2.1. C*-correspondences

Let $A$ and $B$ be unital C $^*$ -algebras. A right $A {-} B$ -correspondence is a right Hilbert C $^*$ -module $K$ over $B$ together with a unital $*$ -homomorphism $\pi \;:\; A \to \mathcal{L}_B(K)$ , where $\mathcal{L}_B(K)$ denotes the C $^*$ -algebra of adjointable $B$ -linear operators on $K$ . When $\pi (A) (K)$ is a dense subspace of $K$ , we say that $K$ is a non-degenerate right $A {-} B$ -correspondence.

The discussion above can be extended to define a 2-C $^*$ -category $\mathsf{C}^*\mathsf{Alg}$ . See [Reference Palomares and Nelson14], Section 1, for more details.

2.2. Unitary tensor categories

A linear category $\mathcal{C}$ is a locally small category such that for every pair $X,Y \in {\mathcal{C}}$ , the space ${\mathcal{C}}(X,Y)$ of morphisms from $X$ to $Y$ is equipped with a structure of a vector space over $\mathbb{C}$ . The model example is the category ${\mathsf {Mod}}(A)$ of left $A$ -modules for an algebra $A$ over $\mathbb{C}$ , where the morphisms between two modules are linear maps between them intertwining the corresponding actions. A dagger, or $*$ -category, is a linear category $\mathcal{C}$ equipped with natural involutions $(-)^*\;:\; {\mathcal{C}}(X,Y) \to {\mathcal{C}}(Y,X)$ . Naturality means that $(-)^*$ can be seen as a functor

\begin{equation*} {\mathcal{C}} \to {\mathcal{C}}^{\textrm {op}} , \end{equation*}

acting as the identity function on the set of objects. In the category $\mathsf {Hilb}_{fd}$ of finite-dimensional Hilbert spaces, for instance, the standard $*$ -structure is given by the adjoint operation on the spaces of bounded linear maps.

Definition 2.5. A C $^*$ -tensor category is a C $^*$ -category $\mathcal{C}$ equipped with a tensor structure $\boxtimes \;:\; {\mathcal{C}} \times {\mathcal{C}} \to {\mathcal{C}}$ such that

\begin{equation*} ((-) \boxtimes (-))^* = (-)^* \boxtimes (-)^* \end{equation*}

on morphisms.

We shall make use of the following terminology. For tensor categories ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ , a tensor functor $F\;:\;{\mathcal{C}}_1 \to {\mathcal{C}}_2$ is understood to have natural isomorphisms as tensor structure. A lax functor $F\;:\;{\mathcal{C}}_1 \to {\mathcal{C}}_2$ is understood to be a functor equipped with associators and unitors, but they are not required to be natural isomorphisms.

We will assume throughout the paper all tensor categories of interest to have been strictified.

Examples of C $^*$ -tensor categories include the category $\mathsf {Rep}(\mathbb{G})$ of unitary representations of a locally compact (quantum) group $\mathbb{G}$ , the category $\mathsf {Bim}(N)$ of bimodules over a von Neumann algebra $N$ , and the category ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}$ of non-degenerate right C $^*$ -correspondences over a C $^*$ -algebra $A$ .

We will find later in this paper unitary functors ${\mathcal{C}} \to {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}$ that are almost unitary tensor functors, Definition 2.6 being broken due to the associators not being adjointable morphisms.

Definition 2.7. A functor $F\;:\;{\mathcal{C}} \to {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}$ is a weak unitary tensor functor if it is unitary and if it is equipped with isometries

\begin{equation*}F^2 = \{ F^2_{X,Y}\;:\; F(X) \boxtimes F(Y) \to F(X \boxtimes Y) \ | \ X,Y \in {\mathcal{C}}\}, \end{equation*}

not necessarily adjointable, satisfying pentagon coherence.

We wish to introduce now the concept of duality in C $^*$ -tensor categories. As a motivating example, let us fix a unital C $^*$ -algebra $A$ and consider the subcategory $\mathsf{Bim}_{\mathsf{fgp}}(A)$ of ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}$ consisting of bi-Hilbertian, finitely generated projective correspondences over $A$ . If $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ , it admits a right Pimsner–Popa basis, consisting of a finite set $\{\xi _i\}_i \subset K$ such that

\begin{equation*} \eta = \sum _i \xi _i \triangleleft {\langle \eta ,\xi _i \rangle }{_A} , \ \forall \ \eta \in K . \end{equation*}

In particular, $\{\xi _i\}$ is a projective basis for $K$ over $A$ , so ‘ $K$ has finite dimension’ over $A$ . Because $K$ also has a left $A$ -valued inner product, the conjugate space $\bar {K}$ can also be equipped with the structure of a right $A$ -correspondence, and moreover $\bar {K} \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ . Denoting by $\underset {A}{\boxtimes }$ the fusion of correspondences over $A$ , it follows that

\begin{equation*} \sum _i \bar {\xi _i} \underset {A}{\boxtimes } \xi _i \in \bar {K} \underset {A}{\boxtimes } K \end{equation*}

is an $A$ -central vector, providing a canonical $A{-}A$ -bimodular linear map $R_K\;:\; A \to \bar {K} \underset {A}{\boxtimes } K$ .

Given a tensor category $\mathcal{C}$ , we will denote a choice of tensor unit for $\mathcal{C}$ by $\unicode{x1D7D9}$

Definition 2.9. Let $\mathcal{C}$ be a C $^*$ -tensor category. An object $X \in {\mathcal{C}}$ is dualizable if there exist: an object $\bar {X} \in {\mathcal{C}}$ , morphisms $R_X \;:\; \unicode{x1D7D9} \to \bar {X} \boxtimes X$ , $\bar {R}_X \;:\; \unicode{x1D7D9} \to X \boxtimes \bar {X}$ such that

commute.

A pair $(R_X, \bar {R}_X)$ is said to be a solution of the conjugate equations for $X$ .

Definition 2.11. Let $\mathcal{C}$ be a rigid C $^*$ -tensor category, and let $X$ be an object of $\;\mathcal{C}$ . The intrinsic dimension $d_X$ of $X$ in $\mathcal{C}$ is defined by

\begin{equation*} d_X \;:\!=\; \underset {(R_X, \bar {R}_X)}{\min } \|R_X\| \cdot \| \bar {R}_X\| , \end{equation*}

where $(R_X, \bar {R}_X)$ runs through the solutions of the conjugate equations for $X$ .

The function defined on the set of objects of $\mathcal{C}$ given by $X \mapsto d_X$ is a dimension function. In particular, $d_{\unicode{x1D7D9}} = 1$ , $d_{X \boxtimes Y} = d_X d_Y$ , and $d_{X \oplus Y} = d_X + d_Y$ .

2.3. Ind-completions and C*-algebra objects

The treatment of infinite index inclusions will require the use of objects that are not dualizable, but that can be decomposed into dualizable objects. We shall now introduce the formalism needed to describe this situation. The reader is referred to [Reference Jones and Penneys20] for more details.

To specify an object $V \in {\mathsf {Vec}}({\mathcal{C}})$ , it suffices, up to isomorphism, to specify its values, or fibers, at the objects in $\textrm {Irr}({\mathcal{C}})$ , that is, a family $\{V(X)\}_{X \in \textrm {Irr}({\mathcal{C}})}$ of vector spaces indexed by $\textrm {Irr}({\mathcal{C}})$ . Another useful perspective is that of understanding the object $V = \{V_X\}_{X \in \textrm {Irr}({\mathcal{C}})} \in {\mathsf {Vec}}({\mathcal{C}})$ as a formal direct sum

\begin{equation*} V = \bigoplus _{X \in \textrm {Irr}({\mathcal{C}})} X \odot V(X) \end{equation*}

of objects in $\mathcal{C}$ , where the irreducible object $X$ appears $\dim V(X)$ times as a direct summand. We say that $V(X)$ is the multiplicity space of $X$ in $V$ . Moreover, given $V_1 = \{V_1(X)\}_X$ and $V_2=\{V_2(X)\}_X$ in ${\mathsf {Vec}}({\mathcal{C}})$ , it follows that

\begin{equation*} {\mathsf {Vec}}({\mathcal{C}})(V_1,V_2) \simeq \prod _{X \in \textrm {Irr}({\mathcal{C}})} {\textrm {Hom}}_{{\mathsf {Vec}}}(V_1(X),V_2(X)) , \end{equation*}

where ${\textrm {Hom}}_{\mathsf {Vec}}(\cdot ,\cdot )$ denotes the space of linear maps.

The category $\mathcal{C}$ embeds into ${\mathsf {Vec}}({\mathcal{C}})$ as a linear full subcategory, and the tensor structure $\boxtimes$ in $\mathcal{C}$ can be extended to a tensor structure $\boxdot$ on ${\mathsf {Vec}}({\mathcal{C}})$ : for $V_1$ and $V_2$ as above,

\begin{equation*} (V_1 \boxdot V_2)(X) \;:\!=\; \bigoplus _{Y,Z \in \textrm {Irr}({\mathcal{C}})} {\mathcal{C}}(X, Y \boxtimes Z) \odot V_1(Y) \odot V_2(Z) . \end{equation*}

An object $H \in \mathsf {Hilb}({\mathcal{C}})$ is determined by a family $\{H(X)\}_{X \in \textrm {Irr}({\mathcal{C}})}$ of Hilbert spaces. We have

\begin{equation*} \mathsf {Hilb}({\mathcal{C}})(H_1,H_2) \simeq \prod _{X \in \textrm {Irr}({\mathcal{C}})}^{\ell ^\infty } \mathcal{L}(H_1(X),H_2(X)) . \end{equation*}

The category $\mathsf {Hilb}({\mathcal{C}})$ is a C $^*$ -category (actually a W $^*$ -category), and $\mathcal{C}$ is a full C $^*$ -subcategory of it. The tensor structure of $\mathcal{C}$ extends to $\mathsf {Hilb}({\mathcal{C}})$ ; seeing ${\mathcal{C}}(X,Y \boxtimes Z)$ as a Hilbert space,

\begin{equation*} (H_1 \boxtimes H_2)(X) = \bigoplus _{Y,Z \in \textrm {Irr}({\mathcal{C}})}^{\ell ^2} {\mathcal{C}}(X, Y \boxtimes Z) \otimes H_1(Y) \otimes H_2(Z) , \end{equation*}

where on the $(Y,Z)$ -direct summand the inner product is scaled by a factor of $(d_Y d_Z)^{-1}$ . The purpose of this renormalization is to have a clean interpretation of the induced inner product by means of graphical calculus (see [Reference Jones and Penneys20], Section 2).

Since ${\mathsf {Vec}}({\mathcal{C}})$ is a tensor category, we can consider algebra objects in it. An algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ consists of a functor $\mathbb{A} \;:\; {\mathcal{C}}^{\textrm {op}} \to {\mathsf {Vec}}$ together with a lax-natural transformation

\begin{equation*}\mathbb{A}^2 = \{ \mathbb{A}^2_{X,Y} \;:\; \mathbb{A}(X) \odot \mathbb{A}(Y) \to \mathbb{A}(X \boxtimes Y) \ | \ X,Y \in {\mathcal{C}}\} ,\end{equation*}

satisfying coherence with respect to the pentagon diagram associated with the orderings of triple tensor products. Furthermore, we say an algebra object $\mathbb{A}$ is connected if $\mathbb{A}(\unicode{x1D7D9})\cong \mathbb{C}$ , and else we say $\mathbb{A}$ is non-connected.

Definition 2.17. A $*$ -structure on an algebra object $\mathbb{A}$ is a conjugate linear natural transformation

\begin{equation*} j^{\mathbb{A}} = \{ j^{\mathbb{A}}_X \;:\; \mathbb{A}(X) \to \mathbb{A}(\bar {X})\}_{X \in {\mathcal{C}}} , \end{equation*}

satisfying

• • involution: $j^{\mathbb{A}}_{\bar {X}} \circ j^{\mathbb{A}}_X = {\textrm {id}}_{\mathbb{A}(X)}$ , under the identification $\bar {\bar {X}} \simeq X$ ;

• • unitality: $j^{\mathbb{A}}_{1_{\mathcal{C}}} = {\textrm {id}}_{\mathbb{A}(1_{\mathcal{C}})}$ , under the identification $\bar {\unicode{x1D7D9}} \simeq \unicode{x1D7D9}$ ;

• • monoidality: the diagram

  

  commutes.

If $\mathbb{A} = (\mathbb{A}, \mathbb{A}^2)$ is an algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ , then the vector spaces $\mathbb{A}(\bar {X} \boxtimes X)$ become algebras, with multiplication given by

\begin{equation*} \mathbb{A}(\bar {X} \boxtimes X) \odot \mathbb{A}(\bar {X} \boxtimes X) \overset {\mathbb{A}^2}{\to } \mathbb{A}( \bar {X} \boxtimes X \boxtimes \bar {X} \boxtimes X) \overset {\mathbb{A}({\textrm {id}}_{\bar {X}} {\boxtimes \bar {R}_X^* \boxtimes {\textrm {id}}_X)}}{\to } \mathbb{A}(\bar {X} \boxtimes X) . \end{equation*}

If $j^{\mathbb{A}}$ is a $*$ -structure on $\mathbb{A}$ , then $\mathbb{A}(\bar {X} \boxtimes X)$ becomes $*$ -algebra, with involution given by $j^{\mathbb{A}}_{\bar {X} \boxtimes X}$ .

Remark 2.19. C*-algebra objects can be equivalently defined in terms of C*-module categories as follows: Given a $*$ -algebra object $\mathbb{A}$ as above, the category $\mathcal{M}_{\mathbb{A}}$ generated under finite direct sums and idempotent completion by objects of the form $\mathbb{A} \odot X$ , with $X \in {\mathcal{C}}$ , and with morphisms

\begin{equation*} \mathcal{M}_{\mathbb{A}}(\mathbb{A} \odot X, \mathbb{A} \odot Y) \;:\!=\; {\mathsf {Vec}}({\mathcal{C}})(X, \mathbb{A} \odot Y) \simeq \mathbb{A}(X \otimes \bar {Y}) \end{equation*}

has a right $\mathcal{C}$ -module structure. $\mathbb{A}$ is a C $^*$ -algebra object if and only if $\mathcal{M}_{\mathbb{A}}$ is a $\mathcal{C}$ -module C $^*$ -category [Reference Jones and Penneys20 , Definition 25]. In particular, if $\mathbb{A}$ is a C $^*$ -algebra object, $\mathbb{A}(X)$ has a Banach space structure for every $X \in {\mathcal{C}}$ .

Definition 2.20 [Reference Jones and Penneys20, Definition 22]. Let $\mathbb{D}_1$ and $\mathbb{D}_2$ be C $^*$ -algebra objects in ${\mathsf {Vec}}({\mathcal{C}})$ . A natural transformation $\theta \;:\; \mathbb{D}_1 \to \mathbb{D}_2$ is called a $*$ -natural transformation if

\begin{equation*} j^{\mathbb{D}_2} \circ \theta = \theta \circ j^{\mathbb{D}_1} . \end{equation*}

A $*$ -natural transformation $\theta \;:\; \mathbb{D}_1 \to \mathbb{D}_2$ is called a ucp map if, for all $X \in {\mathcal{C}}$ , $\theta \;:\; \mathbb{D}_1(\bar {X} \boxtimes X) \to \mathbb{D}_2(\bar {X} \boxtimes X)$ is a ucp map.

3.1. Intrinsic characterization of internal C*-algebra objects

In this subsection, we recast the results in [Reference Hataishi and Yamashita16, Appendix A] and [Reference Jones and Penneys20, section 4.2] concerning Pimsner–Popa inequaltities that arise naturally when considering C $^*$ -algebra objects in UTCs. Later, these results will be our main tools for producing faithful conditional expectations.

Let $\mathcal{C}$ be a UTC and $(\mathbb{D}, \mathbb{D}^2, j^{\mathbb{D}})$ be a C*-algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ . Recall in particular that each fiber $\mathbb{D}(X)$ has a canonical Banach space structure, the norm of which we denote simply by $\| \cdot \|$ , and that $\mathbb{D}(\overline {X} \boxtimes X)$ is a C*-algebra, for all $X \in {\mathcal{C}}$ .

Definition 3.1. Given a C*-algebra object $\mathbb{D}\in {\mathsf {Vec}}({\mathcal{C}})$ , we call $\mathbb{D}(\unicode{x1D7D9})$ the ground C*-algebra of $\mathbb{D}$ . Furthermore, for each $X \in {\mathcal{C}}$ , fix a standard solution to the conjugate equations $(R_X,\overline {R}_X)$ for the pair $(X, \overline {X})$ . Denoting by $d_X$ the dimension of $X$ in $\mathcal{C}$ , we define the map

\begin{equation*} \mathbb{E}^{\mathbb{D}}_X \;:\!=\; d_X^{- 1} \mathbb{D}(R_X) \;:\; \mathbb{D}(\overline {X} \boxtimes X)\to \mathbb{D}(\unicode{x1D7D9}). \end{equation*}

Lemma 3.2 [Reference Jones and Penneys20, Corollary 2]. The map $\mathbb{E}^{\mathbb{D}}_X \;:\; \mathbb{D}(\overline {X} \boxtimes X) \to \mathbb{D}(\unicode{x1D7D9})$ is a conditional expectation. For all $T \in \mathbb{D}(\overline {X} \boxtimes X)_+$ , we have that

\begin{equation*} \| \mathbb{E}^{\mathbb{D}}_X(T) \| \leq \|T\| \leq d_X^2 \| \mathbb{E}^{\mathbb{D}}_X(T) \| . \end{equation*}

That is, the conditional expectation $\mathbb{E}_X^{\mathbb{D}}$ has finite Pimsner–Popa index.

Lemma 3.2 implies that $\mathbb{E}_X^{\mathbb{D}}$ is a faithful conditional expectation. Let $X \in {\mathcal{C}}$ . Then

\begin{align*} \mathbb{D}^2_{X, \unicode{x1D7D9}} & \;:\; \mathbb{D}(X) \odot \mathbb{D}(\unicode{x1D7D9}) \to \mathbb{D}(X \boxtimes \unicode{x1D7D9}) \simeq \mathbb{D}(X) \\ \mathbb{D}^2_{\unicode{x1D7D9},X} & \;:\; \mathbb{D}(\unicode{x1D7D9}) \odot \mathbb{D}(X) \to \mathbb{D}(\unicode{x1D7D9} \boxtimes X) \simeq \mathbb{D}(X) \end{align*}

gives to $\mathbb{D}(X)$ the structure of a $\mathbb{D}(\unicode{x1D7D9})$ -bimodule. We claim that this structure can be further enhanced to that of a right C*-correspondence, allowing us to see $\mathbb{D}(X)$ as an object in ${\mathsf{C}^*\mathsf{Alg}}_{\mathbb{D}(\unicode{x1D7D9}) {-} \mathbb{D}(\unicode{x1D7D9})}$ .

For a proof of the following lemma, see [Reference Jones and Penneys20, section 4.2].

Lemma 3.3. For each $X \in {\mathcal{C}}$ , the composition

is a $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product on $\mathbb{D}(X)$ . For $\xi ,\eta \in \mathbb{D}(X)$ , we shall write

\begin{equation*} \langle \xi , \eta \rangle _{\mathbb{D}(\unicode{x1D7D9})} \;:\!=\; \mathbb{E}^{\mathbb{D}}_X \left ( \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}(\xi ) \odot \eta ) \right ) . \end{equation*}

For $\xi \in \mathbb{D}(X)$ , let us write $\| \xi \|_{\mathbb{D}(\unicode{x1D7D9})}$ for the norm of $\xi$ in $ {_{\mathbb{D}(\unicode{x1D7D9})}}{\mathbb{D}(X)}{_{\mathbb{D}(\unicode{x1D7D9})}}$ , that is, the norm coming from the $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product. Note that, for $\xi \in \mathbb{D}(\unicode{x1D7D9})$ , $\| \xi \| = \|\xi \|_{\mathbb{D}(\unicode{x1D7D9})}$ .

Lemma 3.4. For every $\xi \in \mathbb{D}(X)$ ,

\begin{equation*} \| \xi \|^2 = \| \mathbb{D}^2_{\overline {X},X} (j^{\mathbb{D}}_X(\xi ) \odot \xi ) \| . \end{equation*}

Proof. In terms of the $\mathcal{C}$ -module C $^*$ -category $\mathcal{M}_{\mathbb{D}}$ associated with $\mathbb{D}$ and the identification $\Psi \;:\; \mathbb{D}(X) \simeq \mathcal{M}_{\mathbb{D}}(\mathbb{D} \odot X, \mathbb{D})$ , the right-hand side is just the norm of $\Psi (\xi )^* \circ \Psi (\xi )$ . Since $\mathcal{M}_{\mathbb{D}}$ is a C*-category,

\begin{equation*} \| \Psi (\xi )^* \circ \Psi (\xi ) \| = \| \Psi (\xi ) \|^2 = \|\xi \|^2. \end{equation*}

From Proposition 6 in [Reference Jones and Penneys20], it follows that the norms $\| \cdot \|$ and $\| \cdot \|_{\mathbb{D}(\unicode{x1D7D9})}$ in the vector space $\mathbb{D}(X)$ are equivalent. The proof of this statement will be important for the coend realization of Section 3, given us the motive to write it down here.

Corollary 3.5. Let $\mathcal{C}$ be a UTC, $\mathbb{D}$ a C*-algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ , and let $X \in {\mathcal{C}}$ . For every $\xi \in \mathbb{D}(X)$ , it holds

\begin{equation*} \| \xi \|_{\mathbb{D}(\unicode{x1D7D9})} \leq \| \xi \| \leq d_X \| \xi \|_{\mathbb{D}(\unicode{x1D7D9})} . \end{equation*}

In particular, $\mathbb{D}(X)$ is complete with respect to the $\mathbb{D}(\unicode{x1D7D9})$ -Hilbert C $^*$ -module norm, that is, $\mathbb{D}(X) \in {\mathsf{C}^*\mathsf{Alg}}_{\mathbb{D}(\unicode{x1D7D9}) {-} \mathbb{D}(\unicode{x1D7D9})}$ .

Proof. First, observe that

\begin{align*} \| \xi \|_{\mathbb{D}(\unicode{x1D7D9})}^2 & = \| \mathbb{E}^{\mathbb{D}}_X \left ( \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}_X(\xi ) \odot \xi ) \right ) \| \\ & \leq \| \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}_X(\xi ) \odot \xi ) \| \\ & = \| \xi \|^2 . \end{align*}

The last equality follows from the previous lemma. On the other hand, by Lemma 3.2, we have

\begin{align*} \| \xi \|^2 & = \| \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}_X(\xi ) \odot \xi ) \| \leq d_X^2 \| \mathbb{E}_X ( \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}_X(\xi ) \odot \xi )) \| = d_X^2 \|\xi \|_{\mathbb{D}(\unicode{x1D7D9})}^2 . \end{align*}

Thus, if $\mathbb{D}$ is a C*-algebra object in a UTC $\mathcal{C}$ , then $\mathbb{D}$ has the structure of a lax tensor functor $\mathbb{D} \;:\; {\mathcal{C}}^{\textrm {op}} \to {\mathsf{C}^*\mathsf{Alg}}_{\mathbb{D}(\unicode{x1D7D9}){-}\mathbb{D}(\unicode{x1D7D9})}$ . It can be deduced moreover that the components of the lax structure $\mathbb{D}^2$ are isometries, possibly non-adjointable, that is, $\mathbb{D}^2$ is a weak unitary tensor functor as introduced in the previous section.

Lemma 3.6. Let $\mathbb{D}$ be a C $^*$ -algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ . Then $\mathbb{D}^2$ induces isometric linear maps

\begin{equation*} \mathbb{D}(X) \underset {\mathbb{D}(\unicode{x1D7D9})}{\boxtimes } \mathbb{D}(Y) \to \mathbb{D}(X \boxtimes Y) \end{equation*}

for $X,Y \in {\mathcal{C}}$ .

Proof. Fix $X,Y \in {\mathcal{C}}$ . It follows from the axioms of $\mathbb{D}^2$ that $\mathbb{D}^2_{X,Y} \;:\; \mathbb{D}(X) \odot \mathbb{D}(Y)\to \mathbb{D}(X\boxtimes Y)$ is $\mathbb{D}(\unicode{x1D7D9})$ -balanced. Using the right $\mathcal{C}$ -module C $^*$ -category $\mathcal{M}_{\mathbb{D}}$ associated with $\mathbb{D}$ , given $f \in \mathcal{M}_{\mathbb{D}}(\mathbb{D} \odot X, \mathbb{D})$ and $g \in \mathcal{M}_{\mathbb{D}}(\mathbb{D} \odot Y, \mathbb{D})$ , we have

\begin{equation*} \mathbb{D}^2_{X,Y}( f \odot g) = g (f \odot {\textrm {id}}_Y) \;:\; \mathbb{D} \odot (X \boxtimes Y) \simeq (\mathbb{D} \odot X) \odot Y \to \mathbb{D} , \end{equation*}

and that

\begin{align*} \langle \mathbb{D}^2_{X,Y}( f \odot g), \mathbb{D}^2_{X,Y}( f \odot g) \rangle _{\mathbb{D}(\unicode{x1D7D9})} &= g (f \odot {\textrm {id}}_Y) (f^* \odot {\textrm {id}}_Y) g^* \\ & = \langle g , \langle f,f \rangle _{\mathbb{D}(\unicode{x1D7D9})} \triangleright g \rangle _{\mathbb{D}(\unicode{x1D7D9})} \\ & = \langle f \boxtimes g, f \boxtimes g \rangle _{\mathbb{D}(\unicode{x1D7D9})} . \end{align*}

We summarize our conclusions in the following corollary.

To finish the preliminary section, we have one final observation. This appears in Appendix A of [Reference Hataishi and Yamashita16].

Proof. The action is defined, for $a \in \mathbb{D}(X)$ and $x \in \mathbb{D}(\overline {X} \boxtimes X)$ , by

\begin{equation*} a \triangleleft x \;:\!=\; \mathbb{D}(R_X \boxtimes {\textrm {id}}_X) \left (\mathbb{D}^2_{X, \overline {X} \boxtimes X}( a \odot x) \right ) . \end{equation*}

The $\mathbb{D}(\overline {X} \boxtimes X)$ -valued inner product is given by

\begin{equation*} \langle a,a' \rangle _{\mathbb{D}(\bar {X} \boxtimes X)} \;:\!=\; \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}(a) \odot a') , \end{equation*}

which is shown to be $\mathbb{D}(\overline {X} \boxtimes X)$ -linear by means of the following computation: by definition,

\begin{align*} \langle a, a' \triangleleft x \rangle _{\mathbb{D}(\bar {X} \boxtimes X)} & = \mathbb{D}^2_{\overline {X},X}\left (j^{\mathbb{D}}(a) \odot \mathbb{D}(R_X \boxtimes {\textrm {id}}_X) (\mathbb{D}^2_{X,\overline {X} \boxtimes X}(a' \odot x))\right ) \end{align*}

The right-hand side of the above equation can be written as:

\begin{align*} \mathbb{D}^2_{\bar {X},X} \circ ({\textrm {id}}_{\mathbb{D}(\bar {X})} \odot \mathbb{D}(R_X \boxtimes {\textrm {id}}_X)) \left (j^{\mathbb{D}}(a) \odot (\mathbb{D}^2_{X,\overline {X} \boxtimes X}(a' \odot x))\right ) \end{align*}

Now, by the naturality and lax property of $\mathbb{D}^2$ ,

\begin{align*} \mathbb{D}^2_{\bar {X},X} \circ ({\textrm {id}}_{\mathbb{D}(\bar {X})} \odot \mathbb{D}(R_X \boxtimes {\textrm {id}}_X)) = \mathbb{D}^2_{\bar {X},X} \circ (\mathbb{D}({\textrm {id}}_{\bar {X}}) \odot \mathbb{D}(R_X \boxtimes {\textrm {id}}_X)) = \mathbb{D}({\textrm {id}}_{\overline {X}} \boxtimes R_X \boxtimes {\textrm {id}}_X) \circ \mathbb{D}^2_{\overline {X},X \boxtimes \overline {X} \boxtimes X} \end{align*}

Therefore,

\begin{align*} \langle a, a' \triangleleft x \rangle _{\mathbb{D}(\bar {X} \boxtimes X)} & = \mathbb{D}({\textrm {id}}_{\overline {X}} \boxtimes R_X \boxtimes {\textrm {id}}_X) \mathbb{D}^2_{\overline {X},X \boxtimes \overline {X} \boxtimes X} \left ( j^{\mathbb{D}}(a) \odot \mathbb{D}^2_{X,\overline {X} \boxtimes X}(a' \odot x) \right ) \end{align*}

Due to the associativity axiom,

\begin{align*} \mathbb{D}^2_{\overline {X},X \boxtimes \overline {X} \boxtimes X} \left ( j^{\mathbb{D}}(a) \odot \mathbb{D}^2_{X,\overline {X} \boxtimes X}(a' \odot x) \right ) = \mathbb{D}^2_{\overline {X} \boxtimes X,{\overline {X} \boxtimes X}}( \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}(a) \odot a') \odot x), \end{align*}

implying

\begin{align*} \\ \langle a, a' \triangleleft x \rangle _{\mathbb{D}(\bar {X} \boxtimes X)} & = \mathbb{D}({\textrm {id}}_{\overline {X}} \boxtimes R_X \boxtimes {\textrm {id}}_X) \mathbb{D}^2_{\overline {X} \boxtimes X,{\overline {X} \boxtimes X}}( \mathbb{D}^2_{\overline {X},X}(j^{\mathbb{D}}(a) \odot a') \odot x) \\ & = \langle a,a'\rangle _{\mathbb{D}(\bar {X} \boxtimes X)} \cdot x . \end{align*}

The right Hilbert $\mathbb{D}(\unicode{x1D7D9})$ -module structure of $\mathbb{D}(X)$ is induced from the Hilbert $\mathbb{D}(\overline {X} \boxtimes X)$ -module structure together with the conditional expecation $\mathbb{E}^{\mathbb{D}}_X \;:\; \mathbb{D}(\overline {X} \boxtimes X) \to \mathbb{D}(\unicode{x1D7D9})$ , meaning

\begin{equation*} \langle x,y\rangle _{\mathbb{D}(\unicode{x1D7D9})} = \mathbb{E}_X^{\mathbb{D}}(\langle x,y\rangle _{\mathbb{D}(\bar {X} \boxtimes X)}) .\end{equation*}

Thus the Banach space structure of $\mathbb{D}(X)$ also coincides with the one coming from the right Hilbert $\mathbb{D}(\overline {X} \boxtimes X)$ -module structure. From this, it follows that the topology on $\mathbb{D}(X)$ induced by the $\mathbb{D}(\bar {X} \boxtimes X)$ -valued inner product is the same as the one induced by the $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product due to finiteness of the index of $\mathbb{E}_X^{\mathbb{D}}$ . In particular, $\mathbb{D}(X)$ is closed for the former topology.

3.2. Coend realization

Given an arbitrary UTC $\mathcal{C}$ and C*-algebra objects $\mathbb{A}$ and $\mathbb{B}$ in ${\mathsf {Vec}}({\mathcal{C}}^{\textrm {op}})$ and ${\mathsf {Vec}}({\mathcal{C}})$ , respectively, it was shown in [Reference Jones and Penneys21], Definition4.12 and Corollary 4.13, that the $*$ -algebra

\begin{equation*} |\mathbb{A} \times \mathbb{B}| \;:\!=\; \bigoplus _{X \in \textrm {Irr}({\mathcal{C}})} \mathbb{A}(X) \odot \mathbb{B}(X) \end{equation*}

has a universal C*-completion. We show in this section that there is a canonical minimal C*-completion $\mathbb{A} \bowtie \mathbb{B}$ of $|\mathbb{A} \times \mathbb{B}|$ , analogous to regular representations, featuring a faithful conditional expectation $\mathbb{A} \bowtie \mathbb{B} \to \mathbb{A}(\unicode{x1D7D9}) \otimes \mathbb{B}(\unicode{x1D7D9})$ . This is analogous to the comparison between universal and reduced group C $^*$ -algebras.

For $X \in {\mathcal{C}}$ , as in Definition 3.1 we write $\mathbb{E}^{\mathbb{A}}_X$ and $\mathbb{E}^{\mathbb{B}}_X$ for the conditional expectations $\mathbb{A}(\overline {X} \boxtimes X) \to \mathbb{A}(\unicode{x1D7D9})$ and $\mathbb{B}(\overline {X} \boxtimes X) \to \mathbb{B}(\unicode{x1D7D9})$ , respectively. Let $\mathbb{A}(X) \otimes \mathbb{B}(X)$ be the exterior tensor product of the $\mathbb{A}(\unicode{x1D7D9})$ - $\mathbb{A}(\unicode{x1D7D9})$ -correspondence $\mathbb{A}(X)$ with the $\mathbb{B}(\unicode{x1D7D9})$ - $\mathbb{B}(\unicode{x1D7D9})$ -correspondence $\mathbb{B}(X)$ and define

\begin{equation*} {\mathcal{E}} \;:\!=\; \bigoplus _{X \in \textrm {Irr}({\mathcal{C}})}^{\ell ^2} \mathbb{A}(X) \otimes \mathbb{B}(X) \end{equation*}

to be the completion of $\bigoplus _{X \in \textrm {Irr}({\mathcal{C}})} \mathbb{A}(X) \otimes \mathbb{B}(X)$ as a $(\mathbb{A}(\unicode{x1D7D9}) \otimes \mathbb{B}(\unicode{x1D7D9}))$ - $(\mathbb{A}(\unicode{x1D7D9}) \otimes \mathbb{B}(\unicode{x1D7D9}))$ -correspondence.

Given irreducible objects $X_0,X_1,X_2 \in {\mathcal{C}}$ , there is a Hilbert space structure on ${\mathcal{C}}(X_0, X_1 \boxtimes X_2)$ given by $\langle x, y \rangle \;:\!=\; x^* y$ . Let us fix, for each triple $(X_0,X_1,X_2)$ of irreducible objects, a choice of an orthonormal basis $O(X_0,X_1 \boxtimes X_2)$ for this Hilbert space.

Lemma 3.9. Let $\mathcal{C}$ be a UTC and $\mathbb{D}$ a C*-algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ . For $X_0,X_1,X_2 \in \textrm {Irr}({\mathcal{C}})$ , given $\xi _0 \in \mathbb{D}(X_0)$ , the linear map $\xi _0 \triangleright -:\; \mathbb{D}(X_1) \to \mathbb{D}(X_2)$ given by

\begin{equation*}\xi _0 \triangleright \xi _1 \;:\!=\; \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \mathbb{D}(v) \left ( \mathbb{D}^2_{X_0,X_1}(\xi _0 \odot \xi _1) \right )\end{equation*}

is an adjointable map of $\mathbb{D}(\unicode{x1D7D9})$ -Hilbert C*-modules. Its adjoint is then given by $j^{\mathbb{D}}_{X_0}(\xi _0) \triangleright (-)$ .

Proof. Given $X \in {\mathcal{C}}$ , the definition of the $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product on $\mathbb{D}(X)$ is

\begin{equation*}\langle \xi , \xi ' \rangle _{\mathbb{D}(\unicode{x1D7D9})} \;:\!=\; \frac {1}{d_X}\mathbb{D}(R_X) \left ( \mathbb{D}^2_{\overline {X},X} ( j^{\mathbb{D}}(\xi ) \odot \xi ') \right ) . \end{equation*}

Then, for $\xi _i \in \mathbb{D}(X_i)$ , $i \in \{0,1,2\}$ ,

\begin{align*} \langle \xi _0 \triangleright \xi _1, \xi _2 \rangle _{\mathbb{D}(\unicode{x1D7D9})} & = \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \frac {1}{d_{X_2}}\langle \mathbb{D}(v) \left ( \mathbb{D}^2_{X_0,X_1}(\xi _0\odot \xi _1)\right ),\xi _2\rangle _{\mathbb{D}(\unicode{x1D7D9})} \\ &=\sum _{v \in O(X_2,X_0 \boxtimes X_1)}\frac {1}{d_{X_2}}\mathbb{D}(R_{X_2})\big(\mathbb{D}^2_{\overline {X}_2,X_2} \left(j^{\mathbb{D}}_{X_2}\left(\left(\left(\mathbb{D}(v)\left(\mathbb{D}^2_{X_0,X_1}\left(\xi _0\odot \xi _1\right)\right)\right)\right) \odot \xi _2 \right)\right) , \end{align*}

by definition. Let

\begin{equation*}(v^*)^\vee \;:\!=\; (\bar {R}_{X_2}^* \boxtimes {\textrm {id}}_{\bar {X}_1 \boxtimes \bar {X}_0}) ({\textrm {id}}_{\bar {X}_2} \boxtimes v^* \boxtimes {\textrm {id}}_{\bar {X}_1 \boxtimes \bar {X}_0}) ({\textrm {id}}_{\bar {X}_2} \boxtimes R_{X_0 \boxtimes X_1}). \end{equation*}

From monoidality and conjugate linear naturality of the $*$ -structure $j^{\mathbb{D}}$ , it follows that

\begin{equation*}j^{\mathbb{D}}_{X_2}\big(\left(\left(\mathbb{D}(v)\left(\mathbb{D}^2_{X_0,X_1}(\xi _0\odot \xi _1)\right) \right)\right) = \left ( \mathbb{D}((v^*)^\vee ) \mathbb{D}^2_{\overline {X}_1,\overline {X}_0} (j^{\mathbb{D}}_{X_1}(\xi _1) \odot j^{\mathbb{D}}_{X_0}(\xi _0) )\right ) . \end{equation*}

Consequently,

\begin{align*} \langle \xi _0 \triangleright \xi _1, \xi _2 \rangle _{\mathbb{D}(\unicode{x1D7D9})} & = \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \frac {1}{d_{X_2}}\mathbb{D}(R_{X_2}) \left ( \mathbb{D}^2_{\overline {X}_2,X_2} \left ( \mathbb{D}((v^*)^\vee ) \mathbb{D}^2_{\overline {X}_1,\overline {X}_0} (j^{\mathbb{D}}_{X_1}(\xi _1) \odot j^{\mathbb{D}}_{X_0}(\xi _0) )\right ) \odot \xi _2 \right ) \\ & \overset {(1)}{=} \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \frac {1}{d_{X_2}} \mathbb{D}(R_{X_2}) (\mathbb{D}((v^*)^\vee )\odot {\textrm {id}}) \left ( \mathbb{D}^2_{\overline {X}_1 \boxtimes \overline {X}_0,X_2} \left ( \mathbb{D}^2_{\overline {X}_1,\overline {X}_0} ( j^{\mathbb{D}}_{X_1}(\xi _1) \odot j^{\mathbb{D}}_{X_0}(\xi _0)) \odot \xi _2 \right ) \right ) \\ & \overset {(2)}{=} \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \frac {1}{d_{X_2}}\mathbb{D}(R_{X_2}) \left ( \mathbb{D}((v^*)^\vee \boxtimes {\textrm {id}} ) \left ( \mathbb{D}^2_{\overline {X}_1, \overline {X}_0 \boxtimes X_2}\left ( j^{\mathbb{D}}_{X_1}(\xi _1) \odot \mathbb{D}^2_{\overline {X}_0,X_2} (j^{\mathbb{D}}_{X_0}(\xi _0) \odot \xi _2) \right ) \right ) \right ) \\ & \overset {(3)}{=} \sum _{v \in O(X_2,X_0 \boxtimes X_1)} \frac {1}{d_{X_2}} \mathbb{D}(R_{X_0 \boxtimes X_1}) \left (\mathbb{D}({\textrm {id}} \boxtimes v^* ) \left ( \mathbb{D}^2_{\overline {X}_1, \overline {X}_0 \boxtimes X_2} \left ( j^{\mathbb{D}}_{X_1}(\xi _1) \odot \mathbb{D}^2_{\overline {X}_0,X_2} (j^{\mathbb{D}}_{X_0}(\xi _0) \odot \xi _2) \right ) \right ) \right ) \\ & \overset {(4)}{=} \sum _{w \in O(X_1,\overline {X}_0 \boxtimes X_2)} \frac {1}{d_{X_1}}\mathbb{D}(R_{X_1}) \left ( \mathbb{D}({\textrm {id}} \boxtimes w) \left ( \mathbb{D}^2_{\overline {X}_1, \overline {X}_0 \boxtimes X_2} \left ( j^{\mathbb{D}}_{X_1}(\xi _1) \odot \mathbb{D}^2_{\overline {X}_0,X_2} (j^{\mathbb{D}}_{X_0}(\xi _0) \odot \xi _2) \right ) \right ) \right ) \\ & \overset {(5)}{=} \sum _{w \in O(X_1,\overline {X}_0 \boxtimes X_2)} \frac {1}{d_{X_1}} \mathbb{D}(R_{X_1}) \left ( \left ( \mathbb{D}^2_{\overline {X}_1, \overline {X}_0 \boxtimes X_2} \left ( j^{\mathbb{D}}_{X_1}(\xi _1)\odot \mathbb{D}(w) \mathbb{D}^2_{\overline {X}_0,X_2} (j^{\mathbb{D}}_{X_0}(\xi _0) \odot \xi _2) \right ) \right ) \right ) \\ & \overset {(6)}{=} \langle \xi _1, j^{\mathbb{D}}_{X_0}(\xi _0) \triangleright \xi _2 \rangle _{\mathbb{D}(\unicode{x1D7D9})}. \end{align*}

In the above computation, (1) follows from the lax property of $\mathbb{D}^2$ , (2) from associativiy of $\mathbb{D}^2$ , and (3) from the equality

\begin{equation*} \mathbb{D}(R_{X_2}) (\mathbb{D}((v^*)^\vee \boxtimes {\textrm {id}})) = \mathbb{D}( (v^*)^\vee \boxtimes {\textrm {id}})) R_{X_2}) = \mathbb{D}(({\textrm {id}} \boxtimes v^*) R_{X_0 \boxtimes X_1}) = \mathbb{D}(R_{X_2}) \mathbb{D}({\textrm {id}} \boxtimes v^*), \end{equation*}

which characterizes $(v^*)^\vee$ . For (4), we are making use of the canonical isomorphism

\begin{equation*}{\mathcal{C}}(X_2, X_0 \boxtimes X_1) \overset {\text{Frob.}}{\simeq } {\mathcal{C}}(\overline {X}_0 \boxtimes X_2,X_1) \overset {*}{\simeq } {\mathcal{C}}(X_1, \overline {X}_0 \boxtimes X_2) , \end{equation*}

given by

\begin{equation*} v \mapsto \frac {d_{X_2}}{d_{X_1}} ({\textrm {id}}_{\bar {X}_0} \otimes v^*) \circ (R_{X_0} \otimes {\textrm {id}}_{X_1}). \end{equation*}

Equality (5) uses once again the lax property of $\mathbb{D}^2$ , and finally (6) follows from the definition of the $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product.

Remark 3.10. As a consequence of the above computation, each map

\begin{equation*}\xi _1 \mapsto \mathbb{D}(\omega ) \left (\mathbb{D}^2_{X_0,X_1}(\xi _0 \odot \xi _1 )\right ) , \end{equation*}

with $\omega \in O(X_2, X_0 \boxtimes X_1)$ , is adjointable. We shall make use of this observation later.

Applying the Lemma above to the C $^*$ -algebra objects $\mathbb{A}$ and $\mathbb{B}$ simultaneously, we obtain the following.

Using the notation $\langle \cdot , \cdot \rangle$ for the $\mathbb{A}(\unicode{x1D7D9}) \otimes \mathbb{B}(\unicode{x1D7D9})$ -valued inner product on $\mathcal{E}$ , the formula

\begin{equation*} \mathbb{A} \bowtie \mathbb{B} \ni T \mapsto \mathbb{E}(T) \;:\!=\; \langle T \Omega , \Omega \rangle \end{equation*}

defines a conditional expectation $\mathbb{A} \bowtie \mathbb{B} \to \mathbb{A}(\unicode{x1D7D9}) \otimes \mathbb{B}(\unicode{x1D7D9})$ . Our goal is to show that this conditional expectation is faithful.

Lemma 3.13. Let $\sum _{i=1}^n a_i \odot b_i \in \mathbb{A}(X) \odot \mathbb{B}(X)$ . Then

\begin{equation*} \sum _{i,j=1}^n \mathbb{A}^2_{\overline {X},X}(j^{\mathbb{A}}_X(a_i) \odot a_j) \otimes \mathbb{B}^2_{\overline {X},X} (j^{\mathbb{B}}_X(b_i) \odot b_j) \end{equation*}

is positive in $\mathbb{A}(\overline {X} \boxtimes X) \otimes \mathbb{B}(\overline {X} \boxtimes X)$ .

Proof. Consider on $\mathbb{A}(X)$ the canonical right Hilbert $\mathbb{A}(\overline {X} \boxtimes X)$ -C*-module structure from Lemma 3.8. Observe that

\begin{equation*} \sum _{i,j=1}^n \mathbb{A}^2_{\overline {X},X} (j^{\mathbb{A}}_X(a_i) \odot a_j) \otimes \mathbb{B}^2_{\overline {X},X} (j^{\mathbb{B}}_X(b_i) \odot b_j) \end{equation*}

is the $\mathbb{A}(\overline {X} \boxtimes X) \otimes \mathbb{B}(\overline {X} \boxtimes X)$ -valued inner product of $\sum _i a_i \otimes b_i$ with itself in the exterior tensor product $\mathbb{A}(X) \otimes \mathbb{B}(X)$ .

Lemma 3.14. For $T \in \mathbb{A}(X) \odot \mathbb{B}(X) \subset \mathbb{A} \bowtie \mathbb{B}$ , it holds

\begin{equation*} \| T \Omega \| \leq \| T \|_{\mathbb{A} \bowtie \mathbb{B}} \leq d_X^2 \| T \Omega \| \end{equation*}

.

Proof. Writing $T = \sum _i a_i \odot b_i \in \mathbb{A}(X) \odot \mathbb{B}(X)$ , we have

\begin{equation*} \mathbb{E}(T^*T) = (E^{\mathbb{A}}_X \otimes E^{\mathbb{B}}_X) \left ( \sum _{i,j} \mathbb{A}^2_{\overline {X},X}(j^{\mathbb{A}}_X(a_i) \odot a_j) \otimes \mathbb{B}^2_{\overline {X},X} (j^{\mathbb{B}}_X(b_i) \odot b_j) \right ) . \end{equation*}

Since both $E^{\mathbb{A}}_X$ and $E^{\mathbb{B}}_X$ are conditional expectations with Pimsner–Popa index at most $d_X^2$ , $E^{\mathbb{A}}_X \otimes E^{\mathbb{B}}_X$ has Pimsner–Popa index at most $d_X^4$ . Moreover, the argument of $(E^{\mathbb{A}}_X \otimes E^{\mathbb{B}}_X)$ in the right-hand side of the above inequality is positive by Lemma 3.13. Thus,

\begin{equation*} \|T \Omega \|^2 = \|\mathbb{E}(T^*T) \| \leq \|T^*T\|_{\mathbb{A} \bowtie \mathbb{B}} \leq d_X^4 \| \mathbb{E}(T^*T) \| = d_X^4 \|T \Omega \|^2 . \end{equation*}

Now we prove Theorem B, which generalizes Proposition 3.7 in [Reference Palomares and Nelson14] to arbitrary C $^*$ -algebra objects instead of just connected ones. In that case, the proof relied on the fibers being finite-dimensional, while for a general C*-algebra object, we use the fact that its fibers are finite index over the base algebra.

Proof. Let $T \in \mathbb{A} \bowtie \mathbb{B}$ be such that $\mathbb{E}(T^*T) = 0$ . Let $(T_\lambda )_{\lambda \in \Lambda }$ be a net in $|\mathbb{A} \times \mathbb{B} |$ converging to $T$ in norm. Write

\begin{equation*}T_\lambda = \sum _{X \in \textrm {Irr}({\mathcal{C}})} T_\lambda ^{(X)} ,\end{equation*}

with $T_\lambda ^{(X)} \in \mathbb{A}(X) \odot \mathbb{B}(X)$ . Observe that $E(T^*T) = 0$ if and only if $T \Omega = 0$ . Let $P_X \;:\; {\mathcal{E}} \to \mathbb{A}(X) \otimes \mathbb{B}(X)$ be the canonical projection. Then we have

\begin{equation*} T_\lambda ^{(X)} \Omega = P_X T_\lambda \Omega \to P_X T \Omega = 0 .\end{equation*}

From Lemma 3.14, $T_\lambda ^{(X)} \Omega \to 0$ implies $T_\lambda ^{(X)} \to 0$ in the operator norm. Let $Y,Z \in \textrm {Irr}({\mathcal{C}})$ . For $S \in \mathbb{A}(Y) \odot \mathbb{B}(Y)$ , $T_\lambda ^{(X)} S \Omega$ is supported on the direct summands $\mathbb{A}(W) \otimes \mathbb{B}(W)$ of $\mathcal{E}$ for which $W \leq X \boxtimes Y$ , meaning ${\mathcal{C}}(W,X \boxtimes Y) \neq 0$ . Using Frobenius reciprocity in $\mathcal{C}$ , we have then

\begin{equation*}P_Z T_\lambda S \Omega = \sum _{X \leq Z \boxtimes \overline {Y}} P_Z T_\lambda ^{(X)} S \Omega .\end{equation*}

Observe that the sum above is over a finite index set. It follows that

\begin{equation*}P_Z T S \Omega = \lim _\lambda P_Z T_\lambda \Omega = \sum _{X \leq Z \boxtimes \overline {Y}} \lim _\lambda P_Z T_\lambda ^{(X)} S\Omega = 0 .\end{equation*}

Since

\begin{equation*}T S \Omega = \sum _{Z \in \textrm {Irr}({\mathcal{C}})} P_Z TS \Omega ,\end{equation*}

it follows that $TS \Omega = 0$ . From the density of $\{S \Omega \ | S \in \mathbb{A}(Y) \odot \mathbb{D}(Y) , \ Y \in \textrm {Irr}({\mathcal{C}})\}$ in $\mathcal{E}$ , we conclude that $T = 0$ .

Recall that an action ${\mathcal{C}} \curvearrowright A$ of a UTC $\mathcal{C}$ on a C $^*$ -algebra $A$ can in particular be seen as a C $^*$ -algebra object in ${\mathcal{C}}^{\textrm {op}}$ .

Definition 3.18. Let $\mathcal{C}$ be a unitary tensor category and $A$ a unital C $^*$ -algebra, and let $\alpha \;:\; {\mathcal{C}} \to \mathsf{Bim}_{\mathsf{fgp}}(A)$ be an action. Let $\mathbb{D}$ be a C $^*$ -algebra object in ${\mathsf {Vec}}({\mathcal{C}})$ . The crossed product of $A$ by the pair $(\alpha , \mathbb{D})$ is defined to be the coend realization

\begin{align*} A \rtimes \mathbb{D} \;:\!=\; \alpha \bowtie \mathbb{D}. \end{align*}

By Theorem3.15, there is a canonical faithful conditional expectation $\mathbb{E} \;:\; A \rtimes \mathbb{D} \to A \otimes \mathbb{D}(\unicode{x1D7D9})$ .

3.3. Conditional expectations down to $A$

We say that a conditional expectation $E\;:\;A \rtimes \mathbb{D} \to A$ is compatible with the canonical conditional expectation $\mathbb{E} \;:\; A \rtimes \mathbb{D} \to A \otimes \mathbb{D}(\unicode{x1D7D9})$ if it is the composite of $\mathbb{E}$ with a conditional expectation $E \;:\; A \otimes \mathbb{D}(\unicode{x1D7D9}) \to A$ . The following Lemma will allow us to characterize such conditional expectations by means of states on the ground C $^*$ -algebra $\mathbb{D}(\unicode{x1D7D9})$ .

Proof. Given a state $\omega$ on $D$ , the corresponding ucp map is $E_\omega \;:\!=\; {\textrm {id}}_A \otimes \omega$ . Now, if $E:A \otimes D \to A$ is a ucp $A{-}A$ -bimodular map, for any $d \in D$ , the map

\begin{equation*} A \ni a \mapsto E_d(a) = E(a \otimes d) \end{equation*}

is a bounded $A{-}A$ -bimodular map $A \to A$ . Since ${\textrm {End}}_{A{-}A}(A) \simeq \mathcal{Z}(A) \simeq \mathbb{C}$ , there is $\omega _E \;:\; D \to \mathbb{C}$ such that

\begin{equation*}E_d = \omega _E(d) {\textrm {id}}_{A}. \end{equation*}

From the fact that $E$ is ucp one deduces that $\omega _E$ is a state. Indeed, for all $d \in D$ ,

\begin{equation*}\omega _E(d^*d)1_A = E_{d^*d}(1_A) = E(1_A \otimes d^*d) \geq 0 ,\end{equation*}

and $\omega _E(1_D)1_A = E(1_A \otimes 1_D) = 1$ . It is immediate that the constructions $\omega \mapsto E_\omega$ and $E \mapsto \omega _E$ are mutually inverses.

Observe that, in the proposition above, the conditional expectation $E_\omega \;:\; A \rtimes \mathbb{D} \to A$ is faithful if and only if the corresponding state $\omega$ on $\mathbb{D}(\unicode{x1D7D9})$ is faithful.

4.1. Discrete, PQR, and ind-inclusions

Let $A$ be a unital C*-algebra with trivial center. Suppose we are given a unital inclusion $A \overset {E}{\subset }D$ and suppose $E \;:\; D \to A$ is a faithful conditional expectation. Let $\mathcal{E}$ be the right $A$ - $A$ -correspondence obtained as the completion of $D$ in the $A$ -valued inner product determined by $E$ . We will use the notation ${\mathcal{E}} = \overline {D}^{\| \cdot \|_A}$ . The multiplication in $D$ extends to an action of $D$ on $\mathcal{E}$ , denoted by $(d,\zeta )\mapsto d \triangleright \zeta$ for $d \in D$ and $\zeta \in {\mathcal{E}}$ . Letting $\Omega$ be the unit of $D$ seen as a vector in $\mathcal{E}$ , the canonical embedding $D \to {\mathcal{E}}$ is given by $d \mapsto d \triangleright \Omega$ .

Let $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ . There is a distinguished class of adjointable $A{-}A$ -bilinear maps $K \to {\mathcal{E}}$ , those lying in the diamond space

\begin{equation*}{\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})^{\diamondsuit }\;:\!=\; \{ f \in {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}}) \ | \ f[K] \subset D \} . \end{equation*}

Given $f \in {\mathsf{C}^*\mathsf{Alg}}_A(K,{\mathcal{E}})^{\diamondsuit }$ , there is a unique $A$ - $A$ -bimodular map $\check {f} \;:\; K \to D$ such that $f(\xi ) = \check {f}(\xi ) \triangleright \Omega$ for all $\xi \in K$ . The next theorem, named C $^*$ -Frobenius reciprocity, shows that the diamond space functor ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}((-),D)^\diamondsuit \;:\; \mathsf{Bim}_{\mathsf{fgp}}(A)^{\textrm {op}} \to {\mathsf {Vec}}$ has a canonical structure of a C $^*$ -algebra object.

Theorem 4.1 [Reference Palomares and Nelson14, Theorem 2.6]. Let $A \overset {E}{\subset } D$ be as above. Let ${\mathcal{E}} = \overline {D}^{\| \cdot \|_A}$ and let $\Omega$ be the image of $1_D$ in $\mathcal{E}$ . For $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ , the maps

\begin{equation*} \Psi _K \;:\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,D)^\diamondsuit \leftrightarrow {\mathsf{C}^*\mathsf{Alg}}_{A {-} D} (K \underset {A}{\boxtimes } D,D) \;:\; \Phi _K , \end{equation*}

given by

\begin{equation*} \Psi _K(f)( \xi \boxtimes d) \;:\!=\; \check {f}(\xi )d \ \ \text{and} \ \ \Phi _K(g)(\xi ) \;:\!=\; g( \xi \boxtimes 1_D) \triangleright \Omega , \end{equation*}

are mutually inverses. They are moreover natural in $K$ and therefore are mutually inverses in ${\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ :

\begin{equation*}\Psi \;:\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(({-}),D)^\diamondsuit \leftrightarrow {\mathsf{C}^*\mathsf{Alg}}_{A{-} D} \Big(({-}) \underset {A}{\boxtimes } D,D\Big) \;:\; \Phi \end{equation*}

The isomorphism $\Psi \;:\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(({-}),D)^\diamondsuit \simeq {\mathsf{C}^*\mathsf{Alg}}_{A {-} D} (({-}) \underset {A}{\boxtimes } D,D)$ provides a Banach space structure on the diamond space. From the assumption $\mathcal{Z}(A) \simeq \mathbb{C}$ , it follows, on the other hand, that ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(({-}),D)$ has a canonical Hilbert space structure. Generally, the Banach space structure of ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}((-),{\mathcal{E}})^\diamondsuit$ may not agree with the Hilbert space structure of ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}((-),{\mathcal{E}})$ . This is close in spirit to the distinction between operator norm and $\ell ^2$ -norm in the GNS construction for C $^*$ -algebras.

C $^*$ -discreteness will be characterized by the fact that the images of the maps in ${\mathsf{C}^*\mathsf{Alg}}_{A}(K,{\mathcal{E}})^{\diamondsuit }$ form a dense subspace of $D$ as $K$ runs in $\textrm {Irr}(A)$ . Then C*-Frobenius reciprocity gives a interpretation of the diamond spaces, and hence of discrete inclusions, in terms of categorical data.

Suppose we are given a Hilbert space object $\mathcal{H}$ in the unitary ind-completion $\mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ of $\mathsf{Bim}_{\mathsf{fgp}}(A)$ . Consider, for each $K \in \textrm {Irr}(A)$ , the amplified $A{-}A$ -correspondence $K \otimes \mathcal{H}(K)$ .

Definition 4.3. The realization $|\mathcal{H}|$ of $\mathcal{H} \in \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ is the $A{-}A$ -correspondence

\begin{equation*} |\mathcal{H}| \;:\!=\; \bigoplus _{K \in \textrm {Irr}(A)}^{\ell ^2} K \otimes \mathcal{H}(K) . \end{equation*}

Proof. If $f \;:\; \mathcal{H}_1 \to \mathcal{H}_2$ is a morphism of Hilbert space objects, the morphism $|f| \;:\; |\mathcal{H}_1| \to |\mathcal{H}_2|$ given on the $K$ -th component by ${\textrm {id}}_K \otimes f_K$ defines the action of $|-|$ on morphisms. We have thus a functor $\mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A)) \to {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}$ .

Let us show that this functor is a unitary tensor functor. Given Hilbert space objects $\mathcal{H}_1$ and $\mathcal{H}_2$ , we have

\begin{align*} |\mathcal{H}_1 \boxtimes \mathcal{H}_2| &= \bigoplus _{K \in \textrm {Irr}(A)}^{\ell ^2} K \otimes (\mathcal{H}_1 \boxtimes \mathcal{H}_2)(K) \\ &= \bigoplus _{K \in \textrm {Irr}(A)}^{\ell ^2} K \otimes \left ( \bigoplus _{K_1,K_2 \in \textrm {Irr}(A)}^{\ell ^2} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}\left(K,K_1 \underset {A}{\boxtimes } K_2\right) \otimes \mathcal{H}_1(K_1) \otimes \mathcal{H}_2(K_2) \right ) \\ & \simeq \bigoplus _{K,K_1,K_2 \in \textrm {Irr}(A)}^{\ell ^2} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}\left(K,K_1 \underset {A}{\boxtimes } K_2\right) \otimes K \otimes \mathcal{H}_1(K_1) \otimes \mathcal{H}_2(K_2) \\ &\simeq \bigoplus _{K_1,K_2 \in \textrm {Irr}(A)}^{\ell ^2} K_1 \underset {A}{\boxtimes } K_2 \otimes \mathcal{H}_1(K_1) \otimes \mathcal{H}_2(K_2) \simeq |\mathcal{H}_1| \underset {A}{\boxtimes } |\mathcal{H}_2| , \end{align*}

and all isomorphisms are canonical and unitary, providing $|-|$ with a unitary tensor structure, associativity being straightforward to check.

It remains to show that $|-|$ is fully faithful. We compute

\begin{align*} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(|\mathcal{H}_1|, |\mathcal{H}_2|) &= {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}\left (\bigoplus _{K_1 \in \textrm {Irr}(A)}^{\ell ^2} K_1 \otimes \mathcal{H}_1(K_1),\bigoplus _{K_2 \in \textrm {Irr}(A)}^{\ell ^2} K_2 \otimes \mathcal{H}_2(K_2)\right ) \\ & \simeq \prod _{K \in \textrm {Irr}(A)}^{\ell ^\infty }{\mathsf{C}^*\mathsf{Alg}}_{A{-}A}\left ( K \otimes \mathcal{H}_1(K), K \otimes \mathcal{H}_2(K) \right ) \\ & \simeq \prod _{K \in \textrm {Irr}(A)}^{\ell ^\infty } \mathbb{B}(\mathcal{H}_1(K),\mathcal{H}_2(K)) = \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))(\mathcal{H}_1,\mathcal{H}_2) . \end{align*}

From the data $A \overset {E}{\subset } D$ , we can extract three ind-objects of the UTC $\mathsf{Bim}_{\mathsf{fgp}}(A)$ .

Definition 4.5. The Hilbert space object $\mathcal{H}_{\mathcal{E}}$ associated with $A \overset {E}{\subset } D$ is given by

\begin{equation*} \mathcal{H}_{\mathcal{E}}(K) \;:\!=\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K, {\mathcal{E}}) . \end{equation*}

The C*-algebra object $\mathbb{D}$ associated with $A \overset {E}{\subset } D$ with faithful expectation $E$ is given by

\begin{equation*} \mathbb{D}(K) \;:\!=\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A} (K, {\mathcal{E}})^{\diamondsuit } . \end{equation*}

Canonically, we have $\mathbb{D}(K) \subset \mathcal{H}_{\mathcal{E}}(K)$ for all $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ .

The third ind-object we consider is the fiberwise closure of $\mathbb{D}$ inside $\mathcal{H}_{\mathcal{E}}$ . It can also be characterized as a GNS construction by means of the Lemma below.

Proof. We have the $*$ -algebra inclusions

\begin{equation*} A \subset A \otimes \mathbb{D}(\unicode{x1D7D9}) \subset \bigoplus _{K \in \textrm {Irr}(A)} K \odot \mathbb{D}(K) \subset D. \end{equation*}

The restriction of the conditional expectation $E \;:\; D \to A$ to a conditional expectation $A \otimes \mathbb{D}(\unicode{x1D7D9}) \to A$ is given by $({\textrm {id}}_A \otimes \omega )$ for a unique state $\omega$ on $\mathbb{D}(\unicode{x1D7D9})$ (Lemma 3.19).

The fibers of $L^2_{\omega }\mathbb{D}$ are described as follows. Recall that, for every $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ , $\mathbb{D}(K)$ is a $\mathbb{D}(\unicode{x1D7D9}){-}\mathbb{D}(\unicode{x1D7D9})$ -correspondence. Composing the right $\mathbb{D}(\unicode{x1D7D9})$ -valued inner product with $\omega$ endows $\mathbb{D}(K)$ with a pre-Hilbert space structure, and $L^2_{\omega }\mathbb{D}(K)$ is the completion of this pre-Hilbert space. By applying the realization in Definition 4.3, we obtain an $A$ - $A$ -correspondence $|L^2_\omega \mathbb{D}|$ .

For each $K \in \textrm {Irr}(A)$ , by C $^*$ -Frobenius reciprocity 4.1,

\begin{equation*}{\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})^\diamondsuit \simeq {\mathsf{C}^*\mathsf{Alg}}_{A{-}D}\Big(K \underset {A}{\boxtimes } D, D\Big), \end{equation*}

naturally in the variable $K$ . Using this natural identification, we shall regard ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})^\diamondsuit$ as a Banach space. On the other hand, $\mathcal{H}_{\mathcal{E}}(K) = {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})$ is a Hilbert space, the inner product given by $\langle f, g \rangle _{\mathcal{H}_{\mathcal{E}}(K)} = f^* g$ , and ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})^\diamondsuit \subset {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}})$ as a linear subspace.

Proof. Given $f \in \mathbb{D}(K)$ , its $\mathcal{H}_{\mathcal{E}}(K)$ -norm $\| f\|_{\mathcal{H}_{\mathcal{E}}(K)}$ is defined by

\begin{equation*} f^*f(\xi ) = \| f\|_{\mathcal{H}_{\mathcal{E}}(K)}^2 \xi \qquad \forall \ \xi \in K, \end{equation*}

or equivalently $\langle f(\xi ),f(\xi )\rangle = \| f\|_{\mathcal{H}_{\mathcal{E}}(K)}^2 \| \xi \|^2$ for all $\xi \in K$ . But

\begin{align*} \langle f(\xi ),f(\xi )\rangle & = \langle \check {f}(\xi ) \Omega , \check {f}(\xi ) \Omega \rangle \\ & = \langle (\check {f}(\xi ) )^* \check {f}(\xi ) \Omega , \Omega \rangle \\ & = E ( (\check {f}(\xi ) )^* \check {f}(\xi )) \\ & = ({\textrm {id}} \otimes \omega _E)\mathbb{E}^{\mathbb{D}} (\check {f}(\xi ) )^* \check {f}(\xi )). \end{align*}

In the identification of $A \rtimes \mathbb{D}$ with a C $^*$ -subalgebra of $D$ , the element $\check {f}(\xi ) )^* \check {f}(\xi )$ corresponds to

\begin{equation*} \left ( \bar {\xi } \odot j^{\mathbb{D}}(\check {f}) \right ) \cdot \left (\xi \odot \check {f} \right ), \end{equation*}

and therefore

\begin{equation*} ({\textrm {id}} \otimes \omega _E)\mathbb{E}^{\mathbb{D}} (\check {f}(\xi ) )^* \check {f}(\xi )) = \| \xi \|^2 \omega _E(\mathbb{E}^{\mathbb{D}}(j^{\mathbb{D}}(\check {f}) \cdot \check {f})) , \end{equation*}

where we are writing $\mathbb{D}^2(j^{\mathbb{D}}(\check {f}) \odot \check {f}) = j^{\mathbb{D}}(\check {f}) \cdot \check {f}$ for short. Since this must hold for every $\xi \in K$ , it follows that

\begin{equation*} \| f\|_{\mathcal{H}_{\mathcal{E}}(K)}^2 = \omega _E(\mathbb{E}^{\mathbb{D}}(j^{\mathbb{D}}(\check {f}) \cdot \check {f})) . \end{equation*}

4.2. Characterization of C*-discrete inclusions

In the spirit of categorical duality for compact quantum group actions [Reference Neshveyev and Tuset29] and of subfactor reconstruction, in this section we characterize C $^*$ -discrete inclusions by means of categorical data.

The following theorem is an extension of [Reference Palomares and Nelson14, Theorem 4.3] to non-connected C*-algebra objects.

Theorem 4.15 (Theorem B). There is an equivalence

\begin{equation*} {\mathsf{C}^*\mathsf{Alg}}_\Omega ({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A))) \simeq \mathsf{C}^{\ast}\mathsf{Disc}(A). \end{equation*}

Proof. At the level of objects, the functor ${\mathsf{C}^*\mathsf{Alg}}_\Omega ({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A))) \to \mathsf{C}^{\ast}\mathsf{Disc}(A)$ is given by taking the crossed product. Functoriality is proven as in Theorem 5.7 in [Reference Hataishi11], but we give a sketch of the proof for completeness.

Suppose $\theta \;:\; (\mathbb{D}_1, \omega _1) \to (\mathbb{D}_2,\omega _2)$ is a morphism in ${\mathsf{C}^*\mathsf{Alg}}_\Omega ({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A)))$ . Define

\begin{align*} |\theta |:& |\mathbb{D}_1| \to |\mathbb{D}_2|\\ & \xi \odot \eta \mapsto \xi \odot \theta (\eta ). \end{align*}

It extends to a ucp map ${\textrm {id}}_A \rtimes \theta \;:\; A \rtimes \mathbb{D}_1 \to A \rtimes \mathbb{D}_2$ . Let $\pi _2 \;:\; \mathbb{D}_2 \to \mathbb{B}(L^2_{\omega _2}\mathbb{D}_2)$ be the GNS representation. The composite $\pi _2 \circ \theta$ is a ucp map $\mathbb{D}_1 \to \mathbb{B}(L^2_{\omega _2}\mathbb{D}_2)$ . For $\xi \odot \eta \in K \odot \mathbb{D}_1(K)$ , write

\begin{equation*} |\pi _2 \circ \theta |(\xi \odot \eta ) \;:\!=\; \xi \odot \pi _2 (\theta (\eta )). \end{equation*}

Then $|\pi _2 \circ \theta |$ extends to a ucp map $A \rtimes \mathbb{D}_1 \to \mathcal{L}_{A}(|L^2_{\omega _2} \mathbb{D}_2|)$ in such a way that

commutes; in the diagram, $\text{can.}$ denotes the canonical representation of $A \rtimes \mathbb{D}_2$ on $| L^2_{\omega _2} \mathbb{D}_2|$ . Denoting by $E_i \;:\; A \rtimes \mathbb{D}_i \to A$ the conditional expectations corresponding to $\omega _i$ , since $\omega _2 \circ \theta = \omega _1$ , it follows

\begin{equation*} E_2 \circ ({\textrm {id}}_A \rtimes \theta ) = E_1. \end{equation*}

We remark that, in contrast with [Reference Hataishi11] and [Reference Jones and Penneys21], the above proof of functoriality is shorter because complete positivity implies norm continuity. In the cited papers, the context was a W $^*$ -context, and normality of the realization of the maps had to be shown.

Let us construct the functor $\mathsf{C}^{\ast}\mathsf{Disc}(A) \to {\mathsf{C}^*\mathsf{Alg}}_\Omega ({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A)))$ . At the level of objects, it is given by taking the underlying C*-algebra object associated with the discrete inclusion, together with the faithful state on it provided by the faithful conditional expectation (Proposition 3.20). We are left to check functoriality with respect to morphisms. Suppose $\phi \;:\; (A \subset D_1, E_1) \to (A \subset D_2,E_2)$ is a morphism of discrete inclusions. Using that $\phi$ sends $E_1$ to $E_2$ and Kadisons’ inequality, we obtain

\begin{equation*} E_2 (\phi (x^*) \phi (x)) \leq E_2(\phi (x^*x)) = E_1(x^*x) \qquad \forall \ x \in D_1. \end{equation*}

As a consequence, $\phi$ extends to a bounded $A$ - $A$ -bimodular map $\Phi \;:\; {\mathcal{E}}_1 \to {\mathcal{E}}_2$ , where ${\mathcal{E}}_i \;:\!=\; \overline {D_i}^{\| \cdot \|_{A_i}}$ . Let $\mathbb{D}_i$ be the underlying C*-algebra object of the discrete inclusion $(A \subset D_i, E_i)$ , and let $\omega _i$ be the faithful state on $\mathbb{D}_i$ provided by the conditional expectation $E_i$ . Recall that

\begin{equation*} \mathbb{D}_i(K) = {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}}_i)^\diamondsuit , \end{equation*}

and, since $A \subset D_i$ is assumed C $^*$ -discrete, which implies PQR,

\begin{equation*}L^2_{\omega _i}\mathbb{D}_i(K) = {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}}_i), \end{equation*}

where $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ . For every such $K$ , and for every $f \in {\mathsf{C}^*\mathsf{Alg}}_{A}(K,{\mathcal{E}}_1)$ , $\Phi \circ f$ is a bounded $A{-}A$ -bimodular map, and therefore adjointable since $K$ is fgp. We have thus, for every $K \in \mathsf{Bim}_{\mathsf{fgp}}(A)$ , a well-defined linear map

\begin{equation*} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}}_1) \to {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,{\mathcal{E}}_2) \end{equation*}

given by $f \mapsto \Phi \circ f$ . Since $\Phi$ extends $\phi \;:\; D_1 \to D_2$ , it preserves the diamond spaces and defines therefore a natural transformation $\mathbb{D}_1 \to \mathbb{D}_2$ . Since $\phi$ is an $A{-}A$ -bimodular ucp map, the resulting morphism $\mathbb{D}_1 \to \mathbb{D}_2$ is easily seen to be a ucp map of C*-algebra objects.

It is straightforward to check that the functors ${\mathsf{C}^*\mathsf{Alg}}_\Omega ({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A))) \leftrightarrow \mathsf{C}^{\ast}\mathsf{Disc}(A)$ are mutually quasi-inverses.

4.3. Characterization of ind-inclusions

Let $A \overset {E}{\subset } D$ be an ind-inclusion of unital C*-algebras, where $A$ is assumed to have trivial center. Our goal in this section is to characterize the algebra

\begin{equation*} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}}) \end{equation*}

where $\mathcal{E}$ is as in Section 4, that is, the right $A{-}A$ correspondence obtained from $D$ and the conditional expectation $E \;:\; D \to A$ . We will show in this section that a decomposition of ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}})$ into a direct product of type I factors characterizes $A \overset {E}{\subset }D$ as being an ind-inclusion.

Let $\mathcal{H}_{\mathcal{E}}$ be the object in $\mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ associated with $\mathcal{E}$ . Recall $A \overset {E}{\subset } D$ being an ind-inclusion means

\begin{equation*} {\mathcal{E}} \simeq |\mathcal{H}_{\mathcal{E}}| \;:\!=\; \bigoplus _{K \in \textrm {Irr}(A)}^{\ell ^2} K \otimes \mathcal{H}_{\mathcal{E}}(K) \ \, \end{equation*}

as a right $A{-}A$ -correspondence. In particular, $K \otimes \mathcal{H}_{\mathcal{E}}(K)$ is a complemented $A{-}A$ -subcorrespondence of $\mathcal{E}$ for all $K \in \textrm {Irr}(A)$ . Let us denote by $P_K$ the corresponding orthogonal central projection. The net of projections which associates with every finite subset $\Lambda \subset \textrm {Irr}(A)$ the projection

\begin{equation*} \sum _{K \in \Lambda } P_K \end{equation*}

converges strictly to the identity ${\textrm {id}}_{\mathcal{E}}$ .

Lemma 4.16. For each $K \in \textrm {Irr}(A)$ ,

\begin{equation*} P_K {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}}) \simeq \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K)). \end{equation*}

Proof. It follows from the above orthogonal decomposition of $\mathcal{E}$ that

\begin{equation*}P_K {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}}) = P_K {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}})P_K \simeq {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K \otimes \mathcal{H}_{\mathcal{E}}(K)). \end{equation*}

Since $K$ is a simple $A{-}A$ -bimodule, it follows that the latter is $\mathcal{L}(\mathcal{H}_{\mathcal{E}}(K))$ .

Corollary 4.17. There is an isomorphism

\begin{equation*} {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}}) \simeq \prod _{K \in \textrm {Irr}(A)}^{\ell ^\infty } \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K)). \end{equation*}

Proof. It follows from the discussion above that there is an embedding of ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}})$ into $\prod _K^{\ell ^\infty } \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K))$ . Now an element $x \in \prod _K^{\ell ^\infty } \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K))$ is a bounded family $(x_K)_{K \in \textrm {Irr}(A)}$ where $x_K \in \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K))$ . There is a unique $\tilde {x} \in {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}})$ defined by the condition that

\begin{equation*} P_k \tilde {x} = ({\textrm {id}}_K \otimes x_K) P_K \qquad \forall \ K \in \textrm {Irr}(A). \end{equation*}

Then the image of $\tilde {x}$ under the embedding ${\mathsf{C}^*\mathsf{Alg}}_{A{-}A}({\mathcal{E}}) \hookrightarrow \prod _K^{\ell ^\infty } \mathcal{L}(\mathcal{H}_{\mathcal{E}}(K))$ coincides with $x$ .

We now show the results above characterize ind-inclusions.

Proof. We already know that $A \subset D$ being and ind-inclusion implies the existence of $\{P_i\}_{i \in I}$ with the properties above. Let us prove the converse.

Given $i \in I$ , $(3)$ implies that

\begin{equation*} P_i( {\mathcal{E}}) = \bigoplus _{K \in \textrm {Irr}(A)}^{\ell ^2} K \otimes H_i(K), \end{equation*}

where $H_i \in \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ . Since for $i \neq j$ in $I$ we have $P_i P_j = P_j P_i = 0$ , the supports of $H_i$ and $H_j$ in $\textrm {Irr}(A)$ must be disjoint. ( $K \in \textrm {Irr}(A)$ belongs to the support of $H_i$ means that $H_i(K) \neq 0$ .) Thus, if $K$ maps nontrivially into $\mathcal{E}$ as an $A{-}A$ -bimodule, there is a unique $i \in I$ such that $K$ belongs to the support of $H_i$ . For $i,j \in I$ ,

\begin{equation*} \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))(H_i,H_j) = \prod _{K \in \textrm {Irr}(A)}^{\ell ^\infty } \mathcal{L}(H_i(K),H_j(K)). \end{equation*}

Now the orthogonality of the family $\{P_i\}_i$ together with item $(2)$ imply that for each $i$ there is a unique $K_i$ in the support of $H_i$ . Moreover, there is an embedding

\begin{equation*} \bigoplus _{i \in I} K_i \otimes H_i(K_i) \hookrightarrow {\mathcal{E}}. \end{equation*}

Item $(1)$ implies that this embedding has dense range, that is,

\begin{equation*} {\mathcal{E}} \simeq \bigoplus _{i \in I}^{\ell ^2} K_i \otimes H_i(K_i). \end{equation*}

Thus, defining the Hilbert space object $\mathcal{H}_{\mathcal{E}}$ by $\mathcal{H}_{\mathcal{E}}(K_i) = H_i(K_i)$ , $\mathcal{H}_{\mathcal{E}}(K) = 0$ if $K$ is not in the support of any of the $H_i$ ’s, it follows

\begin{equation*} {\mathcal{E}} \simeq |\mathcal{H}_{\mathcal{E}}|. \end{equation*}

5.1. Compact quantum group actions

Now we explain how the theory of continuous actions of compact quantum groups fits into our framework, providing examples of C $^*$ -discrete inclusions. For the fundamentals of the theory of compact quantum groups and their representation categories, see [Reference Neshveyev and Tuset29].

Let $\mathbb{G}$ be a compact quantum group with reduced algebra of functions $C(\mathbb{G})$ , that is, we consider the Haar state to be faithful on $C(\mathbb{G})$ . Consider the UTC $\mathsf {Rep}(\mathbb{G})$ of finite-dimensional unitary representations of $\mathbb{G}$ , together with its fiber functor $F \;:\; \mathsf {Rep}(\mathbb{G}) \to \mathsf {Hilb}_{fd}$ , which is in particular a C $^*$ -algebra object in ${\mathsf {Vec}}(\mathsf {Rep}(\mathbb{G})^{\textrm {op}})$ . Then the coend realization

\begin{equation*} F \bowtie (-) \;:\; \mathbb{A} \mapsto F \rtimes \mathbb{A} \;=\!:\; A \end{equation*}

can be equipped with the structure of a functor from the category of C $^*$ -algebra objects in ${\mathsf {Vec}}(\mathsf {Rep}(\mathbb{G}))$ to the category of C $^*$ -algebras equipped with a continuous action of $\mathbb{G}$ [Reference Neshveyev28]. There is a canonical isomorphism

\begin{equation*} \mathbb{A}(\unicode{x1D7D9}) \simeq A^{\mathbb{G}}, \end{equation*}

where $A^{\mathbb{G}}$ denotes the fixed point subalgebra of $A$ under the $\mathbb{G}$ -action, and $\unicode{x1D7D9} \in \mathsf {Rep}(\mathbb{G})$ stands for the trivial representation. The canonical conditional expectation

\begin{equation*} \mathbb{E} \;:\; A = F \bowtie \mathbb{A} \to F(\unicode{x1D7D9}) \otimes \mathbb{A}(\unicode{x1D7D9}) \simeq \mathbb{A}(\unicode{x1D7D9}) \simeq A^{\mathbb{G}} \end{equation*}

corresponds to averaging along the $\mathbb{G}$ -action on $A$ by means of the Haar state.

5.2. Semi-circular systems

In this section, we describe when an $A$ -valued semi-circular system gives rise to an ind-inclusion. This is equivalent to asking when the Fock space $\mathcal{F}(\eta )$ of the given covariance matrix is an ind-object in $\mathsf{Bim}_{\mathsf{fgp}}(A)$ . Our main result here is a characterization of that property in terms of the covariance matrix $\eta$ .

Covariance matrices can be produced by means of the following Lemma.

Lemma 5.2. Suppose $\mathcal{E}$ is a right $A{-}A$ -correspondence. Suppose $\{\xi _i\}_{i \in I}$ is a family of vectors in $\mathcal{E}$ with the property that there exists $C \geq 0$ such that

\begin{equation*} \sum _{j \in I} \| \langle \xi _i, a \triangleright \xi _j \rangle \|^2 \leq C \|a\|^2 \ \forall \ a \in A , \ \forall \ i \in I . \end{equation*}

Denoting by $\{e_{i,j}\}_{i,j \in I}$ a choice of a system of matrix units in $\mathcal{L}(\ell ^2(I))$ , it follows that

\begin{align*} \eta \;:\; A & \to A \otimes \mathcal{L}(\ell ^2(I)) \\ a & \mapsto \sum _{i,j \in I} \langle \xi _i, a \triangleright \xi _j \rangle \otimes e_{i,j} \end{align*}

is a covariance matrix on $A$ .

Proof. In the canonical representation $A \otimes \mathcal{L}(\ell ^2(I)) \subset \mathcal{L}_{A}(A \otimes \ell ^2(I))$ , it is straightforward to see that

\begin{equation*}\sum _{j \in I} \| \langle \xi _i, a \triangleright \xi _j \rangle \|^2 \leq C \|a\|^2 \end{equation*}

for all $i \in I$ implies $\|\eta (a)\| \leq C^{1/2} \|a\|$ , and that $\eta (a^*a)$ is positive for all $a$ . To show that $\eta$ is completely positive, observe that $\eta \otimes {\textrm {id}}_{\mathbb{M}_n}$ can be written in terms of the correspondence ${\mathcal{E}} \otimes \mathbb{C}^n$ as

\begin{equation*}(\eta \otimes {\textrm {id}}_{\mathbb{M}_n})( a \otimes m_{kl}) = \sum _{i,j \in I} \sum _{k',l' = 1}^n \langle \xi _i \otimes v_{k'}, (a \otimes m_{k,l}) \triangleright (\xi _j \otimes v_{l'} \rangle \otimes m_{k',l'} \otimes e_{i,j} , \end{equation*}

where $\{v_k\}_k$ is an orthonormal basis for $\mathbb{C}^n$ and $\{m_{k,l}\}_{k,l}$ the corresponding system of matrix units.

The discussion to follow shows in particular that every covariance matrix can be obtained in this way.

Associated with a covariance matrix $\eta$ , there is an $A{-}A \otimes \mathcal{L}(\ell ^2(I))$ -correspondence

\begin{equation*}A \underset {\eta }{\otimes } (A \otimes \mathcal{L}(\ell ^2(I))) . \end{equation*}

It is the separation-completion of the algebraic tensor product $A \odot (A \otimes \mathcal{L}(\ell ^2(I)))$ with respect to the inner product

\begin{equation*} \langle a \otimes x , b \otimes y \rangle \;:\!=\; x^* \eta (a^*b) y . \end{equation*}

Fix $i_0 \in I$ . There is an associated embedding of C*-algebras

\begin{equation*} A \simeq A \otimes \mathbb{C} e_{i_0,i_0} \subset A \otimes \mathcal{L}(\ell ^2(I)) , \end{equation*}

It is clear then that

\begin{equation*} \mathcal{X} \;:\!=\; \Big[A \underset {\eta }{\otimes } (A \otimes \mathcal{L}(\ell ^2(I)))\Big] \triangleleft (1 \otimes e_{i_0,i_0}) \end{equation*}

inherits the structure of a right $A{-}A$ -correspondence. Observe that, if we write

\begin{equation*} \eta (a) = \sum _{i,j} \eta _{i,j}(a) \otimes e_{i,j} , \end{equation*}

and define $\xi _i \;:\!=\; 1 \otimes e_{i,i_0} \in \mathcal{X}$ , then

\begin{equation*} \langle \xi _i, a \triangleright \xi _j \rangle = \eta _{i,j}(a) . \end{equation*}

Definition 5.3. The Fock space associated with the covariance matrix $\eta$ above is

\begin{equation*} \mathcal{F}(\eta ) \;:\!=\; \bigoplus _{n \in \mathbb{Z}_{\geq 0}}^{\ell ^2}\mathcal{X}^{\underset {A}{\boxtimes } n} , \end{equation*}

where $\mathcal{X}^{\underset {A}{\boxtimes }0 } \;:\!=\; A$ .

Given $\xi \in \mathcal{X}$ , let $T_\xi \in \mathcal{L}_A(\mathcal{F}(\eta ))$ be the creation operator associated with $\xi$ :

\begin{equation*} T_{\xi }(\eta ) = \xi \boxtimes \eta \ \forall \ \eta \in \mathcal{F}(\eta ) . \end{equation*}

Definition 5.4. The semi-circular operator $X_\xi \in \mathcal{L}_A(\mathcal{F}(\eta ))$ associated with a vector $\xi \in \mathcal{X}$ is given by

\begin{equation*} X_\xi \;:\!=\; T_\xi + T_\xi ^* . \end{equation*}

For $i \in I$ , write $X_i \;:\!=\; X_{\xi _i}$ . Let

\begin{equation*} \Phi (\eta ) \;:\!=\; C^*(A,\{X_i\}_{i \in I}) \subset \mathcal{L}_A(\mathcal{F}(\eta )) . \end{equation*}

The image $\Omega$ of the unit $1_A \in A$ under the embedding $A \to \mathcal{X} \to \mathcal{F}(\eta )$ is a cyclic vector for the action of $\Phi (\eta )$ . It induces a conditional expectation $E \;:\; \Phi (\eta ) \to A$ via

\begin{equation*} E(x) \;:\!=\; \langle x \Omega , \Omega \rangle . \end{equation*}

From [Reference Palomares and Nelson15, Theorem 5.2], a sufficient condition for the conditional expectation $E$ to be faithful is that $A$ admits a faithful tracial state $\tau$ such that $\tau (\eta _{ij}(x)y) = \tau (x\eta _{ji}(y))$ for all $x,y \in A$ .

Assume from now on that $E$ is faithful. Let $\mathbb{D}_\eta \;:\; \mathsf{Bim}_{\mathsf{fgp}}(A)^{\textrm {op}} \to {\mathsf {Vec}}$ be given by

\begin{equation*} \mathbb{D}_\eta (K) \;:\!=\; {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K,\mathcal{F}(\eta ))^{\diamondsuit } \;:\!=\; \{ f \in {\mathsf{C}^*\mathsf{Alg}}_{A{-}A}(K, \mathcal{F}(\eta )) \ | \ f[K] \subset \Phi (\eta ) \Omega \} . \end{equation*}

Due to C*-Frobenius reciprocity, $\mathbb{D}_\eta \in {\mathsf{C}^*\mathsf{Alg}}({\mathsf {Vec}}(\mathsf{Bim}_{\mathsf{fgp}}(A)))$ . There is a canonical embedding $A \rtimes \mathbb{D}_\eta \hookrightarrow \Phi (\eta )$ , given by

\begin{align*} K \otimes \mathbb{D}_\eta (K) & \to \Phi (\eta ) \\ \zeta \otimes f &\mapsto \check {f}(\zeta ) , \end{align*}

where, for $K$ a simple $A{-}A$ -bimodule, $\check {f}$ denotes the unique factorization of $f$ through $\Phi (\eta ) \ni x \mapsto x \triangleright \Omega \in \mathcal{F}(\eta )$ .

Lemma 5.5. We have

\begin{equation*} \overline {\textrm {span}\{f(\xi ) \ | f \in \mathbb{D}_\eta (K), \xi \in K, K \in \textrm {Irr}(A) \}}^{\|\cdot \|_A} \simeq |L^2_\omega \mathbb{D}_\eta | , \end{equation*}

as $A$ - $A$ correspondences, where $\omega$ is the state associated with the conditional expectation $A \rtimes \mathbb{D} \to A$ obtained by restriction of the conditional expectation on $\Phi (\eta )$ .

Proof. Recall the $*$ -algebra

\begin{equation*} | A \times \mathbb{D}_\eta | \;:\!=\; \bigoplus _{K \in \textrm {Irr}(A)} K \odot \mathbb{D}_\eta (K) , \end{equation*}

and the faithful conditional expectation $\mathbb{E} \;:\; | A \times \mathbb{D}_\eta | \to A \odot \mathbb{D}_\eta (\unicode{x1D7D9}) \subset A \otimes \mathbb{D}_\eta (\unicode{x1D7D9})$ . The state $\omega \in \mathbb{D}_\eta (\unicode{x1D7D9})$ is determined by $E = ({\textrm {id}}_A \otimes \omega ) \circ \mathbb{E}$ . Let $x_\lambda$ be a net in $|A \times \mathbb{D}|$ such that $x_\lambda \triangleright \Omega \to 0$ in $\mathcal{F}(\eta )$ . This is the case if and only if

\begin{equation*} E(x_\lambda ^* x_\lambda ) = ({\textrm {id}} \otimes \omega )(\mathbb{E}(x_\lambda ^* x_\lambda )) \to 0 . \end{equation*}

The right-hand side is exactly the $A$ -valued inner product on $|L^2_\omega \mathbb{D}_\eta |$ .

Proof. From Theorem 4.4, we deduce that $A \underset {\eta _{ii}}{\otimes } A \in \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ if and only if

\begin{equation*} A \underset {\eta _{ii}}{\otimes }A \simeq |{\mathsf{C}^*\mathsf{Alg}}_{A{-}A}((-),A \underset {\eta _{ii}}{\otimes }A )| . \end{equation*}

We also have that $A \underset {\eta _{ii}}{\otimes }A \simeq \overline {A \triangleright \xi _i \triangleleft A}^{\|\cdot \|_A} = \overline {A \triangleright (X_i \triangleright \Omega )\triangleleft A}^{\|\cdot \|_A}$ . Thus, A $\underset {\eta _{ii}}{\otimes } A \in \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ if and only if the $A{-}A$ -correspondence generated by $X_i \triangleright \Omega$ is a Hilbert space object. A simple computation show that

\begin{equation*}\overline {A \triangleright X_j X_i \triangleright \Omega \triangleleft A}^{\|\cdot \|_A} \subset \Big(A \underset {\eta _{jj}}{\otimes } A\Big) \underset {A}{\boxtimes }\Big(A \underset {\eta _{ii}}{\otimes }A\Big) \oplus A .\end{equation*}

Since realization is a unitary tensor functor (Proposition 4.4), the right-hand side of the above inclusion is a Hilbert space object, and therefore the left-hand side also is. By iteration, we conclude that the $A{-}A$ -correspondence generated by $P\triangleright \Omega$ , where $P$ is a polynomial in the semi-circular elements, is a Hilbert space object if and only if the $A \underset {\eta _{ii}}{\otimes } A$ ’s are Hilbert space objects. Passing to the limit, we have that $\overline {\Phi (\eta )\triangleright \Omega }^{\| \cdot \|_A}$ is a Hilbert space object if and only if the $A\underset {\eta _{ii}}{\otimes } A$ ’s are.

Corollary 5.7. Suppose the conditional expectation $E \;:\; \Phi (\eta ) \to A$ is faithful. Then the $A$ -valued semi-circular system $A \subset \Phi (\eta )$ is a ind-inclusion if and only if $\eta$ can be written

\begin{equation*} \eta _{ij}(a) = \langle \xi _i, a \triangleright \xi _j \rangle \end{equation*}

for a family $\{\xi _i\}_{i \in I}$ of vectors in an ind-object $H \in \mathsf {Hilb}(\mathsf{Bim}_{\mathsf{fgp}}(A))$ , satisfying

\begin{equation*} \sum _{j \in I} \| \langle \xi _i, a \triangleright \xi _j \rangle \|^2 \leq C \|a\|^2 \ \forall \ a \in A , \ \forall \ i \in I, \end{equation*}

for some $C \geq 0$ .

Example 5.8. Let $\{\alpha _i\}_{i\in I}$ be a countable collection of automorphisms on a unital C*-algebra $A$ with faithful unique trace $\tau$ and consider the covariance matrix

\begin{equation*} \eta =\mathsf{diag}(\eta _i), \end{equation*}

where $\eta _i= \alpha _i + \alpha _i^{-1}$ . Then, by direct computation we see that for every $i \in I$ and all $x,y\in A$

\begin{equation*}\tau (x\eta _i(y)) = \tau (\eta _i(x)y),\end{equation*}

fulfilling the hypotheses of [Reference Palomares and Nelson15, Theorem 5.2]. Then, the inclusion $A\subset \Phi (\eta )$ has a faithful conditional expectation and is C*-discrete.

5.3. C*-algebraic factorization homology

Factorization homology, as developed in [Reference Ayala and Francis1], is a generalized homology theory for topological manifolds. It is a homology theory in the sense that it is functorial and satisfies an excision property and is generalized in the sense that it assigns categories as invariants of manifolds. In [Reference Hataishi11], it was proven that as a target for factorization homology one can take the symmetric monoidal category $\mathsf{C}^{*}\mathsf{lin}$ , a certain category of C $^*$ -categories, equipped with the max tensor product $\underset {\max }{\boxtimes }$ as introduced in [Reference Antoun and Voigt2]. Our goal here is not to give a detailed account on that subject but briefly explain how it can provide examples of non-connected C $^*$ -algebra objects.

We will be interested in factorization homology for surfaces. A factorization homology theory with values in $({\mathsf{C}^{*}\mathsf{lin}}, \underset {\max }{\boxtimes })$ is a monoidal functor $\mathcal{F} \;:\; {\mathsf{Surf}} \to {\mathsf{C}^{*}\mathsf{lin}}$ , where the source category is the category of surfaces with embeddings as morphisms and disjoint unions as monoidal structure. The functoriality of $\mathcal{F}$ is expressed in 2-categorical terms: smooth-oriented embeddings correspond to unitary functors between C*-tensor categories, and isotopies between embeddings correspond to unitary natural isomorphisms between unitary functors. This implies, in particular, that if $D$ denotes the compact-oriented disk, then