Proposition 4.16. Conversely suppose that $t_{\ast}$ is a $*$-representation
of ${\mathcal{T}}(X)=\mathrm{C}^{*}(\widetilde{t})$ that factors through
$A\times_{X}P$. Without loss of generality, and due to Proposition 4.15, we
can assume that $t$ is a representation of
${\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)$ and thus Fock-
covariant. Proposition 4.17 shows that, if
$b\in[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}$ is such
that $t_{\ast}(b)=0$, then $q_{\operatorname{sc}}(b)=0$, as required. ∎

Finally, one of the important points of Sehnem’s theory is that $A\times_{X}P$
does not depend on the group $G$ where $P$ embeds [68, Lemma 3.9]. The proof
in our case is the same, given that we have the analogues of [68, Proposition
3.5 and Lemma 3.6]. We include it here as we will need to add a comment and
also be explicit that this independence descends also to the reduced version
$A\times_{X,\lambda}P$ which is not discussed in [68].

######  Proposition 4.19.

Let $P$ be a unital subsemigroup and let $X$ be a product system over $P$. Let
$G$ be the enveloping group of $P$ and suppose that $\gamma\colon G\to P$ is a
group homomorphism that is injective on $P$. If $r\in P$ and $g\in G$ satisfy
$K_{\\{r,g\\}}\neq\emptyset$ and $r\notin K_{\\{r,g\\}}$, then
$K_{\\{\gamma(r),\gamma(g)\\}}\neq\emptyset$ and $\gamma(r)\notin
K_{\\{\gamma(r),\gamma(g)\\}}$.

######  Proof..

It is clear that with these assumptions we have that
$K_{\\{\gamma(r),\gamma(g)\\}}\neq\emptyset$ and also there are
$q_{1},q_{2}\in P$ such that $rq_{1}=gq_{2}$ with $q_{2}q_{1}^{-1}\notin P$.
If it were that $\gamma(r)\in K_{\\{\gamma(r),\gamma(g)\\}}$ then
$\gamma(r)=\gamma(rq_{1}q_{2}^{-1}s)$ for some $s\in P$, and thus that
$\gamma(q_{2})=\gamma(sq_{1})$. As $\gamma|_{P}$ is one-to-one we get the
contradiction $q_{2}q_{1}^{-1}=s\in P$. ∎

Although the details of this step are omitted from the proof of [68, Lemma
3.9], we find that they elucidate the independence from the group. Practically
the elements of the group to be considered come from $Pq_{1}q_{2}^{-1}$ with
$q_{1}q_{2}^{-1}\notin P^{-1}$, and this subset of $P\cdot P^{-1}$ cannot give
non-trivial relations in a group generated by $P$.

######  Proposition 4.20.

Let $P$ be a unital subsemigroup and let $X$ be a product system over $P$.
Suppose that $P$ admits two group embeddings $i_{G}\colon P\to G$ and
$i_{H}\colon P\to H$. Then there exists a $*$-isomorphism

$\mathrm{C}^{*}_{\max}({\mathcal{S}}{\mathcal{C}}_{G}X)\longrightarrow\mathrm{C}^{*}_{\max}({\mathcal{S}}{\mathcal{C}}_{H}X);\xi_{i_{G}(p)}\mapsto\xi_{i_{H}(p)},$

which descends to $*$-isomorphism

$\mathrm{C}^{*}_{\lambda}({\mathcal{S}}{\mathcal{C}}_{G}X)\longrightarrow\mathrm{C}^{*}_{\lambda}({\mathcal{S}}{\mathcal{C}}_{H}X);\xi_{i_{G}(p)}\mapsto\xi_{i_{H}(p)}.$

In particular, there exists a Fell bundle isomorphism
${\mathcal{S}}{\mathcal{C}}_{G}X\to{\mathcal{S}}{\mathcal{C}}_{H}X$ such that

$[A\times_{X}P]_{i_{G}(p_{1})^{-1}i_{G}(q_{1})\cdots
i_{G}(p_{n})^{-1}i_{G}(q_{n})}\longrightarrow[A\times_{X}P]_{i_{H}(p_{1})^{-1}i_{H}(q_{1})\cdots
i_{H}(p_{n})^{-1}i_{H}(q_{n})},$

isometrically for every $p_{1},q_{1},\dots,p_{n},q_{n}\in P$.

######  Proof..

Let $G$ be the enveloping group of $P$ and $\gamma\colon G\to H$ be the
canonical epimorphism that fixes $P$. Let us denote by
${\mathcal{I}}_{\operatorname{sc},e_{G}}$ and by
${\mathcal{I}}_{\operatorname{sc},e_{H}}$ the strong covariance relations that
define the $G$-strongly covariant and the $H$-strongly covariant ideals. We
will show that
${\mathcal{I}}_{\operatorname{sc},e_{G}}\subseteq{\mathcal{I}}_{\operatorname{sc},e_{H}}$.
Recall that these ideals are kernels of the maps

$[{\mathcal{T}}(X)]_{e_{G}}\longrightarrow[{\mathcal{T}}_{\lambda}(X)]_{e_{G}}\longrightarrow{\raisebox{1.00006pt}{$\prod\limits_{\textup{fin
}F\subseteq
G}{\mathcal{L}}(X_{F})$}\left/\,\raisebox{-1.00006pt}{$c_{0}({\mathcal{L}}(X_{F})\mid\textup{fin
}F\subseteq G)$}\right.},$

and

$[{\mathcal{T}}(X)]_{e_{H}}\longrightarrow[{\mathcal{T}}_{\lambda}(X)]_{e_{H}}\longrightarrow{\raisebox{1.00006pt}{$\prod\limits_{\textup{fin
}\gamma(F)\subseteq
H}{\mathcal{L}}(X_{\gamma(F)})$}\left/\,\raisebox{-1.00006pt}{$c_{0}({\mathcal{L}}(X_{\gamma(F)})\mid\textup{fin
}\gamma(F)\subseteq H)$}\right.}.$

By Proposition 4.5 we have that
$[{\mathcal{T}}_{\lambda}(X)]_{e_{G}}=[{\mathcal{T}}_{\lambda}(X)]_{e_{H}}$,
and thus it suffices to show that $X_{\gamma(F)}\subseteq X_{F}$ for every
finite $F\subseteq G$. Towards this end it suffices to show that

$I_{\gamma(r)^{-1}K_{\\{\gamma(r),\gamma(g)\\}}}\subseteq
I_{r^{-1}K_{\\{r,g\\}}}\text{ for all }r\in P,g\in G.$

If $K_{\\{r,g\\}}=\emptyset$, or if $r\in K_{\\{r,g\\}}$, then
$I_{r^{-1}K_{\\{r,g\\}}}=A\supseteq
I_{\gamma(r)^{-1}K_{\\{\gamma(r),\gamma(g)\\}}}$. If
$K_{\\{r,g\\}}\neq\emptyset$ and $r\notin K_{\\{r,g\\}}$ then Proposition 4.19
yields $K_{\\{\gamma(r),\gamma(g)\\}}\neq\emptyset$ and $\gamma(r)\notin
K_{\\{\gamma(r),\gamma(g)\\}}$. Since $K_{\\{r,g\\}}\subseteq P$ and
$K_{\\{\gamma(r),\gamma(g)\\}}\subseteq\gamma(P)$ then $\gamma$ defines a
bijection between these two sets. Thus we have

$\displaystyle I_{\gamma(r)^{-1}K_{\\{\gamma(r),\gamma(g)\\}}}$
$\displaystyle=\bigcap_{\gamma(s)\in
K_{\\{\gamma(r),\gamma(g)\\}}}\ker\phi_{\gamma(r)^{-1}\gamma(s)}=\bigcap_{s\in
K_{\\{r,g\\}}}\ker\phi_{r^{-1}s}=I_{r^{-1}K_{\\{r,g\\}}}.$

Therefore, there exists a $*$-epimorphism

$\Phi_{\gamma}\colon\mathrm{C}^{*}_{\max}({\mathcal{S}}{\mathcal{C}}_{G}X)\longrightarrow\mathrm{C}^{*}_{\max}({\mathcal{S}}{\mathcal{C}}_{H}X)$

that fixes every $X_{p}$, and thus
$\Phi_{\gamma}([{\mathcal{S}}{\mathcal{C}}_{G}X]_{g})\subseteq[{\mathcal{S}}{\mathcal{C}}_{H}X]_{\gamma(g)}$.
Hence it descends to a $*$-epimorphism on the reduced C*-algebras

$\dot{\Phi}_{\gamma}\colon\mathrm{C}^{*}_{\lambda}({\mathcal{S}}{\mathcal{C}}_{G}X)\longrightarrow\mathrm{C}^{*}_{\lambda}({\mathcal{S}}{\mathcal{C}}_{H}G).$

By Proposition 4.16 $\dot{\Phi}_{\gamma}|_{A}$ is faithful, and Proposition
4.18 yields that $\dot{\Phi}_{\gamma}$ is faithful on
$[{\mathcal{S}}{\mathcal{C}}_{G}X]_{e}$. By construction $\dot{\Phi}_{\gamma}$
intertwines the faithful conditional expectations, and so
$\dot{\Phi}_{\gamma}$ is faithful. ∎

##  5\. Co-universality

###  5.1. Boundary represenations

Our goal is to show that the reduced strongly covariant representation is
terminal for the class of equivariant injective covariant represenations (and
thus also for the corresponding Fock-covariant). The notion of terminal
representations appears in several contexts and constructions of C*-algebras,
sometimes under disguise. Let us set a general framework and provide some
examples.

######  Definition 5.1.

Let ${\mathcal{F}}$ be a class of $*$-representations of a C*-algebra. We say
that $\Phi$ is _${\mathcal{F}}$ -boundary_ if and only if $\Phi$ is a member
of the class ${\mathcal{F}}$, and $\Phi$ factors through every member $\Psi$
of the class ${\mathcal{F}}$.

Boundary representations generate unique _co-universal C*-algebras_. We will
limit the amount of detail in the following examples. The interested reader
may refer to the cited works for more information.

######  Example 5.2.

Let $A,B$ be C*-algebras. If $\left\|\cdot\right\|_{\gamma}$ is a C*-norm on
the algebraic tensor product $A\odot B$, then we write $A\otimes_{\gamma}B$
for the $\left\|\cdot\right\|_{\gamma}$-completion of $A\odot B$. By
considering “all” the injective $*$-algebraic representations of $A\odot B$ on
Hilbert spaces, we obtain the maximal C*-norm on $A\odot B$ that dominates all
other C*-norms. Then the spatial tensor product norm is boundary for the
$*$-representations of $A\otimes_{\max}B$ that are injective on $A\odot B$.

######  Example 5.3.

Let $I\lhd A$ be a two-sided closed ideal in a C*-algebra $A$ and consider the
short exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
49.68054pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.68054pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.68054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
87.18054pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.18054pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.18054pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A/I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
154.30545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
154.30545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

A short exact sequence $*$-homomorphism of $I\lhd A$ corresponds to a pair
$I^{\prime}\lhd A^{\prime}$ that makes the following diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}/I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

commutative. The maximal tensor product is exact in the sense that it gives a
short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\otimes_{\max}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{\max}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/I\otimes_{\max}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for all C*-algebras $B$. If $A/I$ is nuclear or if $B$ is exact, then the pair
$I\otimes B\lhd A\otimes B$ induces a boundary representation for the short
exact sequence $*$-homomorphisms of $I\otimes_{\max}B\lhd A\otimes_{\max}B$
that are injective on $A\odot B$.

######  Example 5.4.

Let ${\mathfrak{A}}$ be an operator algebra. Let
${\mathcal{F}}_{{\mathfrak{A}}}$ be the class of the $*$-representations of
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})$ that are completely isometric on
${\mathfrak{A}}$. Then the canonical $*$-representation
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})\to\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$
that fixes ${\mathfrak{A}}$ is an ${\mathcal{F}}_{{\mathfrak{A}}}$-boundary
representation. This follows from the existence of the C*-envelope by Hamana
[29]. A similar result holds for operator systems, i.e., unital selfadjoint
subspaces of some ${\mathcal{B}}(H)$. The existence of the maximal C*-algebra
of an operator system is shown by Kirchberg–Wasserman [45], and the existence
of the C*-envelope of an operator system is shown by Arveson [5], simplifying
the arguments of Dritschel-McCullough [21]. For non-unital operator systems
the reader may refer to [44].

######  Example 5.5.

Let ${\mathfrak{A}}$ be an operator algebra and $\alpha$ be an action by a
locally compact Hausdorff group $G$ on ${\mathfrak{A}}$ by completely
isometric automorphisms. Then $\alpha$ extends to
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})$ and to
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$. Let
${\mathcal{F}}_{{\mathfrak{A}}}^{\alpha}$ be the class of the
$*$-representations of $\mathrm{C}^{*}_{\max}({\mathfrak{A}})$ that are
completely isometric on ${\mathfrak{A}}$ and admit an equivariant action of
$G$. Then the $*$-representation
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})\to\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$
that fixes ${\mathfrak{A}}$ is an
${\mathcal{F}}_{{\mathfrak{A}}}^{\alpha}$-boundary representation. This
follows by the C*-envelope property and the automatic extension of $\alpha$ on
$\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})$.

######  Example 5.6.

Let ${\mathfrak{A}}$ be an operator algebra and $\delta$ be a coaction of a
discrete group $G$ on ${\mathfrak{A}}$. Then it induces a coaction on
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})$. Let
${\mathcal{F}}_{{\mathfrak{A}}}^{\delta}$ be the class of the
$*$-representations of $\mathrm{C}^{*}_{\max}({\mathfrak{A}})$ that are
completely isometric on ${\mathfrak{A}}$ and admit an equivariant coaction of
$G$. In [19] it is shown that the $*$-representation
$\mathrm{C}^{*}_{\max}({\mathfrak{A}})\to\mathrm{C}^{*}_{\textup{env}}({\mathfrak{A}})\otimes\mathrm{C}^{*}_{\max}(G)$
that fixes ${\mathfrak{A}}$ is an
${\mathcal{F}}_{{\mathfrak{A}}}^{\delta}$-boundary representation.

######  Example 5.7.

Let ${\mathcal{B}}=\\{{\mathcal{B}}_{g}\\}_{g\in G}$ be a Fell bundle over a
discrete group $G$. Let ${\mathcal{F}}_{\mathcal{B}}^{\delta}$ be the class of
representations of $\mathrm{C}^{*}_{\max}({\mathcal{B}})$ that are injective
on ${\mathcal{B}}_{e}$ and admit an equivariant coaction of $G$. Then the left
regular representation
$\lambda\colon\mathrm{C}^{*}_{\max}({\mathcal{B}})\to\mathrm{C}^{*}_{\lambda}({\mathcal{B}})$
is ${\mathcal{F}}_{\mathcal{B}}^{\delta}$-boundary. This is the co-universal
result of Exel [23]. Fell bundles encode a great number of examples such as
C*-crossed products, partial crossed products and Hilbert bimodules.

######  Example 5.8.

Let $P$ be a subsemigroup of a discrete group of $G$. Let
${\mathcal{F}}_{s}^{\delta}$ be the class of constructible non-zero
representations of $\mathrm{C}^{*}_{s}(P)$ in the sense of Li [52] that admit
an equivariant coaction of $G$. In [33] it is shown that the canonical
$*$-representation
$\mathrm{C}^{*}_{s}(P)\to\mathrm{C}(\partial\Omega_{P})\rtimes_{r}G$ that
fixes $P$ is an ${\mathcal{F}}_{s}^{\delta}$-boundary representation. Here
$\partial\Omega_{P}$ is the smallest $G$-invariant subspace in the spectrum of
$[\mathrm{C}^{*}_{s}(P)]_{e}$. Moreover, it is shown that
$\mathrm{C}(\partial\Omega_{P})\rtimes_{r}G\simeq\mathrm{C}^{*}_{\textup{env}}({\mathcal{A}}(P),G,\delta)$.

The following four examples concern successively stronger results, but each
one with an essentially different approach.

######  Example 5.9.

Let $X$ be a single C*-correspondence. We write ${\mathcal{T}}(X)$ for the
universal C*-algebra with respect to the representations of $X$ and
${\mathcal{O}}(X)$ for the universal C*-algebra with respect to the covariant
representations of $X$ in the sense of Katsura [42]. Let
${\mathcal{F}}_{X}^{\mathbb{T}}$ be the class of representations of
${\mathcal{T}}(X)$ that are injective on $X$ and admit an equivariant action
of $\mathbb{T}$. Then the Cuntz–Pimsner representation
${\mathcal{T}}(X)\to{\mathcal{O}}(X)$ is an
${\mathcal{F}}_{X}^{\mathbb{T}}$-boundary representation. This follows from
the study of the gauge-invariant ideals of ${\mathcal{T}}(X)$ by Katsura [43].
This class contains (topological) graph C*-algebras and C*-algebras of
$\mathbb{Z}$-dynamics.

######  Example 5.10.

Let $(G,P)$ be a quasi-lattice. Let $X$ be a compactly aligned product system
over $P$ in the sense of Fowler [27]. We write ${\mathcal{N}}{\mathcal{T}}(X)$
for the universal C*-algebra with respect to the Nica-covariant
representations of $X$ and ${\mathcal{N}}{\mathcal{O}}(X)$ for the universal
C*-algebra with respect to the Cuntz–Nica–Pimsner representations of $X$ in
the sense of Sims–Yeend [72]. Then ${\mathcal{N}}{\mathcal{O}}(X)$ is
equivariant and let ${\mathcal{N}}{\mathcal{O}}(X)^{r}$ be the reduced
C*-algebra of the bundle induced by this coaction. Let
${\mathcal{F}}_{{\mathcal{N}}X}^{\delta}$ be the class of representations of
${\mathcal{N}}{\mathcal{T}}(X)$ that are injective on $X$ and admit an
equivariant coaction of $G$. Carlsen–Larsen–Sims–Vittadello [9] show that the
Cuntz–Nica–Pimsner representation
${\mathcal{N}}{\mathcal{T}}(X)\to{\mathcal{N}}{\mathcal{O}}(X)^{r}$ is an
${\mathcal{F}}_{{\mathcal{N}}X}^{\delta}$-boundary representation, when $X$ is
injective, or when $P$ is directed and $X$ is $\widetilde{\phi}$-injective.
This follows by studying the structure of the augmented Fock space
representation of Sims–Yeend [72]. This class contains the case
$P=\mathbb{Z}_{+}^{d}$, e.g., C*-algebras from higher rank (topological)
graphs.

######  Example 5.11.

Let $(G,P)$ be an abelian lattice. Let $X$ be a compactly aligned product
system over $P$ in the sense of Fowler [27]. We write
${\mathcal{N}}{\mathcal{T}}(X)$ for the universal C*-algebra with respect to
the Nica-covariant representations of $X$. Let
${\mathcal{F}}_{{\mathcal{N}}X}^{\widehat{G}}$ be the class of representations
of ${\mathcal{N}}{\mathcal{T}}(X)$ that are injective on $X$ and admit an
equivariant action of the dual group $\widehat{G}$. In [20] it is shown that
the representation
${\mathcal{N}}{\mathcal{T}}(X)\to\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+})$
is an ${\mathcal{F}}_{{\mathcal{N}}X}^{\widehat{G}}$-boundary representation.
Furthermore, it is shown that
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+})$ coincides with
$A\times_{X}P$. The latter follows by the co-universal properties of
$A\times_{X}P$ and
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+})$.

######  Example 5.12.

Let $P$ be a right LCM semigroup that is a subsemigroup of a discrete group
$G$. Let $X$ be a compactly aligned product system over $P$ in the sense of
Kwaśniewski–Larsen [47]. We write ${\mathcal{N}}{\mathcal{T}}(X)$ for the
universal C*-algebra with respect to the Nica-covariant representations of
$X$. Let ${\mathcal{F}}_{{\mathcal{N}}X}^{\delta}$ be the class of
representations of ${\mathcal{N}}{\mathcal{T}}(X)$ that are injective on $X$
and admit an equivariant coaction of $G$. In [19] it is shown that the
representation
${\mathcal{N}}{\mathcal{T}}(X)\to\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$
that fixes $X$ is an ${\mathcal{F}}_{{\mathcal{N}}X}^{\delta}$-boundary
representation. Furthermore, it is shown that
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$
coincides with $A\times_{X,\lambda}P$. These representations differ in general
from the augmented Fock representation of Sims–Yeend [72] and
${\mathcal{N}}{\mathcal{O}}(X)^{r}$ of [9]; see for example [9, Example 3.9].

The following example shows that it is not straightforward to develop
weak*-analogues of boundary representations.

######  Remark 5.13.

In analogy to the C*-case, Wiersma [73] introduced a weak* tensor product
theory for von Neumann algebras. The corresponding tensor products for von
Neumann algebras $M$ and $N$ are taken with respect to injective
$*$-representations of $M\odot N$ that are normal on $M$ and $N$. In [73] it
is shown that $(M\otimes_{\max}N)^{**}$ defines a maximal weak* tensor product
$\overline{\otimes}_{\textup{w*-}\max}$. However, the collection of the normal
representations of $M\overline{\otimes}_{\textup{w*-}\max}N$ that are
injective on $M\odot N$ may not have a boundary element. An example is given
in [73, Remark 2.6] by considering $N=M^{\prime}$ for a factor $M$. Then the
multiplication map $m\colon M\odot M^{\prime}\to{\mathcal{B}}(H)$ defines a
weak* tensor product $\overline{\otimes}_{m}$, but
$M\overline{\otimes}_{m}M^{\prime}$ cannot be a normal quotient of
$M\overline{\otimes}M^{\prime}$ when $M$ is not of Type I. Nevertheless, both
$M\overline{\otimes}_{m}M^{\prime}$ and $M\overline{\otimes}M^{\prime}$ give
minimal weak*-tensor products being factors.

###  5.2. Co-universality for product systems through the C*-envelope

We will now show how the C*-envelope of the cosystem on the Fock tensor
algebra can be used to answer the co-universality problem for covariant
representations. This first proof we give follows the strategy of [18, 19]
through the completely contractive maps of the Fock tensor algebra and an
induced coaction by $\mathrm{C}^{*}_{\lambda}(P)$.

Recall that Fell’s absorption principle for groups requires an intertwining
unitary. We are going to establish a similar result for product systems
(actually half of it). In the case of $X_{p}=\mathbb{C}$ for all $p\in P$ we
recover [11, Lemma 5.3].

######  Proposition 5.14.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $t$ is a representation of $X$, then the isometry

$W\colon{\mathcal{F}}X\otimes_{A}H\longrightarrow
H\otimes\ell^{2}(P);\xi_{q}\otimes h\mapsto t(\xi_{q})h\otimes\delta_{q},$

satisfies

$W\lambda(\xi_{p})\otimes I_{H}=(t(\xi_{p})\otimes V_{p})W\text{ for all }p\in
P.$

######  Proof..

A straightforward computation for $\xi_{p},\eta_{q}\in{\mathcal{F}}X$ and
injectivity of $t$ gives that

$\left\langle W\xi_{p}\otimes h_{p},W\xi_{q}\otimes
h_{q}\right\rangle=\delta_{p,q}\left\langle
h_{p},t(\xi_{p})^{*}t(\xi_{q})h_{q}\right\rangle=\left\langle\xi_{p}\otimes
h_{p},\xi_{q}\otimes h_{q}\right\rangle_{{\mathcal{F}}X\otimes_{A}H},$

Thus $W$ is a well-defined isometry. We then compute

$W(\lambda(\xi_{p})\otimes I_{H})(\xi_{q}\otimes
h)=t(\xi_{p}\xi_{q})h\otimes\delta_{pq}=(t(\xi_{p})\otimes
V_{p})W(\xi_{q}\otimes h)$

as required, and the proof is complete. ∎

######  Proposition 5.15.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $t$ is an equivariant representation of $X$ that
is injective on $A$, then there exists a completely contractive homomorphism

$\overline{\operatorname{alg}}\\{t(X_{p})\mid p\in
P\\}\longrightarrow{\mathcal{L}}({\mathcal{F}}X);t(\xi_{p})\mapsto\lambda(\xi_{p}).$

######  Proof..

First we construct a completely contractive map

$\Phi\colon\overline{\operatorname{alg}}\\{t(X_{p})\otimes V_{p}\mid p\in
P\\}\longrightarrow{\mathcal{L}}({\mathcal{F}}X);t(\xi_{p})\otimes
V_{p}\mapsto\lambda(\xi_{p}).$

Let $W$ be the intertwining isometry from Proposition 5.14 with final space
densely spanned by $t(\xi_{q})h\otimes\delta_{q}$. Hence $WW^{*}$ is an
invariant subspace for $\operatorname{alg}\\{t(X_{p})\otimes V_{p}\mid p\in
P\\}$. Moreover, Proposition 5.14 yields

$(t(\xi_{p})\otimes V_{p})|_{WW^{*}}=W\left(\lambda(\xi_{p})\otimes
I_{H}\right)W^{*},$

and injectivity of $t$ on $A$ implies that the map

$\otimes{\operatorname{id}}_{H}\colon{\mathcal{L}}({\mathcal{F}}X)\longrightarrow{\mathcal{L}}({\mathcal{F}}X\otimes_{A}H);S\mapsto
S\otimes{\operatorname{id}}_{H}$

is a faithful $*$-homomorphism [50, p. 42]. Thus we can define the following
completely contractive homomorphism

$\Phi\colon\operatorname{alg}\\{t(X_{p})\otimes V_{p}\mid p\in
P\\}\stackrel{{\scriptstyle|_{WW^{*}}}}{{\longrightarrow}}\operatorname{alg}\\{W(\lambda(X_{p})\otimes
I_{H})W^{*}\mid p\in
P\\}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{T}}_{\lambda}(X)^{+}\otimes
I_{H}\simeq{\mathcal{T}}_{\lambda}(X)^{+},$

so that $\Phi(t(\xi_{p})\otimes V_{p})=\lambda(\xi_{p})$.

Next consider the unital completely positive map

$\phi\colon\mathrm{C}^{*}_{\max}(G)\longrightarrow\mathrm{C}^{*}_{\lambda}(G)\longrightarrow{\mathcal{B}}(\ell^{2}(P));u_{g}\mapsto\lambda_{g}\mapsto
P_{\ell^{2}(P)}\lambda_{g}|_{\ell^{2}(P)},$

which is multiplicative on the subalgebra of $\mathrm{C}^{*}_{\max}(G)$
generated by the $u_{p}$ for $p\in P$, with $\phi(u_{p})=V_{p}$. Then the map
$\Phi({\operatorname{id}}\otimes\phi)\delta_{t}$ for the coaction $\delta_{t}$
of $t$ satisfies

$\Phi({\operatorname{id}}\otimes\phi)\delta_{t}t(\xi_{p})=\Phi({\operatorname{id}}\otimes\phi)(t(\xi_{p})\otimes
u_{p})=\Phi(t(\xi_{p})\otimes V_{p})=\lambda(\xi_{p}),$

and gives the required completely contractive homomorphism. ∎

As a consequence, we obtain a co-action by $\mathrm{C}^{*}_{\lambda}(P)$ on
the Fock tensor algebra where the target space can be over any equivariant
injective representation of ${\mathcal{T}}_{\lambda}(X)$. The following is the
analogue of the key result [19, Proposition 4.5].

######  Proposition 5.16.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $\Psi$ is an equivariant $*$-representation of
${\mathcal{T}}_{\lambda}(X)$ that is injective on $A$, then there exists a
completely isometric homomorphism

${\mathcal{T}}_{\lambda}(X)^{+}\longrightarrow\mathrm{C}^{*}(\Psi)\otimes\mathrm{C}^{*}_{\lambda}(P);\lambda(\xi_{p})\mapsto\Psi\lambda(\xi_{p})\otimes
V_{p}.$

In particular, $\mathrm{C}^{*}(\Psi)$ is a C*-cover of the cosystem
$({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$.

######  Proof..

First we note that $\Psi\lambda\otimes
V\colon\xi_{p}\mapsto\Psi\lambda(\xi_{p})\otimes V_{p}$ defines a
representation of $X$. Moreover it is equivariant as $\Psi$ is so. Let $\Phi$
be the completely contractive homomorphism constructed in the proof of
Proposition 5.15, so that $\Phi(\Psi\lambda(\xi_{p})\otimes
V_{p})=\lambda(\xi_{p})$. Next consider the unital completely positive map

$\phi\colon\mathrm{C}^{*}_{\max}(G)\longrightarrow\mathrm{C}^{*}_{\lambda}(G)\longrightarrow{\mathcal{B}}(\ell^{2}(P));u_{g}\mapsto\lambda_{g}\mapsto
P_{\ell^{2}(P)}\lambda_{g}|_{\ell^{2}(P)}.$

For the coaction $\delta_{\Psi}$ on $\mathrm{C}^{*}(\Psi)$ we directly compute

$({\operatorname{id}}\otimes\phi)\delta_{\Psi}\Psi(\lambda(\xi_{p}))=({\operatorname{id}}\otimes\phi)(\Psi\lambda(\xi_{p})\otimes
u_{p})=\Psi\lambda(\xi_{p})\otimes V_{p}.$

Therefore, the map $({\operatorname{id}}\otimes\phi)\delta_{\Psi}\Psi$ is a
completely contractive left inverse for $\Phi$, and thus $\Phi$ is completely
isometric. On the other hand,
$\Phi({\operatorname{id}}\otimes\phi)\delta_{\Psi}$ is a completely
contractive left inverse of $\Psi$, and thus $\Psi$ is completely isometric. ∎

We next show that the C*-envelope of the cosystem on
${\mathcal{T}}_{\lambda}(X)^{+}$ is the required terminal object.

######  Theorem 5.17.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Then
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$ is a
boundary representation for the equivariant injective covariant
representations of $X$. Moreover, we have that

$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)\simeq
A\times_{X,\lambda}P.$

######  Proof..

Let ${\mathcal{F}}_{\operatorname{c}}$ be the class of equivariant injective
covariant representations of $X$. By definition the relations in
${\mathcal{T}}_{\lambda}(X)$ pass to
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$, and
so the latter is Fock-covariant and thus it is indeed in
${\mathcal{F}}_{\operatorname{c}}$. Now let $t$ be an element of
${\mathcal{F}}_{\operatorname{c}}$. By Proposition 4.10 we can assume without
loss of generality that $t$ is actually Fock-covariant. By Remark 2.2 we get
that $t_{\ast}$ implements a $*$-epimorphism

$\Psi\colon{\mathcal{T}}_{\lambda}(X)\simeq\mathrm{C}^{*}_{\lambda}({\mathcal{F}}{\mathcal{C}}_{G}X)\longrightarrow\mathrm{C}^{*}_{\lambda}(t_{\ast}({\mathcal{F}}{\mathcal{C}}_{G}X)),$

where $t_{\ast}({\mathcal{F}}{\mathcal{C}}_{G}X)$ is the induced Fell bundle
inside $\mathrm{C}^{*}(t)$. Let

$\lambda_{t}\colon\mathrm{C}^{*}(t)\longrightarrow\mathrm{C}^{*}_{\lambda}(t_{\ast}({\mathcal{F}}{\mathcal{C}}_{G}X)).$

be the canonical $*$-epimorphism for the Fell bundle
$t_{\ast}({\mathcal{F}}{\mathcal{C}}_{G}X)$, as $t$ admits a coaction of $G$.
By Proposition 5.16 we get that the map $\Psi$ defines a C*-cover of the
cosystem $({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$, and thus there exists a
canonical $*$-epimorphism

$\Psi^{\prime}\colon\mathrm{C}^{*}(t)\longrightarrow({\mathcal{T}}_{\lambda}(X)^{+},G,\delta).$

Hence we get the following commutative diagram

$\textstyle{{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{\ast}}$$\scriptstyle{\lambda_{\ast}}$$\textstyle{\mathrm{C}^{*}(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{t}\phantom{ooooo}}$$\textstyle{\mathrm{C}^{*}_{\lambda}(t_{\ast}({\mathcal{F}}{\mathcal{C}}_{G}X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{\prime}}$$\textstyle{{\mathcal{T}}_{\lambda}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$$\textstyle{\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)}$

of canonical $*$-epimorphisms, where $\Phi$ is defined by the definition of
the C*-envelope of the cosystem. This shows that the canonical
$*$-representation

${\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)\stackrel{{\scriptstyle\Phi\lambda_{\ast}}}{{\longrightarrow}}\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$

factors through $t_{\ast}$.

For the second part, by Proposition 4.16 we have that $A\times_{X,\lambda}P$
defines an equivariant representation of ${\mathcal{F}}{\mathcal{C}}_{G}X$
that is injective on $A$. Hence there exists an equivariant $*$-epimorphism

$A\times_{X,\lambda}P\longrightarrow\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$

that fixes $X$, and thus it is injective on $A$. By Proposition 4.18 this map
is injective on $[{\mathcal{S}}{\mathcal{C}}_{G}X]_{e}$ by being injective on
$A$. Recall that the coaction on $A\times_{X,\lambda}P$ is normal. Therefore,
by Remark 2.2 we have that the $*$-epimorphism is faithful. ∎

###  5.3. Co-universality for product systems through strong covariance

We can have an independent proof of the co-universality result that goes
through the Fell bundle theory. We have all tools to implement the strategy of
Carlsen–Larsen–Sims–Vittadello [9] and get the existence of the co-universal
representation through $A\times_{X,\lambda}P$. We also provide a Fell bundle
route for Proposition 5.16 and complete the connection with the C*-envelope.

In Proposition 5.16 we saw how a Fell’s absorption principle induces that
representations of ${\mathcal{T}}_{\lambda}(X)^{+}$ that are injective on $A$
are automatically completely isometric. This works also in reverse by using
the Fell bundle technique of [23, Theorem 3.3].

######  Proposition 5.18.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $t$ is an equivariant representation of $X$ that
is injective on $A$, then there exists a completely contractive homomorphism

$\overline{\operatorname{alg}}\\{t(X_{p})\mid p\in
P\\}\longrightarrow{\mathcal{L}}({\mathcal{F}}X);t(\xi_{p})\mapsto\lambda(\xi_{p}).$

######  Proof..

Let $Y_{t}$ be the Hilbert module associated to $t$ from Subsection 3.3, so
that

$\left\langle\eta_{1},\eta_{2}\right\rangle_{Y}:=E_{t_{\ast}}(\eta_{1}^{*}\eta_{2})\text{
for all }\eta_{1},\eta_{2}\in\sum_{p}t(X_{p}),$

where $E_{t_{\ast}}$ is the conditional expectation on $\mathrm{C}^{*}(t)$.
Let $i,j,k\in\\{1,\dots,n\\}$, and let

$t({\xi}^{ij}):=\sum_{p\in
P}t(\xi^{ij}_{p})\quad\text{and}\quad{\eta}^{k}:=\sum_{q\in P}t(\eta^{k}_{q})$

be finitely supported vectors in $\sum_{p}t(X_{p})$. Since
$t(\xi^{ik})\eta_{k}\in\sum_{p}t(X_{p})$, by Bessel’s inequality of Corollary
3.14 we have

$\displaystyle\|[t({\xi}^{ij})][\eta^{k}]\|\leq\left\|[t({\xi}^{ij})]\right\|\cdot\|[\eta^{k}]\|.$

This yields a well-defined completely contractive map

$L\colon\operatorname{alg}\\{t(X_{p})\mid p\in
P\\}\longrightarrow{\mathcal{B}}Y$

by _bounded_ operators on $Y$, such that
$L(t(\xi_{p}))t(\eta_{q})=t(\xi_{p}\eta_{q})$. In particular we see that this
map takes values in ${\mathcal{L}}Y$ with

$L(t(\xi_{p}))^{*}t(\eta_{q})=\begin{cases}t(\xi_{p}^{*}\eta_{q})&\text{if
}q\in pP,\\\ 0&\text{otherwise}.\end{cases}$

Next we note that $Y_{t}$ is unitarily equivalent to the Fock space
${\mathcal{F}}X$ by the map

$U(\sum_{p}t(\xi_{p}))=\sum_{p}\xi_{p}.$

Indeed, first define $U$ on finite sums; equivariance of $t$ implies that
$\\{t(X_{p})\\}_{p\in P}$ is a family of linearly independent spaces in
$\mathrm{C}^{*}(t)$, and injectivity of $t$ yields that $U$ is well defined.
Moreover injectivity of $t$ gives that $U$ is an isometry onto
$\sum_{p}t(X_{p})$. Continuity of $E_{t_{\ast}}$ allows to extend the map $U$
to a unitary from $Y_{t}$ onto ${\mathcal{F}}X$.

Hence we deduce a completely contractive map

$\operatorname{ad}_{U}L\colon\operatorname{alg}\\{t(X_{p})\mid p\in
P\\}\longrightarrow{\mathcal{L}}({\mathcal{F}}X),$

such that

$\operatorname{ad}_{U}L(t(\xi_{p}))\xi_{r}=Ut(\xi_{p})t(\xi_{r})=\xi_{p}\xi_{r}=\lambda(\xi_{p})\xi_{r},$

and the proof is complete. ∎

######  Proposition 5.19.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $\Psi$ is an equivariant $*$-representation of
${\mathcal{T}}_{\lambda}(X)$ that is injective on $A$, then
$\mathrm{C}^{*}(\Psi)$ is a C*-cover of the cosystem
$({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$. Moreover, there exists a
completely isometric homomorphism

${\mathcal{T}}_{\lambda}(X)^{+}\longrightarrow\mathrm{C}^{*}(\Psi)\otimes\mathrm{C}^{*}_{\lambda}(P);\lambda(\xi_{p})\mapsto\Psi\lambda(\xi_{p})\otimes
V_{p}.$

######  Proof..

By applying Proposition 5.18 for $t=\Psi\lambda|_{X}$ we have a completely
contractive map

$\operatorname{ad}_{U}L\colon\operatorname{alg}\\{\Psi\lambda(X_{p})\mid p\in
P\\}\longrightarrow{\mathcal{T}}_{\lambda}(X)^{+},$

such that $\operatorname{ad}_{U}L(\Psi\lambda(\xi_{p}))=\lambda(\xi_{p})$.
Consequently $\operatorname{ad}_{U}L$ is a completely contractive left inverse
to the completely contractive map $\Psi|_{{\mathcal{T}}_{\lambda}(X)^{+}}$,
making the latter a complete isometry.

For the second part, it suffices to show this for $\Psi=\lambda$. As in
Proposition 5.16, let the isometry

$W\colon{\mathcal{F}}X\otimes\ell^{2}(P)\longrightarrow{\mathcal{F}}X\otimes\ell^{2}(P);\xi_{r}\otimes\delta_{s}\mapsto\xi_{r}\otimes\delta_{rs}.$

It has final space densely spanned by $\\{\xi_{r}\otimes\delta_{rs}\mid r,s\in
P;\xi_{r}\in X_{r}\\}$, and thus invariant for
$\operatorname{alg}\\{\lambda(X_{p})\otimes V_{p}\mid p\in P\\}$. Moreover we
directly verify that

$W(\lambda(\xi_{p})\otimes
I_{\ell^{2}(P)})(\xi_{r}\otimes\delta_{s})=\xi_{p}\xi_{r}\otimes\delta_{prs}=(\lambda(\xi_{p})\otimes
V_{p})W(\xi_{r}\otimes\delta_{s}).$

Hence we can define a completely contractive map $\Phi$ by

$\Phi\colon\operatorname{alg}\\{\lambda(X_{p})\otimes V_{p}\mid p\in
P\\}\stackrel{{\scriptstyle|_{WW^{*}}}}{{\longrightarrow}}\operatorname{alg}\\{W(\lambda(X_{p})\otimes
I_{\ell^{2}(P)})W^{*}\mid p\in
P\\}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathcal{T}}_{\lambda}(X)^{+},$

so that $\Phi(\lambda(\xi_{p})\otimes V_{p})=\lambda(\xi_{p})$. It follows
that $\Phi$ has a completely contractive left inverse given by
$({\operatorname{id}}\otimes\phi)\delta$, for the unital completely positive
map $\phi\colon\mathrm{C}^{*}_{\max}(G)\to{\mathcal{B}}(\ell^{2}(P))$ with
$\phi(u_{g})=P_{\ell^{2}(P)}\lambda_{g}|_{\ell^{2}(P)}$, and the coaction
$\delta$ of ${\mathcal{T}}_{\lambda}(X)$. Thus $\Phi$ is completely isometric.
∎

We can apply the Fell bundle arguments that appear in the proof of [9, Theorem
4.1]. There is a key underlying idea that can be traced back to Katayama [36,
Proposition 11] and Raeburn [63, Lemma 3.1]. It relies on the fact that any
coaction induces a reduced coaction to a quotient by an appropriate ideal; and
thus that quotient is minimal for the quotient Fell bundle, hence receiving
also from the original Fell bundle.

######  Theorem 5.20.

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Then $A\times_{X,\lambda}P$ is a boundary
representation for the equivariant injective covariant representations of $X$.
Moreover, we have that

$A\times_{X,\lambda}P\simeq\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta).$

######  Proof..

As in the proof of Theorem 5.17 we can reduce to Fock-covariant
representations. Hence, without loss of generality, let $t$ be an equivariant
injective Fock-covariant representation of $X$. Recall from Proposition 4.17
that

$\ker
t_{\ast}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}\subseteq\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e},$

for the canonical $*$-epimorphism
$q_{\operatorname{sc}}\colon{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)\to
A\times_{X}P$ that fixes every $X_{p}$. Remark 2.3 yields

$t_{\ast}(\ker
q_{\operatorname{sc}})\bigcap[\mathrm{C}^{*}(t)]_{e}=t_{\ast}(\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}),$

since $q_{\operatorname{sc}}$ is equivariant. Let $q_{J}$ be the quotient map
on $\mathrm{C}^{*}(t)$ by the ideal

$J:=\left\langle t_{\ast}(\ker
q_{\operatorname{sc}})\bigcap[\mathrm{C}^{*}(t)]_{e}\right\rangle.$

The goal is to show that

$\ker
q_{J}t_{\ast}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{g}=\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{g}\text{
for all }g\in G.$

It suffices to show this for $g=e_{G}$, and then the equality for all fibres
follows by a standard argument. With that in hand we deduce that the Fell
bundle defined in $\mathrm{C}^{*}(t)/J$ is isomorphic to the strongly
covariant bundle ${\mathcal{S}}{\mathcal{C}}_{G}X$. Consequently, we obtain an
equivariant $*$-epimorphism

$\mathrm{C}^{*}(t)\longrightarrow\mathrm{C}^{*}(t)/J\longrightarrow
A\times_{X,\lambda}P.$

Towards this end first assume that $x\in\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}$.
Then

$t_{\ast}(x)\in t_{\ast}(\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e})=t_{\ast}(\ker
q_{\operatorname{sc}})\bigcap[\mathrm{C}^{*}(t)]_{e}\subseteq J,$

and so $x\in\ker
q_{J}t_{\ast}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}$.
The reverse inclusion will follow by applying Proposition 4.17 on
$q_{J}t_{\ast}$. Towards this end we need to show that $q_{J}t_{\ast}$ is
injective on $\iota(A)$. Let $a\in A$ so that $q_{J}t_{\ast}(\iota(a))=0$, and
so $t_{\ast}(\iota(a))\in J\bigcap[\mathrm{C}^{*}(t)]_{e}$. Since

$\displaystyle J\bigcap[\mathrm{C}^{*}(t)]_{e}$ $\displaystyle\subseteq
t_{\ast}(\ker
q_{\operatorname{sc}})\bigcap[\mathrm{C}^{*}(t)]_{e}=t_{\ast}(\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}),$

there exists a $y\in\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}$
such that $t_{\ast}(\iota(a))=t_{\ast}(y)$. We deduce that

$\iota(a)-y\in\ker
t_{\ast}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e}\subseteq\ker
q_{\operatorname{sc}}\bigcap[{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)]_{e},$

where we applied Proposition 4.17 for the injective Fock-covariant
representation $t$. It follows that $\iota(a)=(\iota(a)-y)+y\in\ker
q_{\operatorname{sc}}$, and thus Proposition 4.16 yields $a=0$ as required.

For the second part, Proposition 4.16 and Proposition 5.16 give a canonical
$*$-epimorphism

$A\times_{X,\lambda}P\longrightarrow\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X),G,\delta)$

that fixes $X$. On the other hand, the co-universal property of
$A\times_{X,\lambda}P$ provides a canonical $*$-epimorphism in the other
direction giving the required $*$-isomorphism. ∎

###  5.4. Applications and remarks

We close with a set of immediate applications and questions that are related
to the theory. First we observe that co-universality of $A\times_{X,\lambda}P$
implies a gauge-invariant uniqueness theorem.

######  Corollary 5.21 (GIUT).

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $t_{\ast}$ be a representation of
$A\times_{X,\lambda}P$. Then $t_{\ast}$ is faithful if and only if $t$ is an
equivariant injective representation of $X$.

######  Proof..

The forward direction is immediate as $A\times_{X,\lambda}P$ shares all these
properties. For the converse, if $t$ satisfies these properties then co-
universality of $A\times_{X,\lambda}P$ gives an inverse to $t_{\ast}$. ∎

One may compare with the following corollary, which is a consequence of the
properties of $A\times_{X}P$.

######  Corollary 5.22 (GIUT II).

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. Let $t$ be a strongly covariant representation of
$X$. Then there is a canonical $*$-isomorphism $\mathrm{C}^{*}(t)\to
A\times_{X,\lambda}P$ if and only if $t$ is injective and equivariant by a
normal coaction.

######  Proof..

The forward direction is immediate as $A\times_{X,\lambda}P$ shares all these
properties. For the converse by universality $t_{\ast}$ defines a
$*$-representation ${\mathcal{S}}{\mathcal{C}}_{G}X$. Since $t$ is injective
we have that $t_{\ast}$ injective on $[{\mathcal{S}}{\mathcal{C}}_{G}X]_{e}$
and thus by Remark 2.2 there is a canonical $*$-isomorphism
$\mathrm{C}^{*}(t)\to A\times_{X,\lambda}P$. ∎

Recall that by construction we have a canonical $*$-epimorphism
$q_{\operatorname{sc}}\colon{\mathcal{T}}_{\operatorname{c}}^{\operatorname{F}}(X)\longrightarrow
A\times_{X}P$. By Remark 2.2 it descends to a canonical $*$-epimorphism

$q_{\operatorname{sc},\lambda}\colon{\mathcal{T}}_{\lambda}(X)\longrightarrow
A\times_{X,\lambda}P.$

By construction $\ker q_{\operatorname{sc},\lambda}$ contains
$\lambda_{\ast}({\mathcal{I}}_{\operatorname{sc}})$. In particular, due to the
(GIUT II), the canonical $*$-epimorphism

${\raisebox{1.00006pt}{${\mathcal{T}}_{\lambda}(X)$}\left/\,\raisebox{-1.00006pt}{$\lambda_{\ast}({\mathcal{I}}_{\operatorname{sc}})$}\right.}\longrightarrow
A\times_{X,\lambda}P$

is faithful if and only if it inherits a normal coaction from
${\mathcal{T}}_{\lambda}(X)$. The reason we are interested in this quotient in
the Fock representation is because the ideal
$\lambda_{\ast}(I_{\operatorname{sc}})$ has been oftenly recognized to give a
boundary space. This still happens in many cases. The following has been
observed for right LCM-semigroups in [19].

######  Corollary 5.23 (Fock boundary quotient).

Let $P$ be a unital subsemigroup of a discrete group $G$ and let $X$ be a
product system over $P$. If $G$ is exact then $\ker
q_{\operatorname{sc},\lambda}=\lambda_{\ast}({\mathcal{I}}_{\operatorname{sc}})$,
i.e., there is a canonical $*$-isomorphism

${\raisebox{1.00006pt}{${\mathcal{T}}_{\lambda}(X)$}\left/\,\raisebox{-1.00006pt}{$\lambda_{\ast}({\mathcal{I}}_{\operatorname{sc}})$}\right.}\simeq
A\times_{X,\lambda}P.$

######  Proof..

Recall that ${\mathcal{T}}_{\lambda}(X)$ admits a normal coaction and that
$\lambda_{\ast}({\mathcal{I}}_{\operatorname{sc}})$ is an induced ideal.
Exactness of $G$ implies that the induced coaction on the quotient is normal
as well. ∎

Hence under exactness we have a hands-on understanding of the algebraic
structure of the boundary quotient.

######  Remark 5.24 (Applications I).

The GIUT has been a central tool in the $P=\mathbb{Z}_{+}$ case. Its ample
applicability is primarily manifested in graph C*-algebras. Yet, there are
many applications to Cuntz–Pimsner algebras that include a 6-term short exact
sequence for computing its K-theory [42], classification [8], and the impact
of shift-type equivalences up to strong Morita equivalence. The reader may
refer to [18] for a list of results which we will not duplicate here. The
question here is to see to what extent the known theorems in the
$\mathbb{Z}_{+}$-case find their analogues for general semigroups.

######  Remark 5.25 (Applications II).

At the same time, recent work on right LCM-semigroups has shown that co-
universality results have an immediate impact on Takai duality [20, 37] and
the reduced Hao-Ng problem [19], with interesting interactions with K-theory
problems [67], but also can feed back to the identification of universal
C*-algebras [33], or in cases can reduce the form of Cuntz-type C*-algebras to
generalized crossed products [33, 37, 70] as in the $\mathbb{Z}_{+}$-case [1].
The use of
$\mathrm{C}^{*}_{\textup{env}}({\mathcal{T}}_{\lambda}(X)^{+},G,\delta)$ is
pivotal for obtaining such results, making the step forwards from right LCM-
semigroups to general semigroups highly plausible.

######  Remark 5.26 (Fock tensor algebra).

We have seen that ${\mathcal{T}}_{\lambda}(X)^{+}$ plays a central role in the
co-universality results. It is a direct generalization of Popescu’s
noncommutative disc algebra [61] which has been under significant study. There
are plenty of directions one can go from here starting with identifying its
completely contractive representations as in [55]. The form of the strong
covariance relations may also shed light into hyperrigidity questions, which
is resolved by Katsoulis and Ramsey for $P=\mathbb{Z}_{+}$ [41]. As they have
shown in [40], such a result will then direct feed back to Takai duality and
the Hao-Ng problem. Finally, there is a lot of work for $P=\mathbb{Z}_{+}$ in
terms of the w*-analogue of the Fock tensor algebra. Questions include
reflexivity, hyperreflixivity as well as a weak*-calculus, with the list being
too long to include it here.

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# An alternative to test experimentally Everett’s theory

Hitoshi Inamori

###### Abstract

It is generally accepted that Everett’s theory of quantum mechanics cannot be
experimentally tested as such experiment would involve operations on the
observer which are beyond our current technology. We propose an alternative to
test Everett’s theory which does not involve any operation on the observer. If
we assume that the observer is of finite dimension, it is shown that Everett’s
theory leads to distinctive properties for the system being observed, and that
such difference can be experimentally tested.

##  1 Introduction

According to quantum theory, performing a measurement on a physical system
leads in general to a truly random outcome which cannot be predicted. Many
theories or interpretations of quantum physics try to explain how this random
outcome arises.

These theories usually agree on how the measurement starts [1]: when an
observer performs a measurement on a system, the two physical systems interact
so that the state of the combined system becomes entangled. It becomes a sum
of states, and in each of these states, the observer witnesses an outcome and
the observed system is in the state corresponding to this outcome.

The theories however diverge from this point. In particular, the Copenhagen
theory [2] asserts that the state of the combined system “jumps” from the
above superposition in which the outcome is not determined, to the final
state, in which only one of the possible outcomes occurs. This irreversible
process is often referred to as the “collapse of the wave function” where
“wave function” is synonym for the state of the system.

On the other hand, Everett’s theory [3] asserts that such irreversible process
does not occur. There are many variants for Everett’s theory but in this paper
we shall consider the original “Relative State” version [3], which asserts
that the combined system remains in a superposition of states in each of which
one version of the observer actually witnesses one possible outcome of the
measurement.

In this paper we will choose to distinguish the above two theories as follows:

###### Definition 1

In the Copenhagen theory, when an observer performs a measurement, the system
state jumps irreversibly from a superposition of states to a single state in
which only one outcome occurs.

In Everett’s theory, such wave function collapse does not occur. The observed
system and the observer remain in an entangled superposition of states.

It is believed that in practice the two theories lead to the same predictions
that can be tested experimentally. However, several thought experiments have
been proposed [4, 5, 6] to test Everett’s theory against the Copenhagen one.
In particular, Deutsch [4] proposed that if the above interaction between the
observer and the observed system could be reversed, then we could
experimentally have the observer perform a measurement on a system, have him
declare that the measurement was done without him revealing the outcome of the
measurement, and then reverse the interaction between the observer and the
observed system. If the initial state for the observed system is in a
superposition of states, then according to Everett’s theory the observed
system should remain in the same superposition of states after this entire
process. In contrast, according to the Copenhagen theory the observed system
would jump into one of the states corresponding to a possible measurement
outcome. Such discrepancy could be tested experimentally. However, Deutsch’s
experiment would require that we can revert the interaction process involving
the observer, which is assumed to be a relatively large system. As such it
remains a Gedanken experiment, not realizable experimentally.

In this paper we propose an alternative to test Everett’s theory that does not
require any transformation to be made on the observer after the measurement.
The key observation is that an observer is a physical system which is finite,
and that this finite dimensionality implies a certain constraint on the state
of the observed system after its interaction with the observer.

More precisely, suppose that we have a very long sequence of qubits, all
prepared in the state $\frac{\left|{0}\right>+\left|{1}\right>}{\sqrt{2}}$.
Assume that we have an observer performing a measurement on each of these
qubits in the $\\{\left|{0}\right>,\left|{1}\right>\\}$ basis. If the observer
had an infinite memory then he could remember the result of all the
measurements and the observed system would be in a fully mixed state,
corresponding to a stream of qubits in which each qubit is in the state
$\left|{0}\right>$ (corresponding to the case in which the observer witnessed
$\left|{0}\right>$) or in the state $\left|{1}\right>$ (corresponding to the
case in which the observer witnessed $\left|{1}\right>$) with equal classical
probability 50%.

Now if the observer is finite, then his memory is finite as well, and
intuitively we can feel that the observer cannot “fully entangle” with all the
qubits of this very long sequence. Now independently of the size of the
memory, according to the Copenhagen theory, the observer would collapse the
wave function of the system, and the very long sequence of the qubits would be
in a fully probabilistic mixture of $\left|{0}\right>$ and
$\left|{1}\right>$’s. The purpose of this paper is to show that with Everett’s
theory the state of the observed system is provably different from what the
Copenhagen theory predicts. More precisely, we give a sufficient condition on
the length of the sequence of the qubits compared to the observer’s
dimensionality so that Everett’s theory gives different experimental
predictions compared to the Copenhagen theory.

Such experiment may be easier to perform than Deustch’s Gedanken experiment
described above. In particular we do not need to know the inner working of the
observer as we do not need to revert any process involving the observer. The
only requirement is that we can perform measurement on the observed system in
different bases so that one of such bases leads to outcome statistics
different from what is predicted by the Copenhagen theory.

##  2 Main result

The following constitutes the main result of this paper:

###### Property 1

Let $N$ and $m$ be positive integers, and let a source of qubits emit $N\times
m$ qubits, each of them in the state
$(\left|{0}\right>+\left|{1}\right>)/\sqrt{2}$. The source is therefore
emitting $m$ streams of $N$ qubits which are all in the same initial state.
Let us ask the observer $O$ to perform a measurement on each of the $N\times
m$ qubits in the basis $\\{\left|{0}\right>,\left|{1}\right>\\}$ and let us
ask the observer to send through the observed qubits. We suppose that the
observer interacts with each qubit one at a time, although it is sufficient to
assume that the observer interacts with one N-qubit stream at a time.

Let $\rho_{1},\rho_{2},\ldots,\rho_{m}$ represent the reduced density matrix
of the $m$ streams, where for each $i=1,\ldots,m$, $\rho_{i}$ is a $2^{N}$
dimensional density matrix obtained from the density matrix representing the
entire experimental setup by tracing out the space complementary to the
Hilbert space for the $i$-th stream.

We want to consider a density matrix representing each stream of $N$-qubits,
which we obtain by shuffling the $m$ streams in a random order. In this case,
the reduced density matrix is identical for all the streams and is equal to
the average of the reduced density matrices for the $m$ streams of qubits
before the shuffling. We denote by
$\rho_{S}=\frac{1}{m}(\rho_{1}+\cdots+\rho_{m})$ this reduced density matrix.
Then,

  * •

according to the Copenhagen theory, the reduced density matrix $\rho_{S}$ is
fully mixed, i.e. $\rho_{S}=\frac{1}{2^{N}}{\mathbf{1}}_{2^{N}}$.

  * •

according to Everett’s theory, if the dimension of the observer is finite then
for a sufficiently large $N$, the reduced density matrix $\rho_{S}$ is not
fully mixed.

As we have $m$ instances of the $N$-qubit streams having the same reduced
density matrix $\rho_{S}$, one can experimentally distinguish whether the
density matrix $\rho_{S}$ is in accordance with the Copenhagen theory or
Everett’s theory.

The first part of the property giving the reduced density matrix $\rho_{S}$
according to the Copenhagen theory is straightforward to prove. Indeed, the
observer is measuring each qubit in the basis
$\\{\left|{0}\right>,\left|{1}\right>\\}$ and the initial state of the qubit
is $(\left|{0}\right>+\left|{1}\right>)/\sqrt{2}$. According to the Copenhagen
theory, each qubit jumps to the state $\left|{0}\right>$ or $\left|{1}\right>$
with probability $\frac{1}{2}$ when the measurement is performed. As a result,
the density matrix for any stream of $N$ qubits is
$\frac{1}{2^{N}}{\mathbf{1}}_{2^{N}}$ after the measurement performed by the
observer.

Proving the second part of the property is less straightforward and is done in
the following two sections.

##  3 Modeling the experiment

In this section we give a mathematical description of the experiment and in
particular the description of the observer as a quantum system interacting
with each stream of $N$ qubits. This description will allow us to precise what
we mean by the dimensionality of the observer. It will later also allow us to
give a constraint on the dimensionality of the reduced density state for the
$N$-qubit systems after interaction with the observer.

Let $H_{O}$ be the Hilbert space describing the observer and let $H_{S}$ be
the Hilbert space describing one stream of $N$ qubits. Without loss of
generality suppose that the observer is initially in a state
$\left|{\phi_{0}}\right>\in H_{O}$. As we have seen, all the $m$ streams of
$N$ qubits are initially in the state
$(\left|{0}\right>+\left|{1}\right>)^{N}/\sqrt{2^{N}}$. We have assumed that
the observer interacts with one $N$-qubit stream at a time: the observer
performs a measurement on the first stream of $N$ qubits, and its interaction
can be represented by an unitary operator $U(1)$ on $H_{S}\otimes H_{O}$ where
$H_{S}$ is the Hilbert space for the first $N$-qubit stream. Then the observer
performs a measurement on the second stream, and the interaction is
represented by an unitary operator $U(2)$ on $H_{S}\otimes H_{O}$ where
$H_{S}$ is now the Hilbert space for the second stream, etc. (we could
explicitly perform a swap of the $k$-th stream of qubits with an auxiliary
system of $N$ qubits and perform the unitary operator on this auxiliary system
combined with the observer, swapping back the $k$-th stream after the
measurement is done. This would lead to an equivalent setup.) A priori, there
is no reason why $U(1)$ should be identical to $U(2)$ as we do not know the
inner working of the observer, except the fact that it is of finite dimension
(Figure 1).

$\frac{(\left|{0}\right>+\left|{1}\right>)^{N}}{\sqrt{2^{N}}}$$U(1)$$\left|{\phi_{0}}\right>$$\frac{(\left|{0}\right>+\left|{1}\right>)^{N}}{\sqrt{2^{N}}}$$U(2)$$\Large\vdots$$\Large\ddots$$\Large\cdots$
Figure 1: Model for the experimental setup

Note however that a time dependent operator $U(i)$ is equivalent to a time
independent operator coupled with an additional physical system, playing the
role of a “clock” internal to the observer. In other words, the time dependent
$U(i)$ can be replaced by a time independent operator $U$ applying $U(i)$
depending on an additional variable $i$ on the observer side, which is
incremented each time a measurement is performed. The internal variable $i$
can be represented by $\lceil\log_{2}m\rceil$ qubits. For instance a time
dependent operator $U(i)$ for $i=1,2,3,4$ can be replaced by a constant
operator $U$ and 2 additional qubits for the observer’s state as in Figure 2.
We will assume henceforth that the additional qubits to encode the internal
variable $i$ is part of the observer $O$.

$H_{S}$$H_{O}$$U(1)$$U(2)$$U(3)$$U(4)$$\left|{0}\right>$$\left|{0}\right>$
Figure 2: A time dependent operator $U(i)$, $i=1,2,3,4$ can be replaced by a
constant operator and $\log_{2}4=2$ additional qubits. The operator $U(i)$ is
applied only when each auxiliary qubit is $\left|{0}\right>$ if the circle is
white or $\left|{1}\right>$ if the circle is black. The $\oplus$ represents
the NOT operator. The last two gates on the auxiliary qubits increment the
counter formed by the two auxiliary qubits.

We have therefore shown that the experiment can be described by a successive
interaction of the observer $O$ with the $N$-qubit streams $i=1,\ldots,m$ and
that the interaction can be represented by an identical unitary operator $U$
acting on $H_{S}\otimes H_{O}$ where $H_{S}$ is the Hilbert space
corresponding to one $N$-qubit stream.

##  4 Demonstration of the main result

We are now ready to prove Property 1. We denote by $D$ the dimension of the
Hilbert space $H_{O}$ describing the observer. By hypothesis of the
Proposition 1, $D$ is finite. Suppose that the size of the streams $N$ is so
that $2^{N}$ is strictly larger than $D^{2}$. Then we can show the following
property:

###### Property 2

There exists a subspace $F_{S}$ of $H_{S}$ of dimension less than $D^{2}$ such
that whichever the initial state for the observer, the density matrix
$\rho_{S}$ for the observed $N$ qubit stream has its support in the subspace
$F_{S}$. In particular, for any state $\left|{\chi}\right>$ which is
orthogonal to $F_{S}$, we have
$\left<{\chi}\right|\rho_{S}\left|{\chi}\right>=0$.

To prove this property we first remind the Schmidt decomposition theorem (the
reader may find a proof for instance in [7]):

###### Property 3

Let $H=H_{A}\otimes H_{B}$ be the product space of Hilbert spaces $H_{A}$ and
$H_{B}$. Then for any state $\left|{\psi_{AB}}\right>\in H$, there exist
orthonormal bases $\\{\left|{a_{i}}\right>\\}_{i}$ for $H_{A}$ and
$\\{\left|{b_{i}}\right>\\}_{i}$ for $H_{B}$ such that:

$\left|{\psi_{AB}}\right>=\sum_{k=1}^{n}\alpha_{k}\left|{a_{k}}\right>\otimes\left|{b_{k}}\right>$
(1)

where $n\leq\min(\dim(H_{A}),\dim(H_{B}))$ and $\alpha_{k}$ are non-negative
real numbers.

Let $\\{\left|{e_{i}}\right>\\}_{i=1,\ldots,D}$ be an orthonormal basis for
$H_{O}$. If the observer’s initial state is $\left|{e_{i}}\right>$ then after
the observer performs the measurement on an $N$-qubit system initially
prepared in the state $(\left|{0}\right>+\left|{1}\right>)^{N}/\sqrt{2^{N}}$,
the combined system is in the state:

$\frac{(\left|{0}\right>+\left|{1}\right>)^{N}}{\sqrt{2^{N}}}\otimes\left|{e_{i}}\right>\mapsto
U\left[\frac{(\left|{0}\right>+\left|{1}\right>)^{N}}{\sqrt{2^{N}}}\otimes\left|{e_{i}}\right>\right]\in
H_{S}\otimes H_{O},$ (2)

which, according to Schmidt decomposition theorem, reads:

$U\left[\frac{(\left|{0}\right>+\left|{1}\right>)^{N}}{\sqrt{2^{N}}}\otimes\left|{e_{i}}\right>\right]=\sum_{j=1}^{D}\alpha^{i}_{j}\left|{\psi^{i}_{j}}\right>\otimes\left|{\phi^{i}_{j}}\right>$
(3)

for some non-negative real numbers $\alpha^{i}_{j}$ (some of them possibly
zero), and orthonormal bases $\\{\left|{\psi^{i}_{j}}\right>\\}_{j}$ of
$H_{S}$ and $\\{\left|{\phi^{i}_{j}}\right>\\}_{j}$ of $H_{O}$ with
$j=1,\ldots,D$. The superscript $i$ reminds us that these orthonormal bases
may depend on the observer state $\left|{e}\right>_{i}$ initially chosen.

Let $F_{S}$ be the subspace of $H_{S}$ spanned by the $D^{2}$ states
$\\{\left|{\psi^{i}_{j}}\right>\\}_{i,j}$, $i,j=1,\ldots,D$. The dimension of
$F_{S}$ is therefore less than, or equal to, $D^{2}$. Now whichever the state
$\left|{\phi}\right>$ for the observer, it reads as a linear combination of
the orthonormal states $\left|{e_{i}}\right>$. Thus, after the observer
performs the measurement on the $m$ streams of $N$-qubits, the state
describing each stream of qubits lies within the subspace $F_{S}$, in the
sense that, for each stream $k\in\\{1,\ldots,m\\}$, the state
$\left|{\psi}\right>$ of the entire experimental setup reads:

$\left|{\psi}\right>=\sum_{i,j}\left|{\eta^{k}_{i,j}}\right>\otimes\left|{\psi^{i}_{j}}\right>$
(4)

where the $\left|{\psi^{i}_{j}}\right>$ are states defined above describing
the $k$-th stream, and $\left|{\eta^{k}_{i,j}}\right>$ are states describing
the remaining $(m-1)$ streams and the observer (the states
$\left|{\eta^{k}_{i,j}}\right>$ are possibly not normalized nor orthogonal
with each others).

We deduce that for any stream of $N$ observed qubits, the reduced density
matrix $\rho_{S}$ describing the stream is such that

$\left<{\chi}\right|\rho_{S}\left|{\chi}\right>=0$ (5)

for any state $\left|{\chi}\right>$ which is orthogonal to $F_{S}$. In other
words, the density matrix $\rho_{S}$ has support in the subspace $F_{S}$,
which completes the proof.

We have shown that in the absence of wave function reduction, the density
matrix describing the stream of $N$ qubits have support in $F_{S}$ of
dimension which is limited by $D^{2}$. If $D^{2}$ is lower than $2^{N}$ then
the density matrix of the $N$ qubits cannot be fully mixed. In particular, for
any state $\left|{\chi}\right>$ in the subspace $F_{S}^{\perp}$ orthogonal to
$F_{S}$ ($F_{S}^{\perp}$ has dimension larger than, or equal to,
$2^{N}-D^{2}$), the probability to observe $\left|{\chi}\right>$ from the
stream of $N$ qubits is zero while the probability is $1/2^{N}$ with the fully
mixed state. Another example, if you take a random state $\left|{\psi}\right>$
for the $N$-qubit system and measure whether the $N$-qubit systems are in this
state then we find that the probability would be between 0 and $q$ with $q\geq
1/2^{(2d)}$ while you should have this probability equal to $1/2^{N}$ with the
fully mixed state.

We have shown that it is sufficient that $2^{N}$ is larger than $D^{2}$ so
that a theory without wave function reduction, such as Everett’s, leads to a
density state for the observed $N$ qubits which is distinct from the fully
mixed state. This completes the proof for Property 1.

Let us illustrate the above discussion with a toy example. Assume that the
observer can be described by a single qubit, meaning $D=2$. Let us take $N=3$,
thus $2^{N}=8>D^{2}$. The unitary operator $U$ representing the interaction
between the streams of $N=3$ qubits and the observer can be of any form, but
let us for instance assume that it is given by:

$\displaystyle
U\left[\frac{(\left|{0}\right>+\left|{1}\right>)^{3}}{\sqrt{8}}\left|{\psi_{0}}\right>\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|{A_{1}}\right>\left|{\psi_{0}}\right>+\frac{1}{\sqrt{2}}\left|{A_{2}}\right>\left|{\psi_{1}}\right>$
(6) $\displaystyle
U\left[\frac{(\left|{0}\right>+\left|{1}\right>)^{3}}{\sqrt{8}}\left|{\psi_{1}}\right>\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|{A_{3}}\right>\left|{\psi_{0}}\right>+\frac{1}{\sqrt{2}}\left|{A_{4}}\right>\left|{\psi_{1}}\right>$
(7)

where $\\{\left|{\psi_{0}}\right>,\left|{\psi_{1}}\right>\\}$ is an
orthonormal basis for $H_{O}$ (we have chosen a particular $U$ for which the
Schmidt decomposition of the mapped states can be done in the same basis of
$H_{O}$), and where

$\displaystyle\left|{A_{1}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{000}\right>+\left|{001}\right>+\left|{010}\right>+\left|{011}\right>}{\sqrt{4}}$
(8) $\displaystyle\left|{A_{2}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{100}\right>+\left|{101}\right>+\left|{110}\right>+\left|{111}\right>}{\sqrt{4}}$
(9) $\displaystyle\left|{A_{3}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{000}\right>-\left|{001}\right>+\left|{010}\right>-\left|{011}\right>}{\sqrt{4}}$
(10) $\displaystyle\left|{A_{4}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{100}\right>-\left|{101}\right>+\left|{110}\right>-\left|{111}\right>}{\sqrt{4}}$
(11)

Suppose that the observer is initially in the state $\left|{\psi_{0}}\right>$
and performs successive measurements on 3-qubit streams. We see that the
reduced density matrix for each stream will be a linear combination of density
matrices
$\rho_{S}^{1}=\frac{1}{2}\left(\left|{A_{1}}\right>\left<{A_{1}}\right|+\left|{A_{2}}\right>\left<{A_{2}}\right|\right)$
and
$\rho_{S}^{2}=\frac{1}{2}\left(\left|{A_{3}}\right>\left<{A_{3}}\right|+\left|{A_{4}}\right>\left<{A_{4}}\right|\right)$.
If we measure the outgoing 3-qubit streams in the canonical basis
$\\{\left|{abc}\right>\\}_{a,b,c=0,1}$, then for each stream, the probability
to observe $\left|{abc}\right>$ is 1/8 for any combination of $a$, $b$ and
$c$. Therefore, in this example, measurement in the canonical basis is not
sufficient to show that the 3-qubit system is not in the fully mixed state.

Now, we also note that the states $\left|{A_{i}}\right>$ are all orthogonal to
the following states in $H_{S}$:

$\displaystyle\left|{B_{1}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{000}\right>-\left|{010}\right>}{\sqrt{2}}=\left|{0}\right>\left|{-}\right>\left|{0}\right>$
(12) $\displaystyle\left|{B_{2}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{001}\right>-\left|{011}\right>}{\sqrt{2}}=\left|{0}\right>\left|{-}\right>\left|{1}\right>$
(13) $\displaystyle\left|{B_{3}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{100}\right>-\left|{110}\right>}{\sqrt{2}}=\left|{1}\right>\left|{-}\right>\left|{0}\right>$
(14) $\displaystyle\left|{B_{4}}\right>$ $\displaystyle=$
$\displaystyle\frac{\left|{101}\right>-\left|{111}\right>}{\sqrt{2}}=\left|{1}\right>\left|{-}\right>\left|{1}\right>$
(15)

where $\left|{-}\right>=(\left|{0}\right>-\left|{1}\right>)/\sqrt{2}$. The
successive interaction of the observer and the $m$ streams of qubits will keep
the state of the streams orthogonal to the subspace spanned by the states
$\left|{B_{i}}\right>$. Therefore, measurement of the streams of 3 qubits in
the basis $\\{\left|{A_{i}}\right>,\left|{B_{i}}\right>\\}_{i}$ will lead to
zero observation of the states $\left|{B_{i}}\right>$ and will thus reveal
that the 3-qubit streams are obviously not in the fully mixed state.

In general, the transformation given by the equations (6,7) is not known, but
as we have many streams of qubits sharing the same density matrix, one can
find an observation basis for which the measurement outcome is not compatible
with a fully mixed state.

##  5 Discussion

We have shown that Everett’s theory and the Copenhagen theory give different
predictions when the dimension of the observed $N$-qubit system is larger than
$D^{2}$ where $D$ is the “dimension” of the observer. If we were to express
the dimension of the observer in qubits, this would give $d=\log_{2}D$ so our
result states that it is sufficient that $N>2d$ to test Everett’s theory
against the Copenhagen theory. Expressed in loose terms, Everett’s predictions
and Copenhagen’s diverge when the observer is performing a measurement on a
system whose information content is twice the information content that he can
store.

Another important point to note is that the condition $N>2d$ is a sufficient
condition and not a necessary one. An experiment for a given observer could
show that the reduced density matrix of the observed system is different from
the fully mixed state for $N$ well below $2d$. This is because the inner
working of the observer is not known and it may turn out that every
dimensionality describing the observer is not optimally coupled with the
information conveyed by the observed system. In particular, Chao. R et al. [8]
have shown that a system of $d$ qubits for which an operation on one qubit can
slightly affect the other qubits can be described in a space of dimension
polynomial in $d$, and not $2^{d}$ as we would expect from a system of $d$
fully independent qubits. Experiment may show that the density matrix of the
observed $N$-qubit stream may diverge from the fully mixed one for $N$ being a
tiny fraction of $d$.

A statistically significant evidence that the observed $N$-qubit system is not
in a fully mixed state would be a sufficient proof that the observer does not
“collapse the wave function” as postulated by the Copenhagen theory.

##  6 Acknowledgements

The author would like to thank Artur Ekert and Lev Vaidman for their comments,
and in particular to David Deutsch for the helpful discussions and also for
pointing out the work by R. Chao et al. on overlapping qubits.

## References

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  * [2] Heisenberg, W. (1958) Physics and Philosophy, Harper
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  * [4] Deutsch, D. (1986) ‘Three experimental implications of the Everett interpretation’, in Quantum Concepts of Space and Time, Oxford, Clarendon Press
  * [5] Lockwood, M. (1989) ‘Mind, Brain and the Quantum’, Oxford: Basil Blackwell
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