}(b)$ for some $b\in{\mathcal{L}}_{F}$.
Note that

$\|a+b\|=\|\phi_{\underline{0}}(a)+\phi_{\underline{0}}(b)\|=\|\phi_{\underline{0}}(a)+k_{\underline{0}}\|<\varepsilon.$

Fix $i\in F^{c}$. Since $b\in{\mathcal{L}}_{F}$, we have that
$\phi_{\underline{i}}(b)\in{\mathcal{K}}(X_{\underline{i}})$ by assumption,
and $\left\langle
X_{\underline{i}},bX_{\underline{i}}\right\rangle\subseteq{\mathcal{L}}_{F}$
by invariance of ${\mathcal{L}}$. Thus an application of (2.1) gives that
$\phi_{\underline{i}}(b)\in{\mathcal{K}}(X_{\underline{i}}{\mathcal{L}}_{F})$,
and

$\|\phi_{\underline{i}}(a)+\phi_{\underline{i}}(b)\|=\|\phi_{\underline{i}}(a+b)\|\leq\|a+b\|<\varepsilon.$

So we may assume that ${\underline{m}}\neq{\underline{0}}$.

Next, take ${\underline{n}}\geq{\underline{m}}$ with ${\underline{n}}\perp F$.
We will show that
$\|\phi_{\underline{n}}(a)+{\mathcal{K}}(X_{\underline{n}}{\mathcal{L}}_{F})\|<\varepsilon$.
We write ${\underline{n}}={\underline{m}}+{\underline{r}}$ for some
${\underline{r}}\perp F$, and without loss of generality we may assume that
${\underline{r}}\neq{\underline{0}}$. By Proposition 2.5.2, we have that
$\phi_{\underline{r}}({\mathcal{L}}_{F})\subseteq{\mathcal{K}}(X_{\underline{r}}{\mathcal{L}}_{F})$.
An application of Corollary 2.2.5 then yields that
$k_{\underline{m}}\otimes\text{id}_{X_{\underline{r}}}\in{\mathcal{K}}((X_{\underline{m}}\otimes_{A}X_{\underline{r}}){\mathcal{L}}_{F})$,
and so
$\iota_{\underline{m}}^{\underline{n}}(k_{\underline{m}})\in{\mathcal{K}}(X_{\underline{n}}{\mathcal{L}}_{F})$.
We then obtain that

$\displaystyle\|\phi_{\underline{n}}(a)+{\mathcal{K}}(X_{\underline{n}}{\mathcal{L}}_{F})\|$
$\displaystyle\leq\|\phi_{\underline{n}}(a)+\iota_{\underline{m}}^{\underline{n}}(k_{\underline{m}})\|=\|\iota_{\underline{m}}^{\underline{n}}(\phi_{\underline{m}}(a)+k_{\underline{m}})\|\leq\|\phi_{\underline{m}}(a)+k_{\underline{m}}\|<\varepsilon,$

from which it follows that $\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\|=0$,
as required. ∎

The limit condition appears naturally in the study of ideals of
${\mathcal{N}}{\mathcal{T}}_{X}$ that are induced by invariant, partially
ordered relative $2^{d}$-tuples that consist of ideals.

######  Proposition 3.3.4.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a relative $2^{d}$-tuple of $X$ that is
invariant, partially ordered and consists of ideals. Fix $F\subsetneq[d]$ and
let
$a\in\bigcap\\{\phi_{{\underline{i}}}^{-1}({\mathcal{K}}(X_{{\underline{i}}}))\mid
i\in[d]\\}$. Then the following are equivalent:

  1. (i)

$\overline{\pi}_{X}(a)\overline{q}_{X,F}\in{\mathfrak{J}}_{\mathcal{L}}^{(\overline{\pi}_{X},\overline{t}_{X})}$;

  2. (ii)

$\overline{\pi}_{X}(a)\overline{q}_{X,F}\in\sum\\{{\mathfrak{J}}_{{\mathcal{L}},D}^{(\overline{\pi}_{X},\overline{t}_{X})}\mid
F\subseteq D\subseteq[d]\\}$.

Furthermore, if one (and hence both) of (i) and (ii) holds, then

$\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\|=0.$

######  Proof..

Without loss of generality, we may replace
$(\overline{\pi}_{X},\overline{t}_{X})$ by the Fock representation
$(\overline{\pi},\overline{t})$ and write
$\mathfrak{J}_{\mathcal{L}}\equiv{\mathfrak{J}}_{\mathcal{L}}^{(\overline{\pi},\overline{t})}$,
and
$\mathfrak{J}_{{\mathcal{L}},F}\equiv{\mathfrak{J}}_{{\mathcal{L}},F}^{(\overline{\pi},\overline{t})}$
for all $F\subseteq[d]$. The implication [(ii)$\Rightarrow$(i)] is immediate.
Conversely, if
$\overline{\pi}(a)\overline{q}_{F}\in{\mathfrak{J}}_{\mathcal{L}}$, then we
have that

$\overline{\pi}(a)\overline{q}_{F}=\overline{q}_{F}(\overline{\pi}(a)\overline{q}_{F})\overline{q}_{F}\in\overline{q}_{F}{\mathfrak{J}}_{{\mathcal{L}}}\overline{q}_{F}\subseteq\sum\\{\mathfrak{J}_{{\mathcal{L}},D}\mid
F\subseteq D\subseteq[d]\\},$

using Proposition 2.5.9 (and the fact that $\overline{q}_{F}$ is a projection)
in the first equality, and Proposition 3.3.2 in the final inclusion. This
establishes the equivalence of (i) and (ii).

Now suppose that one (and hence both) of (i) and (ii) holds. Condition (ii)
then yields that

$\displaystyle\overline{\pi}(a)\overline{q}_{F}$
$\displaystyle\in\sum\\{\overline{q}_{F}{\mathfrak{J}}_{{\mathcal{L}},D}\overline{q}_{F}\mid
F\subseteq D\subseteq[d]\\}$
$\displaystyle\subseteq\overline{\operatorname{span}}\\{\overline{t}_{\underline{n}}(X_{\underline{n}})\overline{\pi}({\mathcal{L}}_{D})\overline{q}_{D}\overline{t}_{\underline{m}}(X_{\underline{m}})^{*}\mid{\underline{n}},{\underline{m}}\perp
F,F\subseteq D\subseteq[d]\\},$

using Proposition 3.3.2 for the containment. Let $\gamma$ be the gauge action
of $(\overline{\pi},\overline{t})$ and let
$E_{\gamma}\colon\mathrm{C}^{*}(\overline{\pi},\overline{t})\to\mathrm{C}^{*}(\overline{\pi},\overline{t})^{\gamma}$
be the associated faithful conditional expectation. Therefore, since
$\overline{\pi}(a)\overline{q}_{F}\in\mathrm{C}^{*}(\overline{\pi},\overline{t})^{\gamma}$,
applying $E_{\gamma}$ yields that

(3.2)
$\overline{\pi}(a)\overline{q}_{F}\in\overline{\operatorname{span}}\\{\overline{t}_{\underline{n}}(X_{\underline{n}})\overline{\pi}({\mathcal{L}}_{D})\overline{q}_{D}\overline{t}_{\underline{n}}(X_{\underline{n}})^{*}\mid{\underline{n}}\perp
F,F\subseteq D\subseteq[d]\\}.$

Fix $\varepsilon>0$. By Proposition 3.3.3, it is sufficient to find
${\underline{m}}\perp F$ and
$k_{\underline{m}}\in{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})$ such
that $\|\phi_{\underline{m}}(a)+k_{\underline{m}}\|<\varepsilon$. By (3.2),
for each $F\subseteq D\subseteq[d]$ there exists

$c_{D}\in\operatorname{span}\\{\overline{t}_{\underline{n}}(X_{\underline{n}})\overline{\pi}({\mathcal{L}}_{D})\overline{q}_{D}\overline{t}_{\underline{n}}(X_{\underline{n}})^{*}\mid{\underline{n}}\perp
F\\}$

such that

$\|\overline{\pi}(a)\overline{q}_{F}-\sum\\{c_{D}\mid F\subseteq
D\subseteq[d]\\}\|<\varepsilon.$

Moreover, we may assume that

$c_{D}=\sum\\{\sum_{j=1}^{N}\overline{t}_{\underline{n}}(\xi_{{\underline{n}},D}^{j})\overline{\pi}(b_{{\underline{n}},D}^{j})\overline{q}_{D}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},D}^{j})^{*}\mid{\underline{0}}\leq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}}\\},$

for some $m,N\in\mathbb{N}$,
$\xi_{{\underline{n}},D}^{j},\eta_{{\underline{n}},D}^{j}\in
X_{\underline{n}}$ and $b_{{\underline{n}},D}^{j}\in{\mathcal{L}}_{D}$, for
all $j\in[N]$ and ${\underline{0}}\leq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}}$. We may use the same $m$ and $N$ for every
$c_{D}$ by padding with zeros, if necessary. Fix the projection

$P\colon{\mathcal{F}}X\to X_{(m+1)\cdot{\underline{1}}_{F^{c}}}.$

First we claim that $c_{D}P=0$ whenever $F\subsetneq D\subseteq[d]$, so that

$\displaystyle\|\overline{\pi}(a)\overline{q}_{F}P-c_{F}P\|$
$\displaystyle=\|(\overline{\pi}(a)\overline{q}_{F}-\sum\\{c_{D}\mid
F\subseteq D\subseteq[d]\\})P\|$
$\displaystyle\leq\|\overline{\pi}(a)\overline{q}_{F}-\sum\\{c_{D}\mid
F\subseteq D\subseteq[d]\\}\|<\varepsilon.$

Indeed, for each ${\underline{0}}\leq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}}$ and each $j\in[N]$, we have that
$\overline{t}_{\underline{n}}(\eta_{{\underline{n}},D}^{j})^{*}P$ has image in
$X_{\underline{r}}$ for some ${\underline{r}}\neq{\underline{0}}$ with
$\operatorname{supp}{\underline{r}}=F^{c}$, since $m<m+1$. Note also that
$D\cap F^{c}\neq\emptyset$ since $F\subsetneq D$, and thus

$\overline{q}_{D}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},D}^{j})^{*}P({\mathcal{F}}X)\subseteq\overline{q}_{D}\sum\\{X_{\underline{r}}\mid\operatorname{supp}{\underline{r}}=F^{c}\\}=\\{0\\}.$

Consequently

$\overline{t}_{\underline{n}}(\xi_{{\underline{n}},D}^{j})\overline{\pi}(b_{{\underline{n}},D}^{j})\overline{q}_{D}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},D}^{j})^{*}P=0,\text{
for all }{\underline{0}}\leq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}},j\in[N],$

and hence $c_{D}P=0$.

Next we examine the terms of $c_{F}P$. To this end, fix
${\underline{0}}\neq{\underline{n}}\leq m\cdot{\underline{1}}_{F^{c}}$ and
$j\in[N]$. Take $\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}}\in
X_{(m+1)\cdot{\underline{1}}_{F^{c}}}$ such that

$\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}}=\zeta_{\underline{n}}\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}\textup{
for some }\zeta_{\underline{n}}\in
X_{\underline{n}},\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}\in
X_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}.$

Note that
$\operatorname{supp}((m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}})=F^{c}$,
and so we have that

$\displaystyle\overline{t}_{\underline{n}}(\xi_{{\underline{n}},F}^{j})\overline{\pi}(b_{{\underline{n}},F}^{j})\overline{q}_{F}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},F}^{j})^{*}P\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}}$
$\displaystyle=\overline{t}_{\underline{n}}(\xi_{{\underline{n}},F}^{j})\overline{\pi}(b_{{\underline{n}},F}^{j})\overline{q}_{F}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},F}^{j})^{*}\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}}$
$\displaystyle=\xi_{{\underline{n}},F}^{j}\big{(}\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}(b_{{\underline{n}},F}^{j}\langle\eta_{{\underline{n}},F}^{j},\zeta_{\underline{n}}\rangle)\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}\big{)}$
$\displaystyle=\big{(}\Theta_{\xi_{{\underline{n}},F}^{j}b_{{\underline{n}},F}^{j},\eta_{{\underline{n}},F}^{j}}(\zeta_{\underline{n}})\big{)}\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}$
$\displaystyle=\iota_{\underline{n}}^{(m+1)\cdot{\underline{1}}_{F^{c}}}(\Theta_{\xi_{{\underline{n}},F}^{j}b_{{\underline{n}},F}^{j},\eta_{{\underline{n}},F}^{j}})\zeta_{(m+1)\cdot{\underline{1}}_{F^{c}}}.$

We deduce that

$\overline{t}_{\underline{n}}(\xi_{{\underline{n}},F}^{j})\overline{\pi}(b_{{\underline{n}},F}^{j})\overline{q}_{F}\overline{t}_{\underline{n}}(\eta_{{\underline{n}},F}^{j})^{*}P=\iota_{\underline{n}}^{(m+1)\cdot{\underline{1}}_{F^{c}}}(\Theta_{\xi_{{\underline{n}},F}^{j}b_{{\underline{n}},F}^{j},\eta_{{\underline{n}},F}^{j}}),$

where we view the right hand side as an element of
${\mathcal{L}}({\mathcal{F}}X)$ in the way described in Subsection 2.3.
Arguing in a similar (and simpler) way when ${\underline{n}}={\underline{0}}$,
we deduce that

$c_{F}P=\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}}(b)+\sum\\{\sum_{j=1}^{N}\iota_{\underline{n}}^{(m+1)\cdot{\underline{1}}_{F^{c}}}(\Theta_{\xi_{{\underline{n}},F}^{j}b_{{\underline{n}},F}^{j},\eta_{{\underline{n}},F}^{j}})\mid{\underline{0}}\neq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}}\\},$

for some $b\in{\mathcal{L}}_{F}$. By Proposition 2.5.2, we have that

$\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}}(b)\in{\mathcal{K}}(X_{(m+1)\cdot{\underline{1}}_{F^{c}}}{\mathcal{L}}_{F})$

and

$\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}(b_{{\underline{n}},F}^{j})\in{\mathcal{K}}(X_{(m+1)\cdot{\underline{1}}_{F^{c}}-{\underline{n}}}{\mathcal{L}}_{F})\text{
for all }{\underline{0}}\neq{\underline{n}}\leq
m\cdot{\underline{1}}_{F^{c}},j\in[N].$

An application of Lemma 2.2.4 gives that all of the summands of $c_{F}P$
belong to
${\mathcal{K}}(X_{(m+1)\cdot{\underline{1}}_{F^{c}}}{\mathcal{L}}_{F})$.
Therefore we have that
$c_{F}P\in{\mathcal{K}}(X_{(m+1)\cdot{\underline{1}}_{F^{c}}}{\mathcal{L}}_{F})\hookrightarrow{\mathcal{L}}({\mathcal{F}}X)$,
which satisfies

$\|\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}}(a)-c_{F}P\|=\|\overline{\pi}(a)\overline{q}_{F}P-c_{F}P\|<\varepsilon,$

where we use that

$\phi_{(m+1)\cdot{\underline{1}}_{F^{c}}}(a)=\overline{\pi}(a)P=\overline{\pi}(a)\overline{q}_{F}P$

in the identification
${\mathcal{L}}(X_{(m+1)\cdot{\underline{1}}_{F^{c}}})\hookrightarrow{\mathcal{L}}({\mathcal{F}}X)$.
Hence we have found ${\underline{m}}:=(m+1)\cdot{\underline{1}}_{F^{c}}\perp
F$ and
$k_{\underline{m}}:=-c_{F}P\in{\mathcal{K}}(X_{{\underline{m}}}{\mathcal{L}}_{F})$
such that $\|\phi_{{\underline{m}}}(a)+k_{\underline{m}}\|<\varepsilon$, as
required. ∎

######  Remark 3.3.5.

Proposition 3.3.4 holds for $F=[d]$ as well, though the second claim requires
a different argument. Specifically, we compress
$\overline{\pi}(a)\overline{q}_{[d]}\in{\mathfrak{J}}_{{\mathcal{L}},[d]}$ at
the ${\underline{0}}$-th entry to deduce that $a\in{\mathcal{L}}_{[d]}$. Then
$\phi_{\underline{0}}(a)\in{\mathcal{K}}(A{\mathcal{L}}_{[d]})$, and so
$\lim_{{\underline{m}}\perp[d]}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{[d]})\|=0$,
as required.

###  3.4. Constructing (M)-$2^{d}$-tuples

In this subsection we show how the maximal $2^{d}$-tuple inducing
${\mathfrak{J}}_{{\mathcal{L}}}^{(\overline{\pi}_{X},\overline{t}_{X})}$, for
${\mathcal{L}}$ an (E)-$2^{d}$-tuple, is constructed. The following definition
is motivated in part by Proposition 3.3.4.

######  Definition 3.4.1.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$ and let ${\mathcal{L}}$ be a $2^{d}$-tuple of $X$ that consists
of ideals. Fixing $\emptyset\neq F\subsetneq[d]$, we define

${\mathcal{L}}_{{\operatorname{inv}},F}:=\bigcap_{{\underline{m}}\perp
F}X_{\underline{m}}^{-1}(\cap_{F\subsetneq
D}{\mathcal{L}}_{D})\quad\text{and}\quad{\mathcal{L}}_{\lim,F}:=\\{a\in
A\mid\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\|=0\\}.$

If ${\mathcal{L}}$ is in addition an (E)-$2^{d}$-tuple of $X$, then we define
the $2^{d}$-tuple ${\mathcal{L}}^{(1)}$ of $X$ by

${\mathcal{L}}_{F}^{(1)}:=\begin{cases}\\{0\\}&\text{ if }F=\emptyset,\\\
{\mathcal{I}}_{F}\cap{\mathcal{L}}_{{\operatorname{inv}},F}\cap{\mathcal{L}}_{\lim,F}&\text{
if }\emptyset\neq F\subsetneq[d],\\\ {\mathcal{L}}_{[d]}&\text{ if
}F=[d].\end{cases}$

Observe that each ${\mathcal{L}}_{F}^{(1)}$ is non-empty, as
$0\in{\mathcal{L}}_{F}^{(1)}$. Note also that ${\mathcal{L}}^{(1)}$ is an
(E)-$2^{d}$-tuple by construction. We will show that ${\mathcal{L}}^{(1)}$
consists of ideals, and so we can write
${\mathcal{L}}^{(k+1)}:=({\mathcal{L}}^{(k)})^{(1)}$ for $k\in\mathbb{N}$ (by
convention we set ${\mathcal{L}}^{(0)}:={\mathcal{L}}$). To this end, first we
note that ${\mathcal{L}}_{{\operatorname{inv}},F}$ is an intersection of
ideals and is thus an ideal itself. To address ${\mathcal{L}}_{\lim,F}$, we
have the following proposition.

######  Proposition 3.4.2.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be a $2^{d}$-tuple of $X$ that consists of
ideals and fix $\emptyset\neq F\subsetneq[d]$. Then we have that
${\mathcal{L}}_{\lim,F}$ is an ideal.

######  Proof..

Firstly, we define

$B_{F}:=\bigoplus_{{\underline{m}}\perp
F}{\mathcal{L}}(X_{\underline{m}})/{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\quad\text{and}\quad
c_{0}(B_{F}):=\\{(S_{{\underline{m}}})_{{\underline{m}}\perp F}\in
B_{F}\mid\lim_{{\underline{m}}\perp F}\|S_{{\underline{m}}}\|=0\\},$

and observe that $c_{0}(B_{F})$ is an ideal in $B_{F}$. Consider the
$*$-homomorphism $\psi_{F}$ defined by

$\displaystyle\psi_{F}\colon A\to\bigoplus_{{\underline{m}}\perp
F}{\mathcal{L}}(X_{\underline{m}})\to B_{F}\to B_{F}/c_{0}(B_{F});$
$\displaystyle\psi_{F}\colon
a\mapsto(\phi_{\underline{m}}(a))_{{\underline{m}}\perp
F}\mapsto(\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F}))_{{\underline{m}}\perp
F}\mapsto(\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F}))_{{\underline{m}}\perp
F}+c_{0}(B_{F}).$

It is now a standard C*-result that

$\ker\psi_{F}=\\{a\in A\mid\lim_{{\underline{m}}\perp
F}\|\phi_{\underline{m}}(a)+{\mathcal{K}}(X_{\underline{m}}{\mathcal{L}}_{F})\|=0\\},$

and thus ${\mathcal{L}}_{\lim,F}=\ker\psi_{F}$ is an ideal, as required. ∎

######  Proposition 3.4.3.

Let $X$ be a strong compactly aligned product system with coefficients in a
C*-algebra $A$. Let ${\mathcal{L}}$ be an $(E)$-$2^{d}$-tuple of $X$ that
consists of ideals. Then ${\mathcal{L}}^{(1)}$ is an (E)-$2^{d}$-tuple of $X$
that consists of ideals. If ${\mathcal{L}}$ is invariant, then so is
${\mathcal{L}}^{(1)}$.

######  Proof..

We have already remarked that ${\mathcal{L}}^{(1)}$ is an (E)-$2^{d}$-tuple.
The fact that ${\mathcal{L}}^{(1)}$ consists of ideals follows from
Proposition 3.4.2 and the discussion preceding it.

Now assume that ${\mathcal{L}}$ is invariant. The invariance condition for
${\mathcal{L}}^{(1)}$ clearly holds when $F=\emptyset$ or $F=[d]$, so assume
that $\emptyset\neq F\subsetneq[d]$. Fix ${\underline{n}}\perp F$,
$\xi_{\underline{n}},\eta_{\underline{n}}\in X_{\underline{n}}$, and
$a\in{\mathcal{L}}_{F}^{(1)}$. It suffices to show that

$\left\langle\xi_{\underline{n}},\phi_{\underline{n}}(a)\eta_{\underline{n}}\right\rangle\in{\mathcal{L}}_{F}^{(1)}={\mathcal{I}}_{F}\cap{\mathcal{L}}_{{\operatorname{inv}},F}\cap{\mathcal{L}}_{\lim,F}.$

If ${\underline{n}}={\underline{0}}$ there is nothing to show, so assume that
${\underline{n}}\neq{\underline{0}}$. First we have that
$\left\langle\xi_{\underline{n}},\phi_{\underline{n}}(a)\eta_{\underline{n}}\right\rangle\in{\mathcal{I}}_{F}$
by [16, Proposition 2.7], since $a\in{\mathcal{I}}_{F}$. Now fix
${\underline{m}}\perp F$ and $\xi_{\underline{m}},\eta_{\underline{m}}\in
X_{\underline{m}}$. Then

$\displaystyle\left\langle\xi_{\underline{m}},\phi_{\underline{m}}(\left\langle\xi_{\underline{n}},\phi_{\underline{n}}(a)\eta_{\underline{n}}\right\rangle)\eta_{\underline{m}}\right\rangle$
$\displaystyle=\left\langle\xi_{\underline{n}}\xi_{\underline{m}},(\phi_{\underline{n}}(a)\eta_{\underline{n}})\eta_{\underline{m}}\right\rangle=\left\langle\xi_{\underline{n}}\xi_{\underline{m}},\phi_{{\underline{n}}+{\underline{m}}}(a)(\eta_{\underline{n}}\eta_{\underline{m}})\right\rangle\in\cap_{F\subsetneq
D}{\mathcal{L}}_{D},$