it can be adapted to the operator algebras category; the reader is directed to
[13] for the full details.

###  4.2. Strong covariant representations

We review the key elements of the strong covariant C*-algebra of a product
system established by Sehnem [47, 48], and its Fell bundle considered by Dor-
On, the first author, Katsoulis, Laca and Li [11].

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Set $A:=X_{e}$, and for each $p\in P$ let
$\varphi_{p}\colon A\to{\mathcal{L}}(X_{p})$ be the $*$-homomorphism that
implements the left action of $A$ on the C*-correspondence $X_{p}$. For a
finite set $F\subseteq G$ let

$K_{F}:=\bigcap_{g\in F}gP.$

For $r\in P$ and $g\in F$ define the ideal of $A$ given by

$I_{r^{-1}K_{\\{r,g\\}}}:=\begin{cases}\bigcap\limits_{s\in
K_{\\{r,g\\}}}\ker\varphi_{r^{-1}s}&\text{if }K_{\\{r,g\\}}\neq\emptyset\text{
and }r\notin K_{\\{r,g\\}},\\\ \hskip
14.22636ptA&\text{otherwise},\end{cases}$

and set

$I_{r^{-1}(r\vee F)}:=\bigcap_{g\in F}I_{r^{-1}K_{\\{r,g\\}}}.$

We have that $I_{r^{-1}(r\vee F)}=I_{(pr)^{-1}(pr\vee pF)}$ for all $r,p\in
P$, and $I_{r^{-1}(r\vee F)}=I_{(s^{-1}r)^{-1}(s^{-1}r\vee s^{-1}F)}$ for all
$r\in sP$. Moreover

$I_{r^{-1}(r\vee F_{1})}\subseteq I_{r^{-1}(r\vee F_{2})}\;\text{ when
}\;F_{1}\supseteq F_{2}.$

We declare $K_{\emptyset}=\emptyset$ and $I_{r^{-1}(r\vee\emptyset)}=A$. For a
finite set $F\subseteq G$, let the C*-correspondences

$X_{F}:=\operatornamewithlimits{\sum^{\oplus}}_{r\in P}X_{r}I_{r^{-1}(r\vee
F)}\quad\text{and}\quad
X_{F}^{+}:=\operatornamewithlimits{\sum^{\oplus}}_{g\in G}X_{gF}.$

We declare $X_{\emptyset}=X_{\emptyset}^{+}={\mathcal{F}}X$. Each $X_{F}^{+}$
is reducing for the coaction

$\delta\colon{\mathcal{T}}_{\lambda}(X)\longrightarrow{\mathcal{T}}_{\lambda}(X)\otimes\mathrm{C}^{*}_{\max}(G);\lambda_{p}(\xi_{p})\mapsto\lambda_{p}(\xi_{p})\otimes
u_{p},$

giving rise to a $*$-representation

$\Phi_{F}\colon{\mathcal{T}}_{\lambda}(X)\longrightarrow{\mathcal{L}}(X_{F}^{+});\lambda_{p}(\xi_{p})\mapsto(\lambda_{p}(\xi_{p})\otimes
u_{p})|_{X_{F}^{+}}.$

Here we make the identification

$X_{gF}\longrightarrow
X_{gF}\otimes\delta_{g};\xi_{r}a_{r}\mapsto\xi_{r}a_{r}\otimes\delta_{g},\text{
for }\xi_{r}\in X_{r},a_{r}\in I_{r^{-1}(r\vee gF)}.$

Moreover $X_{F}\subseteq X_{F}^{+}$ is reducing for $[{\mathcal{T}}(X)]_{e}$
and so we obtain the representation

$\bigoplus\limits_{\textup{fin }F\subseteq
G}\Phi_{F}(\cdot)|_{X_{F}}\colon[{\mathcal{T}}_{\lambda}(X)]_{e}\longrightarrow\prod\limits_{\text{fin
}F\subseteq G}{\mathcal{L}}(X_{F}).$

We fix the ideal ${\mathcal{I}}_{\operatorname{sc},e}$ in
$[{\mathcal{T}}(X)]_{e}$ by using the corona universe, namely

$b\in{\mathcal{I}}_{\operatorname{sc},e}\quad\text{if and only
if}\quad\bigoplus\limits_{\textup{fin }F\subseteq
G}\Phi_{F}(\lambda_{\ast}(b))|_{X_{F}}\in
c_{0}\left({\mathcal{L}}(X_{F})\mid\textup{fin }F\subseteq G\right).$

We write $A\times_{X}P$ for the equivariant quotient of ${\mathcal{T}}(X)$ by
the induced ideal
${\mathcal{I}}_{\operatorname{sc}}:=\left\langle{\mathcal{I}}_{\operatorname{sc},e}\right\rangle$.
It is shown in [47] that this construction does not depend on $G$. Moreover,
we define the _strong covariant_ bundle of $X$ to be the Fell bundle

${\mathcal{S}}{\mathcal{C}}X:=\Big{\\{}[A\times_{X}P]_{g}\Big{\\}}_{g\in G},$

given by the coaction by $G$ on $A\times_{X}P$. A representation of $X$ that
promotes to a representation of ${\mathcal{S}}{\mathcal{C}}X$ will be called a
_strong covariant representation of $X$_. By combining Proposition 2.2 with
Proposition 2.7 we have that $A\times_{X}P$ is the universal C*-algebra of
${\mathcal{S}}{\mathcal{C}}X$. We further write $A\times_{X,\lambda}P$ for the
reduced C*-algebra of ${\mathcal{S}}{\mathcal{C}}X$. As the strong covariant
relations are graded and induced by representations on
${\mathcal{T}}_{\lambda}(X)$ it follows that a strong covariant representation
is Fock covariant.

For a product system $X$ we will write ${\mathcal{T}}_{\lambda}(X)^{+}$ for
the _tensor algebra of $X$_, i.e., for the norm-closed subalgebra generated by
$\\{\lambda_{p}(X_{p})\\}_{p\in P}$ inside ${\mathcal{T}}_{\lambda}(X)$. We
will require the following consequence of [48, Corollary 3.5]. We note that
although only product systems in the sense of Fowler are considered in [48],
the arguments apply in our setting as well.

######  Corollary 4.1.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $t$ is a Fock covariant injective representation
of $X$ and $t$ admits a coaction by $G$ which is normal, then the map

${\mathcal{T}}_{\lambda}(X)^{+}\to\overline{\operatorname{alg}}\\{t_{p}(X_{p})\mid
p\in P\\};\lambda_{p}(\xi_{p})\mapsto t_{p}(\xi_{p}),$

is a (well-defined) completely isometric isomorphism.

######  Proof..

Let the canonical $*$-epimorphism
$\Phi\colon{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)\to\mathrm{C}^{*}(t)$.
Since $t$ is equivariant, then $\Phi$ induces a $*$-epimorphism between the
reduced C*-algebras of the induced Fell bundles. However, since the coaction
on $\mathrm{C}^{*}(t)$ is normal, it follows that $\mathrm{C}^{*}(t)$
coincides with the reduced C*-algebra of its induced Fell bundle. Hence there
exists a canonical $*$-epimorphism
$\Phi_{\lambda}\colon{\mathcal{T}}_{\lambda}(X)\to\mathrm{C}^{*}(t)$, which is
injective on $A$ since $t$ is injective. By [48, Corollary 3.5] it follows
that the restriction of $\Phi_{\lambda}$ on ${\mathcal{T}}_{\lambda}(X)^{+}$
is completely isometric. ∎

Since $A\times_{X,\lambda}P$ satisfies the conditions of Corollary 4.1, it is
a C*-cover of ${\mathcal{T}}_{\lambda}(X)^{+}$. In [48, Theorem 5.1], Sehnem
establishes that $A\times_{X,\lambda}P$ is the C*-envelope of
${\mathcal{T}}_{\lambda}(X)^{+}$.

###  4.3. Crossed products

The theory of crossed products of C*-algebras is well-known, see for example
[50]. Katsoulis and Ramsey [27] have extended this to group actions over
possibly nonselfadjoint operator algebras. Here we will comment only on the
parts that are relevant to the reduced Hao–Ng isomorphism problem.

Let ${\mathfrak{H}}$ be a locally compact group that acts on an operator
algebra ${\mathfrak{A}}$ by completely isometric automorphisms. The action of
${\mathfrak{H}}$ then extends also to a group action $\dot{\alpha}$ of the
C*-envelope of ${\mathfrak{A}}$, and we can form the reduced crossed product
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}$.
By considering the copy of ${\mathfrak{A}}$ inside
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$, the reduced crossed product
${\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ is defined as the norm-
closed subalgebra of
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}$
generated by the ${\mathfrak{A}}$-valued functions [27, Definition 3.17]. One
of the main questions in the theory is whether this inclusion induces a
canonical $*$-isomorphism with the C*-envelope of
${\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$, i.e., whether it holds
that

(4.1)
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\stackrel{{\scriptstyle?}}{{\simeq}}\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}.$

When ${\mathfrak{A}}$ admits a contractive approximate identity, this has been
answered to the affirmative when ${\mathfrak{H}}$ is discrete [24], and when
${\mathfrak{H}}$ is abelian [27]. By using the maximal representations of [14]
we can remove the contractive approximate identity hypothesis when
${\mathfrak{H}}$ is discrete.

######  Proposition 4.2.

Let ${\mathfrak{H}}$ be a discrete group acting by $\alpha$ on an operator
algebra ${\mathfrak{A}}$. Then

$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\simeq\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}.$

######  Proof..

We identify ${\mathfrak{A}}$ with its copy in
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$. Let
$\pi\colon{\mathfrak{A}}\to{\mathcal{B}}(H)$ be a maximal completely isometric
representation of ${\mathfrak{A}}$. Then $\pi$ has a unique extension to a
faithful $*$-representation on
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$, denoted by the same symbol.
Therefore $\overline{\pi}\rtimes U$ is a faithful $*$-representation of
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}$,
and thus its restriction on
${\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ is completely
isometric. It suffices to show that $\overline{\pi}\rtimes U$ is maximal on
${\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$, as then it will follow
that

$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\simeq\mathrm{C}^{*}(\overline{\pi}\rtimes
U)\simeq\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}$

by $*$-isomorphisms fixing
${\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$.

Towards this end, let
$\rho\colon{\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}\to{\mathcal{B}}(K)$
be a maximal dilation of $\overline{\pi}\rtimes
U|_{{\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}}$, and let us denote
by the same symbol the unique extension of $\rho$ to a $*$-representation on

$\mathrm{C}^{*}({\mathfrak{A}}\rtimes_{\alpha,\lambda}{\mathfrak{H}})=\overline{\operatorname{span}}\\{\overline{\pi}(c)U_{{\mathfrak{h}}}\mid
c\in\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}}),{\mathfrak{h}}\in{\mathfrak{H}}\\}.$

By Arveson’s Extension Principle, each $\pi\circ\alpha_{\mathfrak{h}}$ is a
maximal representation of ${\mathfrak{A}}$, and it is a standard argument that
$\overline{\pi}$ is maximal as the discrete direct sum of maximal
representations, e.g., see [4, Proposition 4.4]. Therefore $\rho$ on
$\overline{\pi}({\mathfrak{A}})$ takes up the form

$\rho(\overline{\pi}(a))=\begin{bmatrix}\overline{\pi}(a)&0\\\
0&\sigma(a)\end{bmatrix}\text{ for all }a\in{\mathfrak{A}},$

for a representation $\sigma$ of ${\mathfrak{A}}$. Next consider
$\overline{\pi}(a)U_{{\mathfrak{h}}}$ for $a\in{\mathfrak{A}}$ and
${\mathfrak{h}}\in{\mathfrak{H}}$, and write

$\rho(\overline{\pi}(a)U_{\mathfrak{h}})=\begin{bmatrix}\overline{\pi}(a)U_{{\mathfrak{h}}}&x\\\
y&z\end{bmatrix}.$

By using the unique extension property of $\rho$ we have that

$\displaystyle\begin{bmatrix}\overline{\pi}(a)\overline{\pi}(a)^{*}+xx^{*}&\ast\\\
\ast&\ast\end{bmatrix}$
$\displaystyle=\rho(\overline{\pi}(a)U_{\mathfrak{h}})\rho(\overline{\pi}(a)U_{\mathfrak{h}})^{*}=\rho(\overline{\pi}(a)U_{\mathfrak{h}}U_{{\mathfrak{h}}}^{*}\overline{\pi}(a)^{*})$
$\displaystyle=\rho(\overline{\pi}(a)\overline{\pi}(a)^{*})=\rho(\overline{\pi}(a))\rho(\overline{\pi}(a))^{*}=\begin{bmatrix}\overline{\pi}(a)\overline{\pi}(a)^{*}&0\\\
0&\sigma(a)\sigma(a)^{*}\end{bmatrix}.$

By equating the $(1,1)$-entries we get that $x=0$. On the other hand, by using
the covariance in the C*-crossed product we can write

$\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))^{*}\overline{\pi}(\alpha_{{\mathfrak{h}}}^{-1}(a))=\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))^{*}U_{{\mathfrak{h}}}^{*}U_{{\mathfrak{h}}}\overline{\pi}(\alpha_{{\mathfrak{h}}}^{-1}(a))=U_{\mathfrak{h}}^{*}\overline{\pi}(a)^{*}\overline{\pi}(a)U_{{\mathfrak{h}}}.$

Then a similar computation gives that

$\displaystyle\begin{bmatrix}\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))^{*}\overline{\pi}(\alpha_{{\mathfrak{h}}}^{-1}(a))+y^{*}y&\ast\\\
\ast&\ast\end{bmatrix}$
$\displaystyle=\rho(\overline{\pi}(a)U_{\mathfrak{h}})^{*}\rho(\overline{\pi}(a)U_{\mathfrak{h}})=\rho\left(U_{\mathfrak{h}}^{*}\overline{\pi}(a)^{*}\overline{\pi}(a)U_{{\mathfrak{h}}}\right)$
$\displaystyle=\rho\left(\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))^{*}\overline{\pi}(\alpha_{{\mathfrak{h}}}^{-1}(a))\right)=\rho\left(\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))\right)^{*}\rho\left(\overline{\pi}(\alpha_{{\mathfrak{h}}}^{-1}(a))\right)$
$\displaystyle=\begin{bmatrix}\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))^{*}\overline{\pi}(\alpha_{\mathfrak{h}}^{-1}(a))&0\\\
0&\sigma(\alpha_{\mathfrak{h}}^{-1}(a))^{*}\sigma(\alpha_{\mathfrak{h}}^{-1}(a))\end{bmatrix}.$

By equating the $(1,1)$-entries we get that $y=0$. Therefore $\rho$ is a
trivial dilation, as required. ∎

###  4.4. Hao–Ng

Let us return to the product system discussion. Let $P$ be a unital
subsemigroup of a discrete group $G$ and let $X$ be a product system over $P$.
Let ${\mathfrak{H}}$ be a discrete group, and let $\alpha$ be a _generalised
gauge action_ of ${\mathfrak{H}}$ on the Fock C*-algebra
${\mathcal{T}}_{\lambda}(X)$, i.e., every $\lambda_{p}(X_{p})$ is
$\alpha$-invariant in the sense that

$\alpha_{{\mathfrak{h}}}(\lambda_{p}(X_{p}))=\lambda_{p}(X_{p})\text{ for all
}p\in P\text{ and }{\mathfrak{h}}\in{\mathfrak{H}}.$

Let $\pi$ be a faithful $*$-representation of ${\mathcal{T}}_{\lambda}(X)$ in
some ${\mathcal{B}}(H)$ and consider the faithful $*$-representation
$\overline{\pi}\rtimes U$ on $\ell^{2}({\mathfrak{H}},H)$ that gives rise to
${\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$. Since
$\alpha$ restricts to an action on ${\mathcal{T}}_{\lambda}(X)^{+}$, from [27,
Corollary 3.16] we have a canonical completely isometric copy of the crossed
product ${\mathcal{T}}_{\lambda}(X)^{+}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$
in ${\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$. We can
now consider the family $X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ of the
subspaces

$(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})_{p}:=\overline{\operatorname{span}}\\{\overline{\pi}(\lambda_{p}(\xi_{p}))U_{{\mathfrak{h}}}\mid\xi_{p}\in
X_{p},{\mathfrak{h}}\in{\mathfrak{H}}\\}\text{ for }p\in P.$

It is accustomed to use the short form
$X_{p}\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ instead of
$(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})_{p}$. We will show that this
construction gives rise to a product system.

######  Proposition 4.3.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $\alpha$ be a generalised gauge action of a
discrete group ${\mathfrak{H}}$ on ${\mathcal{T}}_{\lambda}(X)$. Then
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ is a product system over $P$.

######  Proof..

For condition (i) we have by definition that

$X_{e}\rtimes_{\alpha,\lambda}{\mathfrak{H}}:=\overline{\operatorname{span}}\\{\overline{\pi}(\lambda_{e}(\xi_{e}))U_{{\mathfrak{h}}}\mid\xi_{e}\in
X_{e},{\mathfrak{h}}\in{\mathfrak{H}}\\}\simeq
A\rtimes_{\alpha,\lambda}{\mathfrak{H}},$

which is a C*-algebra. The other two conditions follow by the fact that
$\alpha_{{\mathfrak{h}}}(\lambda_{p}(X_{p}))=\lambda_{p}(X_{p})$ for all $p\in
P$ and ${\mathfrak{h}}\in{\mathfrak{H}}$, the covariant relations of the
crossed product and that $\lambda$ is a product system representation. Indeed,
for condition (ii) we have that

$\displaystyle\overline{\pi}(\lambda_{p}(X_{p}))U_{{\mathfrak{h}}_{1}}\overline{\pi}(\lambda_{q}(X_{q}))U_{{\mathfrak{h}}_{2}}$
$\displaystyle=\overline{\pi}\left(\lambda_{p}(X_{p})\alpha_{{\mathfrak{h}}_{1}}(\lambda_{q}(X_{q}))\right)U_{{\mathfrak{h}}_{1}{\mathfrak{h}}_{2}}$
$\displaystyle=\overline{\pi}\left(\lambda_{p}(X_{p})\lambda_{q}(X_{q})\right)U_{{\mathfrak{h}}_{1}{\mathfrak{h}}_{2}}\subseteq\overline{\pi}\left(\lambda_{pq}(X_{p}X_{q})\right)U_{{\mathfrak{h}}_{1}{\mathfrak{h}}_{2}}\subseteq
X_{pq}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

By considering finite linear combinations and their norm-limits we obtain

$(X_{p}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\cdot(X_{q}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\subseteq
X_{pq}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

For condition (iii) we have that

$\displaystyle\left(\overline{\pi}(\lambda_{p}(X_{p}))U_{{\mathfrak{h}}_{1}}\right)^{*}\overline{\pi}(\lambda_{pq}(X_{pq}))U_{{\mathfrak{h}}_{2}}$
$\displaystyle=\overline{\pi}\circ\alpha_{{\mathfrak{h}}_{1}}^{-1}\left(\lambda_{p}(X_{p})^{*}\lambda_{pq}(X_{pq})\right)U_{{\mathfrak{h}}_{1}^{-1}{\mathfrak{h}}_{2}}$
$\displaystyle\subseteq\overline{\pi}\left(\lambda_{q}(X_{q})\right)U_{{\mathfrak{h}}_{1}^{-1}{\mathfrak{h}}_{2}}\subseteq
X_{q}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

By considering finite linear combinations and their norm-limits we obtain that

$(X_{p}\rtimes_{\alpha,\lambda}{\mathfrak{H}})^{*}\cdot(X_{pq}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\subseteq
X_{q}\rtimes_{\alpha,\lambda}{\mathfrak{H}},$

and the proof is complete. ∎

In order to facilitate comparisons we will use the superscript $\rtimes$ for
the representations of $X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$. That is, we
write $\lambda^{\rtimes}$ for the Fock representation of
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$, and we fix a representation
$\hat{t}^{\rtimes}$ of $X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ such that

${\mathcal{T}}(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})=\mathrm{C}^{*}(\hat{t}^{\rtimes}).$

We will also write $\iota^{\rtimes}$ for the identity representation
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}\hookrightarrow{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$.

Due to the properties of the generalised gauge action we have a canonical
identification of the $\mathbf{K}_{\bullet}$-cores. Note that, since every
$\lambda_{p}(X_{p})$ is $\alpha$-invariant, then so is every C*-subalgebra
$\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}$ for ${\mathbf{x}}\in{\mathcal{J}}$.
By construction we have that

$\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}\subseteq{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}\text{
for all }{\mathbf{x}}\in{\mathcal{J}}.$

######  Proposition 4.4.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $\alpha$ be a generalised gauge action of a
discrete group ${\mathfrak{H}}$ on ${\mathcal{T}}_{\lambda}(X)$ and let the
identity representation $\iota^{\rtimes}\colon
X\rtimes_{\alpha,\lambda}{\mathfrak{H}}\to{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$.
If ${\mathbf{x}}\in{\mathcal{J}}$, then

$\iota^{\rtimes}_{\ast}(\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}})=\mathbf{K}_{{\mathbf{x}},\iota^{\rtimes}_{\ast}}=\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

Consequently, if
$E_{\mathfrak{H}}\colon{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}\to{\mathcal{T}}_{\lambda}(X)$
is the faithful conditional expectation of the reduced crossed product, then

$E_{{\mathfrak{H}}}(\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}}))\in\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\text{
for all
}b_{{\mathbf{x}}}\in\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}}.$

######  Proof..

We will make the standard identification of ${\mathcal{T}}_{\lambda}(X)$ with
$\overline{\pi}({\mathcal{T}}_{\lambda}(X))$, and of
${\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ with
$\mathrm{C}^{*}(\overline{\pi},U)$. In order to make a distinction we write
$e_{G}$ for the unit of $G$ and $e_{\mathfrak{H}}$ for the unit of
${\mathfrak{H}}$. For the first part note that
$\iota^{\rtimes}_{\ast}(\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}})=\mathbf{K}_{{\mathbf{x}},\iota^{\rtimes}_{\ast}}$
since $\iota^{\rtimes}$ is a representation of
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$. Moreover by checking the generators
we have that

$\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}\subseteq\mathbf{K}_{{\mathbf{x}},\iota^{\rtimes}_{\ast}}.$

For the reverse inclusion, let $p_{1},q_{1},\dots,p_{n},q_{n}\in P$ and
$\varepsilon,\varepsilon^{\prime}\in\\{0,1\\}$ such that
$p_{1}^{-\varepsilon}q_{1}\cdots p_{n}^{-1}q_{n}^{\varepsilon^{\prime}}=e_{G}$
and $q_{n}^{-\varepsilon^{\prime}}p_{n}\dots
q_{1}^{-1}p_{1}^{\varepsilon}P={\mathbf{x}}$, and consider an element
$b_{\mathbf{x}}$ in $\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}}$ of
the form:

$(\hat{t}^{\rtimes}\left(\overline{\pi}\left(\lambda_{p_{1}}(\xi_{p_{1}})\right)U_{{\mathfrak{h}}_{1}}\right)^{*})^{\varepsilon}\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{q_{1}}(\xi_{q_{1}})\right)U_{{\mathfrak{h}}_{2}})\cdots\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{p_{n}}(\xi_{p_{n}})\right)U_{{\mathfrak{h}}_{2n-1}})^{*}\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{q_{n}}(\xi_{q_{n}})\right)U_{{\mathfrak{h}}_{2n}})^{\varepsilon^{\prime}}.$

By the covariance relation of $(\overline{\pi},U)$, and the fact that
$\alpha_{\mathfrak{h}}(\lambda_{p}(X_{p}))=\lambda_{p}(X_{p})$ for all $p\in
P$ and ${\mathfrak{h}}\in{\mathfrak{H}}$, we can write

$\displaystyle\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}})$
$\displaystyle=(\left(\overline{\pi}\left(\lambda_{p_{1}}(\xi_{p_{1}})\right)U_{{\mathfrak{h}}_{1}}\right)^{*})^{\varepsilon}\overline{\pi}\left(\lambda_{q_{1}}(\xi_{q_{1}})\right)U_{{\mathfrak{h}}_{2}}\cdots(\overline{\pi}\left(\lambda_{p_{n}}(\xi_{p_{n}})\right)U_{{\mathfrak{h}}_{2n-1}})^{*}(\overline{\pi}\left(\lambda_{q_{n}}(\xi_{q_{n}})\right)U_{{\mathfrak{h}}_{2n}})^{\varepsilon^{\prime}}$
$\displaystyle=\overline{\pi}\left(\lambda_{p_{1}}(\eta_{p_{1}})^{*}\right)^{\varepsilon}\overline{\pi}\left(\lambda_{q_{1}}(\eta_{q_{1}})\right)\cdots\overline{\pi}\left(\lambda_{p_{n}}(\eta_{p_{n}})^{*}\right)\overline{\pi}\left(\lambda_{q_{n}}(\eta_{q_{n}})\right)^{\varepsilon^{\prime}}U_{{\mathfrak{h}}}$
$\displaystyle=\overline{\pi}\left(\left(\lambda_{p_{1}}(\eta_{p_{1}})^{*}\right)^{\varepsilon}\lambda_{q_{1}}(\eta_{q_{1}})\cdots\lambda_{p_{n}}(\eta_{p_{n}})^{*}\lambda_{q_{n}}(\eta_{q_{n}})^{\varepsilon^{\prime}}\right)U_{{\mathfrak{h}}}$

for
${\mathfrak{h}}:={\mathfrak{h}}_{1}^{-\varepsilon}{\mathfrak{h}}_{2}\cdots{\mathfrak{h}}_{2n-1}^{-1}{\mathfrak{h}}_{2n}^{\varepsilon^{\prime}}$,
and some appropriate choice of $\eta_{p_{i}}\in X_{p_{i}}$ and
$\eta_{q_{i}}\in X_{q_{i}}$ for $i=1,\dots,n$. Therefore we have that

$\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}})\in\overline{\pi}(\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}})\cdot
U_{{\mathfrak{h}}}\subseteq\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

By considering linear combinations and their norm-limits we have that

$\mathbf{K}_{{\mathbf{x}},\iota^{\rtimes}_{\ast}}\subseteq\mathbf{K}_{\mathbf{x},\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}},$

as required.

For the second part, let $p_{1},q_{1},\dots,p_{n},q_{n}\in P$ and
$\varepsilon,\varepsilon^{\prime}\in\\{0,1\\}$ such that
$p_{1}^{-\varepsilon}q_{1}\cdots p_{n}^{-1}q_{n}^{\varepsilon^{\prime}}=e_{G}$
and $q_{n}^{-\varepsilon^{\prime}}p_{n}\dots
q_{1}^{-1}p_{1}^{\varepsilon}P={\mathbf{x}}$, and consider an element
$b_{\mathbf{x}}$ in $\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}}$ of
the form:

$(\hat{t}^{\rtimes}\left(\overline{\pi}\left(\lambda_{p_{1}}(\xi_{p_{1}})\right)U_{{\mathfrak{h}}_{1}}\right)^{*})^{\varepsilon}\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{q_{1}}(\xi_{q_{1}})\right)U_{{\mathfrak{h}}_{2}})\cdots\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{p_{n}}(\xi_{p_{n}})\right)U_{{\mathfrak{h}}_{2n-1}})^{*}\hat{t}^{\rtimes}(\overline{\pi}\left(\lambda_{q_{n}}(\xi_{q_{n}})\right)U_{{\mathfrak{h}}_{2n}})^{\varepsilon^{\prime}}.$

Due to the crossed product covariance and the definition of the action we can
write

$\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}})=\left(\overline{\pi}\left(\lambda_{p_{1}}(\eta_{p_{1}})\right)^{*}\right)^{\varepsilon}\overline{\pi}\left(\lambda_{q_{1}}(\eta_{q_{1}})\right)\cdots\overline{\pi}\left(\lambda_{p_{n}}(\eta_{p_{n}})\right)^{*}\overline{\pi}\left(\lambda_{q_{n}}(\eta_{q_{n}})\right)^{\varepsilon^{\prime}}U_{{\mathfrak{h}}}$

for
${\mathfrak{h}}:={\mathfrak{h}}_{1}^{-\varepsilon}{\mathfrak{h}}_{2}\cdots{\mathfrak{h}}_{2n-1}^{-1}{\mathfrak{h}}_{2n}^{\varepsilon^{\prime}}$,
and some $\eta_{p_{i}}\in X_{p_{i}}$ and $\eta_{q_{i}}\in X_{q_{i}}$ for
$i=1,\dots,n$. By definition we have that

$E_{{\mathfrak{H}}}(\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}}))=\begin{cases}\overline{\pi}\left(\left(\lambda_{p_{1}}(\eta_{p_{1}})^{*}\right)^{\varepsilon}\lambda_{q_{1}}(\eta_{q_{1}})\cdots\lambda_{p_{n}}(\eta_{p_{n}})^{*}\lambda_{q_{n}}(\eta_{q_{n}})^{\varepsilon^{\prime}}\right)&\text{if
}{\mathfrak{h}}=e_{{\mathfrak{H}}},\\\ \hskip
2.84544pt0&\text{otherwise},\end{cases}$

and thus
$E_{{\mathfrak{H}}}(\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}}))\in\overline{\pi}\left(\mathbf{K}_{{\mathbf{x}},\lambda_{\ast}}\right)$
as required. The proof is completed by considering finite linear combinations
of elements of this form in
$\mathbf{K}_{{\mathbf{x}},\hat{t}^{\rtimes}_{\ast}}$, and their norm-limits. ∎

The following proposition is the key result of this section.

######  Proposition 4.5.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $\alpha$ be a generalised gauge action of a
discrete group ${\mathfrak{H}}$ on ${\mathcal{T}}_{\lambda}(X)$. Then the
identity representation $\iota^{\rtimes}\colon
X\rtimes_{\alpha,\lambda}{\mathfrak{H}}\to{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$
is equivariant, Fock covariant, and injective.

######  Proof..

We will show that the identity representation satisfies the conditions of
Theorem 3.2. By definition it is injective. Moreover, by [50, Lemma 7.16] we
have that

$({\mathcal{T}}_{\lambda}(X)\otimes\mathrm{C}^{*}_{\max}(G))\rtimes_{\alpha\otimes{\operatorname{id}},\lambda}{\mathfrak{H}}\simeq({\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}})\otimes\mathrm{C}^{*}_{\max}(G),$

by a canonical $*$-isomorphism, and hence we get that $\iota^{\rtimes}$ is
equivariant. It remains to show that $\iota^{\rtimes}$ satisfies conditions
(i) and (ii) of Theorem 3.2. We will make the standard identification of
${\mathcal{T}}_{\lambda}(X)$ with
$\overline{\pi}({\mathcal{T}}_{\lambda}(X))$, and of
${\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$ with
$\mathrm{C}^{*}(\overline{\pi},U)$.

For condition (i), an application of Proposition 4.4 and using that
$\mathbf{K}_{\emptyset,\lambda_{\ast}}=(0)$ yield

$\mathbf{K}_{\emptyset,\iota^{\rtimes}_{\ast}}=\mathbf{K}_{\emptyset,\lambda_{\ast}}\rtimes_{\alpha,\lambda}{\mathfrak{H}}=(0).$

For condition (ii), let
${\mathcal{F}}=\\{{\mathbf{x}}_{1},\dots,{\mathbf{x}}_{n}\\}$ be a finite
$\cap$-closed subset of ${\mathcal{J}}$ such that
$\bigcup_{i=1}^{n}\mathbf{x}_{i}\neq\emptyset$ and let
$b_{{\mathbf{x}}_{i}}\in\mathbf{K}_{{\mathbf{x}}_{i},\hat{t}^{\rtimes}_{\ast}}$
for $i=1,\dots,n$ that satisfy

$\sum_{i:r\in{\mathbf{x}}_{i}}\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}_{i}})\iota^{\rtimes}_{r}\left(\overline{\pi}\left(\lambda_{r}(X_{r}\right)U_{{\mathfrak{h}}}\right)=(0)\text{
for all }r\in\bigcup_{i=1}^{n}{\mathbf{x}}_{i}\text{ and
}{\mathfrak{h}}\in{\mathfrak{H}}.$

We will show that
$\sum_{i=1}^{n}\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}_{i}})=0$. For notational
convenience, set

$c:=\sum_{i=1}^{n}\iota^{\rtimes}_{\ast}(b_{{\mathbf{x}}_{i}})\quad\text{and}\quad
c_{r}:=\sum_{i:r\in\mathbf{x}_{i}}\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}})\text{
for each }r\in\bigcup_{i=1}^{n}{\mathbf{x}}_{i}.$

Since
$b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}}\in\mathbf{K}_{{\mathbf{x}}_{i}\cap{\mathbf{x}}_{j},\hat{t}^{\rtimes}_{\ast}}$,
by Proposition 4.4 we have that

$E_{{\mathfrak{H}}}\left(\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}})\right)\in\overline{\pi}(\mathbf{K}_{{\mathbf{x}}_{i}\cap{\mathbf{x}}_{j},\lambda_{\ast}}),$

and in particular we can write

$E_{{\mathfrak{H}}}\left(\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}})\right)=\overline{\pi}\left(\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\right),\text{
for some
}d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}}\in\mathbf{K}_{\mathbf{x}_{i}\cap\mathbf{x}_{j},\hat{t}_{\ast}}.$

Therefore we have that

$E_{{\mathfrak{H}}}(c^{*}c)=\sum_{i,j=1}^{n}E_{{\mathfrak{H}}}(\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}}))=\overline{\pi}\left(\sum_{i,j=1}^{n}\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\right)\in\overline{\pi}\left(\mathbf{B}_{{\mathcal{F}},\lambda_{\ast}}\right),$

where we used that
$\\{\mathbf{x}_{i}\cap\mathbf{x}_{j}:i,j=1,\dots,n\\}={\mathcal{F}}$ for the
inclusion, since ${\mathcal{F}}$ is $\cap$-closed. Now let

$r\in\bigcup_{i,j=1}^{n}(\mathbf{x}_{i}\cap\mathbf{x}_{j})=\bigcup_{i=1}^{n}\mathbf{x}_{i}.$

By using that ${\mathcal{F}}$ is $\cap$-closed we obtain that

$\\{(i,j)\in[n]^{2}\mid r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}\\}=\\{i\in[n]\mid
r\in\mathbf{x}_{i}\\}\times\\{j\in[n]\mid r\in\mathbf{x}_{j}\\},$

and hence

$\displaystyle E_{{\mathfrak{H}}}(c_{r}^{*}c_{r})$
$\displaystyle=\sum_{i:r\in\mathbf{x}_{i}}\sum_{j:r\in\mathbf{x}_{j}}E_{{\mathfrak{H}}}\left(\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}})\right)=\sum_{i,j:r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}}E_{{\mathfrak{H}}}\left(\iota^{\rtimes}_{\ast}(b_{\mathbf{x}_{i}}^{*}b_{\mathbf{x}_{j}})\right)=\sum_{i,j:r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}}\overline{\pi}\left(\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\right).$

Since $r\in\bigcup_{i=1}^{n}\mathbf{x}_{i}$, by using the assumption we have
that $c_{r}\overline{\pi}(\lambda_{r}(\xi_{r}))=0$ for all $\xi_{r}\in X_{r}$.
Thus we get

$\displaystyle\overline{\pi}(\lambda_{r}(\xi_{r}))^{*}E_{{\mathfrak{H}}}(c_{r}^{*}c_{r})\overline{\pi}(\lambda_{r}(\xi_{r}))=E_{{\mathfrak{H}}}\left(\overline{\pi}(\lambda_{r}(\xi_{r}))^{*}c_{r}^{*}c_{r}\overline{\pi}(\lambda_{r}(\xi_{r}))\right)=0,$

where we used that ${\mathcal{T}}_{\lambda}(X)$ is in the multiplicative
domain of $E_{\mathfrak{H}}$. Consequently, we get that
$E_{{\mathfrak{H}}}(c_{r}^{*}c_{r})^{1/2}\overline{\pi}(\lambda_{r}(\xi_{r}))=0$
and in particular

$\overline{\pi}\left(\sum_{i,j:r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}}\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\lambda_{r}(\xi_{r})\right)=\sum_{i,j:r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}}\overline{\pi}\left(\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\right)\overline{\pi}(\lambda_{r}(\xi_{r}))=E_{{\mathfrak{H}}}(c_{r}^{*}c_{r})\overline{\pi}(\lambda_{r}(\xi_{r}))=0.$

Since $\overline{\pi}$ is faithful we deduce that

$\sum_{i,j:r\in\mathbf{x}_{i}\cap\mathbf{x}_{j}}\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\lambda_{r}(X_{r})=(0)\text{
for all }r\in\bigcup_{i,j=1}^{n}(\mathbf{x}_{i}\cap\mathbf{x}_{j}).$

Since $\lambda$ satisfies condition (ii) of Theorem 3.2 we get that
$\sum_{i,j=1}^{n}\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})=0$, and
therefore

$E_{{\mathfrak{H}}}(c^{*}c)=\overline{\pi}\left(\sum_{i,j=1}^{n}\lambda_{\ast}(d_{\mathbf{x}_{i}\cap\mathbf{x}_{j}})\right)=0.$

Faithfulness of $E_{{\mathfrak{H}}}$ then implies that $c=0$, as required. ∎

Since $A\times_{X,\lambda}P$ is the C*-envelope of
${\mathcal{T}}_{\lambda}(X)^{+}$ it inherits a group action $\dot{\alpha}$
from ${\mathfrak{H}}$. The reduced Hao–Ng isomorphism problem asks if there
exists a canonical $*$-isomorphism such that

$(A\rtimes_{\alpha,\lambda}{\mathfrak{H}})\times_{X\rtimes_{\alpha,\lambda}{\mathfrak{H}},\lambda}P\stackrel{{\scriptstyle?}}{{\simeq}}(A\times_{X,\lambda}P)\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}}.$

By [48, Theorem 5.1] this is equivalent to asking for a canonical
$*$-isomorphism

(4.2)
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})^{+})\stackrel{{\scriptstyle?}}{{\simeq}}\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}},$

which by Proposition 4.2 is equivalent to the existence of a canonical
completely isometric isomorphism

(4.3)
${\mathcal{T}}_{\lambda}(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})^{+}\simeq{\mathcal{T}}_{\lambda}(X)^{+}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

A careful investigation of the arguments in [11, 12, 24, 25, 28] suggests that
it is enough to show that the embedding
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}\hookrightarrow{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$
is Fock covariant, without passing through a $*$-isomorphism of their ambient
C*-algebras.

######  Theorem 4.6.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $\alpha$ be a generalised gauge action of a
discrete group ${\mathfrak{H}}$ on ${\mathcal{T}}_{\lambda}(X)$. Then the
identity representation $\iota^{\rtimes}\colon
X\rtimes_{\alpha,\lambda}{\mathfrak{H}}\to{\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}}$
lifts to a completely isometric isomorphism

${\mathcal{T}}_{\lambda}(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})^{+}\simeq{\mathcal{T}}_{\lambda}(X)^{+}\rtimes_{\alpha,\lambda}{\mathfrak{H}}.$

Consequently, the reduced Hao–Ng isomorphism problem has an affirmative
answer, i.e.,

$(A\rtimes_{\alpha,\lambda}{\mathfrak{H}})\times_{X\rtimes_{\alpha,\lambda}{\mathfrak{H}},\lambda}P\simeq(A\times_{X,\lambda}P)\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}},$

by a canonical $*$-isomorphism, where $\dot{\alpha}$ is the induced action of
${\mathfrak{H}}$ on $A\times_{X,\lambda}P$.

######  Proof..

By Proposition 4.5 the identity representation is an equivariant Fock
covariant injective representation of
$X\rtimes_{\alpha,\lambda}{\mathfrak{H}}$. Moreover, by [50, Lemma 7.16] we
have that

$({\mathcal{T}}_{\lambda}(X)\otimes\mathrm{C}^{*}_{\lambda}(G))\rtimes_{\alpha\otimes{\operatorname{id}},\lambda}{\mathfrak{H}}\simeq({\mathcal{T}}_{\lambda}(X)\rtimes_{\alpha,\lambda}{\mathfrak{H}})\otimes\mathrm{C}^{*}_{\lambda}(G),$

by a canonical $*$-isomorphism, and hence we get that the coaction that
$\iota^{\rtimes}$ admits is normal. Since

${\mathcal{T}}_{\lambda}(X)^{+}\rtimes_{\alpha,\lambda}{\mathfrak{H}}=\overline{\operatorname{alg}}\\{\iota^{\rtimes}_{p}(X_{p}\rtimes_{\alpha,\lambda}{\mathfrak{H}})\mid
p\in P\\},$

the required completely isometric isomorphism is induced by Corollary 4.1. By
Proposition 4.2 we then have that

$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X\rtimes_{\alpha,\lambda}{\mathfrak{H}})^{+})\simeq\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+})\rtimes_{\dot{\alpha},\lambda}{\mathfrak{H}},$

and [48, Theorem 5.1] finishes the proof. ∎

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