2.1 $C^*$ -modules and correspondences

We refer the reader to [Reference LanceLan95, Reference Raeburn and WilliamsRW98] for background on $C^*$ -modules. Throughout this paper, A and B denote separable $C^*$ -algebras. Given a right $C^*$ -module over A (also known as a right Hilbert A-module or right A-module) X—written $X_A$ when we want to remember the coefficient algebra—we denote the $C^*$ -algebra of adjointable operators on X by $ {\mathrm {End}}_A(X)$ , the $C^*$ -ideal of compact operators by $ {\mathrm {End}}^0_A(X)$ , and the finite-rank operators by $ {\mathrm {End}}_A^{00}(X)$ . The finite-rank operators are generated by rank-one operators $\Theta _{x,y}$ with $x,\,y \in X$ satisfying $\Theta _{x,y}(z) = x \cdot \langle y \mid z\rangle _A$ . If $A_A$ is the trivial A-module, then we identify $ {\mathrm {End}}_A(X)$ with the multiplier algebra $ {\mathrm {Mult}}(A)$ of A.

A right $C^*$ -module $X_A$ is full if $A = \overline { {\mathrm {span}}}\langle X \mid X \rangle _A$ . We also have occasion to consider left A-modules, which behave analogously.

We discuss the more general class of bi-Hilbertian bimodules in Section 5.

Frames are important technical and computational tools in the theory and applications of $C^*$ -modules. These are as close as we can get to an orthonormal basis in a $C^*$ -module, and serve similar purposes.

If $(u_j)_{j \ge 1}$ is a frame for X, then X is generated as a right A-module by the $u_j$ , so X is countably generated. Conversely, every countably generated right A-module admits a frame [Reference Raeburn and ThompsonRT03, Corollary 3.3].

We may compose $C^*$ -correspondences using the internal tensor product. Given correspondences $(\phi ,{}_AX_B)$ and $(\psi ,{}_BY_C)$ , we obtain a new correspondence $(\phi \otimes \mathrm {Id}_Y,{}_A(X\otimes _BY)_C)$ , where $X\otimes _B Y$ is the quotient of $X\otimes _{\mathbb {C}} Y$ by the closed submodule generated by

$$ \begin{align*} \{xb\otimes y-x\otimes\psi(b)y:x\in X, y\in Y, b\in B\} \end{align*} $$

[Reference LanceLan95, Proposition 4.5]. It is a nontrivial result that $X\otimes _BY$ is a right $C^*$ -module with inner product for all $x_1,x_2\in X$ and $y_1,y_2\in Y$ .

For a $C^{*}$ -correspondence ( $\phi,X_A$ ), let . Following Katsura [Reference KatsuraKat04a], the covariance ideal of $(\phi ,{}_AX_A)$ is given by

The covariance ideal is the largest ideal of A on which $\phi $ is injective with image contained in $ {\mathrm {End}}_A^0(X)$ .

We set the scene for a restricted class of covariant morphisms that respect the covariance ideal. The following well-known characterisation of the compact operators on $X_A$ will be helpful in formulating covariance.

Correspondence morphisms between $C^*$ -correspondences induce $*$ -homomor- phisms between the associated Toeplitz algebras, and covariant morphisms descend to $*$ -homomorphisms of the associated Cuntz–Pimsner algebras [Reference BrenkenBre10, Proposition 1.4]. We now turn to a description of these algebras.

2.2 Toeplitz–Pimsner and Cuntz–Pimsner algebras

Fix a $C^*$ -correspondence $(\phi ,{}_AX_A)$ . Define , and for $n \ge 1$ . The Fock module of X is the $\ell ^2$ -direct sum regarded as a correspondence over A with diagonal left action. If $x \in X^{\otimes n}$ for some $n \ge 0$ , we call n the length of x and write $|x| = n$ .

As in [Reference Pimsner and VoiculescuPim97], the Toeplitz algebra $\mathcal {T}_X$ is the $C^*$ -subalgebra of $ {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ generated by the creation operators $T_x$ , $x\in X$ , which satisfy

on elementary tensors $x_1\otimes x_2\otimes \cdots \otimes x_k \in {\mathrm {Fock}}_X$ . The adjoint $T^*_x$ satisfies

$$ \begin{align*} T^*_x(x_1 \otimes \cdots \otimes x_k) = \langle x\mid x_1\rangle_A\cdot x_2 \otimes \cdots \otimes x_k \end{align*} $$

for $k \ge 1$ , and $T^*_x|_{X^{\otimes 0}} = 0$ . For $a \in A$ , we let $T_a$ denote the operator of left multiplication by a, given on simple tensors by $T_a(x_1\otimes \cdots \otimes x_k)=a\cdot x_1\otimes \cdots \otimes x_k$ .

Following [Reference KatsuraKat04a], with $J_X$ the covariance ideal of X, the algebra of compact operators $ {\mathrm {End}}_A^0( {\mathrm {Fock}}_X\cdot J_X)$ is an ideal of the Toeplitz–Pimsner algebra $\mathcal {T}_X$ . The Cuntz–Pimsner algebra $\mathcal {O}_X$ is defined to be the quotient $\mathcal {T}_X/ {\mathrm {End}}_A^0( {\mathrm {Fock}}_X\cdot J_X)$ . Thus, we have an exact sequence

$$ \begin{align*} 0\to {\mathrm{End}}_A^0({\mathrm{Fock}}_X\cdot J_X) \longrightarrow \mathcal{T}_X \stackrel{q}{\longrightarrow} \mathcal{O}_X\to 0. \end{align*} $$

The maps $j_X \colon x \mapsto T_x$ and $j_A \colon a \mapsto T_a$ constitute a representation of $(\phi ,{}_A X_A)$ whose image generates $\mathcal {T}_X$ . This representation is universal: for any representation $(\psi , \pi )\colon (\phi ,{}_A X_A) \to B$ , there is a unique $*$ -homomorphism $\psi \times \pi \colon \mathcal {T}_X \to B$ such that $(\psi \times \pi ) \circ j_X = \psi $ and $(\psi \times \pi ) \circ j_A = \psi $ (see [Reference Pimsner and VoiculescuPim97, Theorem 3.4]).

The Cuntz–Pimsner algebra $\mathcal {O}_X$ is generated by the covariant representation of $(\phi ,{}_A X_A)$ given by and . This representation is universal: for every covariant representation $(\psi , \pi ) \colon (\phi ,{}_A X_A) \to B$ , there is a unique $*$ -homomorphism $\psi \times \pi \colon \mathcal {O}_X \to B$ such that $(\psi \times \pi ) \circ i_X = \psi $ and $(\psi \times \pi ) \circ i_A = \pi $ .

2.3 A- $C^*$ -algebras

For a locally compact Hausdorff space X, the notion of a $C_0(X)$ -algebra has been used for many years to handle localisation and limits in homological contexts, for example, [Reference KasparovKas88]. Here, we relax and extend the notion by allowing a noncommutative algebra in place of $C_0(X)$ and we do not assume a central image.

When A is commutative, our notion of an A-algebra is more relaxed than the established notion of a $C_0(X)$ -algebra. However, the advantage is that a Cuntz–Pimsner algebra over an A-module is an A-algebra in our sense, as we show in the next example.

Example 2.9. Suppose $(\phi ,{}_{A}X_A)$ is a $C^*$ -correspondence and let $j_A \colon A \to \mathcal {T}_X$ be the universal inclusion, which we think of as the diagonal representation of A on $ {\mathrm {Fock}}_X$ . Since $ {\mathrm {End}}_{J_X}^0( {\mathrm {Fock}}_X \cdot J_X)$ is an ideal in $\mathcal {T}_X$ , the map $j_A$ induces the structure of an A- $C^*$ -algebra on $ {\mathrm {End}}_{J_X}^0( {\mathrm {Fock}}_X \cdot J_X)$ . In particular, the defining exact sequence for Cuntz–Pimsner algebras,

(2-1)

may be interpreted as an exact sequence of A- $C^*$ -algebras.

A $C^*$ -algebra C is an A-algebra with $\phi \colon A \to {\mathrm {Mult}}(C)$ nondegenerate if and only if $(\phi ,{}_A C_C)$ is a nondegenerate $C^*$ -correspondence. We record the following facts about the relationship between $C^*$ -correspondences and A-algebras.

Lemma 2.10. Let $(\phi ,{}_A F_B)$ be a nondegenerate $C^*$ -correspondence and let C be a B-algebra. Then the following are true:

1. (i) $F\otimes _BC$ is an $ {\mathrm {End}}^0_C(F \otimes _B C)$ –C-Morita equivalence bimodule;

2. (ii) $(F \otimes _B C)^* \cong C {}_B\otimes F^*$ , where ${}_B\otimes $ denotes the tensor product of left correspondences;

3. (iii) $ {\mathrm {End}}^0_C(F \otimes _B C) \cong (F \otimes _B C) \otimes _C (C {}_B\otimes F^*) $ is an A-algebra; and

4. (iv) $(F \otimes _B C) \otimes _C (C {}_B\otimes F^*)$ can be identified with $F\otimes C\otimes F^*$ via the map

   $$ \begin{align*} (f_1 \otimes c_1) \otimes (c_2 \otimes f_2^*) \mapsto f_1 \otimes c_1 c_2 \otimes f_2^*, \end{align*} $$

with the adjoint satisfying $(f_1\otimes c_1\otimes f_2^*)^*=f_2\otimes c_1^*\otimes f_1^*$ and the product satisfying

$$ \begin{align*} (f_1\otimes c_1\otimes f_2^*)(f_3\otimes c_2\otimes f_4^*) =f_1\otimes c_1\langle f_2|f_3\rangle_Bc_2\otimes f_4^* \end{align*} $$

for all $f_1,f_2,f_3,f_4\in F$ and $c_1,c_2\in C$ .

Lemma 2.11. Let $F_B$ be a right B-module and let C be a B-algebra. Suppose $(\phi ,{}_{C}X_D)$ is a nondegenerate $C^*$ -correspondence with $\phi $ injective. Then the action of $ {\mathrm {End}}_C^0 (F \otimes _B C)$ on , defined on rank-one operators by

$$ \begin{align*} \Theta_{f_1 \otimes c_1, f_2 \otimes c_2} (f_3 \otimes x) = f_1 \otimes \phi(c_1 \langle f_2 \mid f_3\rangle_B c_2^*) x \end{align*} $$

for all $f_1,f_2,f_3\in F$ and $c_1,c_2\in C$ , and $x\in X$ , is adjointable and injective.

3.1 The KSGNS construction

In this section, we recall the details of the KSGNS (Kasparov–Stinespring–Gelfand–Naimark–Segal) construction, and we refer to [Reference KasparovKas80] and [Reference LanceLan95, Ch. 5] who we follow here.

Let $X_A$ and $Y_A$ be right A-modules and let $ {\mathrm {End}}_A(X,Y)$ denote the adjointable A-linear operators from $X_A$ to $Y_A$ . Recall that the strong-* topology on $ {\mathrm {End}}_A(X, Y)$ is generated by the seminorms

$$ \begin{align*} T\mapsto || Tx ||_Y \quad \textrm{and} \quad T\mapsto || T^*y ||_X \end{align*} $$

for all $x\in X$ and $y\in Y$ . The strong-* topology agrees with the strict topology on norm-bounded sets [Reference LanceLan95, Proposition 8.1]. We only consider these topologies on bounded sets and so only refer to the strict topology. A bounded net $(T_i)_{i \in \mathbb {N}}$ converges strictly to T in $ {\mathrm {End}}_A(X, Y)$ precisely when

$$ \begin{align*} || (T_i - T)x ||_Y \to 0 \quad \textrm{and} \quad || (T^*_i - T^*)y ||_X \to 0 \end{align*} $$

for every $x\in X$ and $y\in Y$ .

Example 3.7. If $B \subseteq A$ and $\rho \colon A \to B$ is contractive and idempotent, then $\rho $ is strict. Indeed, by Tomiyama’s theorem [Reference Brown and OzawaBO08, Theorem 1.5.10], $\rho $ is a conditional expectation, so if $(a_i)$ is an approximate unit for A, then

$$ \begin{align*} \|\rho(a_i)b - \rho(a_j)b\| = \|\rho(a_ib - a_j b)\| \to 0 \end{align*} $$

for $b\in B$ , so $\rho $ is strict.

We now recall the KSGNS construction.

Construction 3.8 (KSGNS)

Let A and B be $C^*$ -algebras, let $X_B$ be a B-module and let $\rho \colon A \to {\mathrm {End}}_B(X)$ be a strict completely positive map. The algebraic tensor product $A \odot X$ is a right B-module and a semi inner-product space with respect to the sesquilinear form defined by

for all $a_1, a_2\in A$ and $x_1, x_2\in X_B$ . The KSGNS correspondence $A \otimes _\rho X$ is the completion of the quotient of $A \odot X$ by the ideal of tensors whose norm (coming from the semi-inner product) vanishes. The product $A \otimes _\rho X$ carries the obvious right B-module structure, and there is a left action by a $*$ -homomorphism $\pi _\rho \colon A \to {\mathrm {End}}_B(A \otimes _\rho X)$ satisfying

$$ \begin{align*} \pi_\rho(a') (a \otimes x) = a' a\otimes x \end{align*} $$

for all $a', a\in A$ and $x\in X$ , which makes $(\pi _{\rho }, A\otimes _\rho X)$ a nondegenerate A–B- correspondence. There is an adjointable map $V_\rho \colon X \to A \otimes _\rho X$ satisfying

$$ \begin{align*} V_\rho(x) = \lim_i a_i \otimes x, \end{align*} $$

where $x\in X$ and $(a_i)_i$ is an approximate unit for A. Note that $V_\rho ^*(a \otimes x) = \rho (a) x$ for $a\in A$ and $x\in X$ . Then,

$$ \begin{align*} \rho(a) = V_\rho^* \pi_\rho(a) V_\rho \end{align*} $$

for $a\in A$ , and $\pi _\rho (A)V_\rho (X) \subseteq A \otimes _\rho X$ is dense.

Moreover, if $(\phi ,{}_A Y_B)$ is an A–B-correspondence and there exists an adjointable operator $W \colon X \to Y$ such that $\rho (a) = W^* \phi (a) W$ for $a\in A$ , and $\phi (A) W(X) \subseteq Y$ is dense, then

(3-1) $$ \begin{align} (\pi_{\rho},A \otimes_{\rho} X) \cong (\phi,Y ) \end{align} $$

as correspondences. We refer to this property as the uniqueness of the KSGNS construction.

Note that if $\rho $ is nondegenerate, then $V^*_\rho V_\rho (x)=\lim \rho (a_i)x=x$ , so $V_\rho $ is an isometry.

Our aim is to construct a category similar to the Enchilada category of [Reference Echterhoff, Kalisewski, Quigg and RaeburnEKQR06], where instead of $*$ -homomorphisms $\phi \colon A\to {\mathrm {End}}_B(X)$ , we consider strict completely positive maps $\rho \colon A \to {\mathrm {End}}_B(X)$ .

Strictness of a positive correspondence ensures that the KSGNS construction yields a nondegenerate $C^*$ -correspondence that is unique up to isomorphism [Reference LanceLan95, Theorem 5.6].

3.2 The Quesadilla semi-category

The next lemma is key to defining a semi-category using strict completely positive maps. The reason we do not obtain a category is that the ‘obvious’ identity elements only behave as identities from one side (see Lemma 3.14 below).

Lemma 3.12. Suppose that $(\rho , {}_A X_B)$ is a strict positive correspondence and that $(\phi , {}_BY_C)$ is a $C^*$ -correspondence. There is a strict completely positive map $\rho \otimes {\mathrm {Id}} \colon A \to {\mathrm {End}}_C(X\otimes _B Y)$ satisfying

(3-2) $$ \begin{align} (\rho\otimes {\mathrm{Id}})(a)(x \otimes y) = \rho(a)x \otimes y \end{align} $$

for all $a\in A$ , $x\in X$ and $y\in Y$ , and an isomorphism of $C^*$ -correspondences,

(3-3) $$ \begin{align} (\pi_{\rho} \otimes {\mathrm{Id}}_Y, (A \otimes_\rho X) \otimes_B Y )\cong (\pi_{\rho \otimes {\mathrm{Id}}} , A \otimes_{\rho \otimes {\mathrm{Id}}} (X \otimes_B Y)). \end{align} $$

Proof. Applying the KSGNS construction to $\rho $ , we obtain the A–B-correspondence $A\otimes _\rho X$ with left action and an adjointable operator satisfying $\rho (a) = V^*\pi (a)V$ and $\overline {\pi (A)V(X)} = A \otimes _\rho X$ for all $a\in A$ . Since the $*$ -homomorphism $\rho \otimes {\mathrm {Id}}$ is the composition of $\rho $ with the nondegenerate $*$ -homomorphism $ {\mathrm {End}}_B(X) \to {\mathrm {End}}_C (X \otimes _B Y)$ given by $T \mapsto T \otimes {\mathrm {Id}}_{Y}$ , it follows that $\rho \otimes {\mathrm {Id}}$ is a strict completely positive map. So we also obtain the A–C-correspondence $(\pi _{\rho \otimes {\mathrm {Id}}}, A \otimes _{\rho \otimes {\mathrm {Id}}}(X\otimes _B Y))$ .

To establish the isomorphism (3-3), we consider the adjointable operator $ V \otimes {\mathrm {Id}}_Y \colon X \otimes _B Y \to (A \otimes _\rho X) \otimes _B Y. $ Note that $(V \otimes {\mathrm {Id}}_Y)^* = V^* \otimes {\mathrm {Id}}_Y$ . By the uniqueness of the KSGNS construction (3-1), it suffices to show that for all $a \in A$ ,

(3-4) $$ \begin{align} (\rho\otimes{\mathrm{Id}}) (a) =(V^* \otimes {\mathrm{Id}}_Y)\pi(a)(V \otimes {\mathrm{Id}}_Y) \quad\text{and}\quad \overline{\pi(A)(V \otimes {\mathrm{Id}}_Y)(X \otimes_B\! Y)} = (A \otimes_\rho X) \otimes_B\! Y. \end{align} $$

Fix an approximate unit $(a_i)_i$ for A. Let $a\in A$ , $x\in X$ and $y\in Y$ . For the first equality of (3-4), we compute

$$ \begin{align*} (V^* \otimes {\mathrm{Id}}_Y)\pi(a)(V \otimes {\mathrm{Id}}_Y) (x \otimes y) & = \lim_i (V^* \otimes {\mathrm{Id}}_Y) (aa_i \otimes x) \otimes y = V^*(a \otimes x) \otimes y \\ &= \rho(a)x \otimes y = (\rho \otimes {\mathrm{Id}}) (x \otimes y). \end{align*} $$

For the second equality of (3-4), we compute

$$ \begin{align*} \pi(a) (V \otimes {\mathrm{Id}}_Y) (x \otimes y) = \lim_i (aa_i \otimes x) \otimes y = (a \otimes x) \otimes y. \end{align*} $$

So $\pi (a) (V \otimes {\mathrm {Id}}_Y) (X \otimes _B Y)$ is dense in $(A \otimes _{\rho } X) \otimes _B Y$ .

If A and B are $C^*$ -algebras, and $(\rho , {}_A X_B)$ is a strict positive A–B-correspondence, then we write $(\rho , {}_A X_B)\colon A \to B$ to indicate that we are thinking of it as a mapping, and below as a morphism, from A to B. Lemma 3.12 allows us to define the composition of strict positive correspondences. Given two strict positive correspondences $(\rho , {}_A X_B) \colon A \to B$ and $(\sigma , {}_B Y_C) \colon B \to C$ , we define

so that $(\rho \otimes {\mathrm {Id}}_{B \otimes _{{\sigma }} Y}, X \otimes _{{\sigma }} Y) \colon A \to C$ is a strict positive correspondence. The KSGNS space $B \otimes _{\sigma } Y$ is unique up to isomorphism since ${\sigma }$ is strict, and (3-2) implies that

$$ \begin{align*} \rho \otimes {\mathrm{Id}}_{B \otimes_{\sigma} Y}(a)(x \otimes (b \otimes y)) = \rho(a)x \otimes (b \otimes y) \end{align*} $$

for all $a\in A$ , $b\in B$ , $x\in X$ and $y\in Y$ . To simplify notation, we write

The following result gives us associativity of composition of isomorphism classes of positive correspondences.

Lemma 3.13. Let $(\rho , {}_A X_B) \colon A \to B$ , $(\sigma , {}_B Y_C) \colon B \to C$ and $(\tau ,{}_C Z_D) \colon C \to D$ be strict positive correspondences. Then,

$$ \begin{align*} ( (\rho, {}_A X_B) \otimes (\sigma, {}_B Y_C) ) \otimes (\tau,{}_C Z_D) \cong (\rho, {}_A X_B) \otimes ( (\sigma, {}_B Y_C) \otimes (\tau,{}_C Z_D)). \end{align*} $$

Proof. The compositions

$$ \begin{align*} &( (\rho, {}_A X_B) \otimes (\sigma, {}_B Y_C) ) \otimes (\tau,{}_C Z_D)\\ &\quad= ((\rho \otimes {\mathrm{Id}}_{B\otimes_{\sigma}Y} )\otimes {\mathrm{Id}}_{C \otimes_T Z}, {}_{A} (X \otimes_B (B \otimes_{\sigma} Y)) \otimes_C (C \otimes_T Z)) \end{align*} $$

and

$$ \begin{align*} (\rho, {}_A X_B) \otimes ( (\sigma, {}_B Y_C) \otimes (\tau,{}_C Z_D)) = (\rho\otimes {\mathrm{Id}}_{(B \otimes_{\sigma} Y) \otimes_C ( C \otimes_T Z)} ,{}_{A} X \otimes_B ( (B \otimes_{\sigma}\! Y) \otimes_C ( C \otimes_T Z) )) \end{align*} $$

are isomorphic by two applications of Lemma 3.12.

With associative composition of morphisms, it seems obvious that the next step is to define a positive correspondence category. Sadly, this is not quite the case.

Lemma 3.14. Let $(\rho ,{}_AX_B)$ be a strict positive correspondence. Then,

$$ \begin{align*} (\rho,{}_AX_B)\otimes(\mathrm{Id},{}_BB_B)\cong(\rho,{}_AX_B) \quad \text{but} \quad (\mathrm{Id},{}_AA_A)\otimes(\rho,{}_AX_B)\cong (\pi_\rho,A\otimes_\rho X). \end{align*} $$

Lemma 3.14 says that typically, we do not have identity morphisms, though we do always have right identities. The usual name for a ‘category’ without identities is a semi-category or a semi-groupoid [Reference TilsonTil86, Appendix B]. However, having right identities gives us more structure than a general semi-category. For instance, we can still work just with morphisms by identifying objects with their right identities, though we will not explore the ramifications of the semi-category structure here.

Following the Tex-Mex precedent of [Reference Echterhoff, Kalisewski, Quigg and RaeburnEKQR06], we introduce the following.

Example 3.18. Let $\rho \colon A \to B$ be a conditional expectation with nondegenerate inclusion $\iota \colon B \to A$ . We consider the positive correspondence $(\rho ,{}_AB_B)$ and the $C^*$ -correspondence $(\iota ,{}_BA_A)$ . There are two possible compositions,

$$ \begin{align*} (\rho\otimes\mathrm{Id},{}_AB \otimes_{\iota } A_A) \cong (\iota\circ\rho,{}_AA_A) \quad\text{and}\quad (\iota\otimes\mathrm{Id}_B,{}_BA\otimes_{\rho}B_B) \cong (\iota,{}_BL^2_B(A,\rho)). \end{align*} $$

The latter correspondence is that of Example 3.9, with the left action restricted to B. As $\iota \circ \rho $ is always an adjointable projection on $L^2_B(A,\rho )$ , there is a copy of the identity correspondence $(\mathrm {Id}_B,{}_BB_B)$ contained as a complemented sub-correspondence in $(\iota ,{}_BL^2_B(A,\rho ))$ .

The positive correspondence semi-category seems to be the right domain for the KSGNS construction. Since every $C^*$ -correspondence is in particular a strict positive correspondence, ${ \mathsf {Enchilada}}$ is a wide subcategory of ${ \mathsf {Quesadilla}}$ . Further, we may interpret the proposition below as saying that ${ \mathsf {Enchilada}}$ is a retract of ${ \mathsf {Quesadilla}}$ .

Proposition 3.19. The $ {\mathrm {KSGNS}}$ construction provides a semi-functor (that is, preserves composition)

$$ \begin{align*} {\mathrm{KSGNS}} \colon { \mathsf{Quesadilla}} \to { \mathsf{Enchilada}} \end{align*} $$

such that:

1. (i) $ {\mathrm {KSGNS}}(A) = A$ for every $C^*$ -algebra A; and

2. (ii) for a positive correspondence $(\rho , {}_A X_B)$ , $ {\mathrm {KSGNS}}(\rho , {}_A X_B)$ is the class of the correspondence $(\pi _\rho ,{}_A(A \otimes _\rho X)_B)$ . In particular, if $\rho $ is a nondegenerate $*$ -homomorphism, then $ {\mathrm {KSGNS}}(\rho , {}_A X_B)=(\rho , {}_A X_B)$ .

If $U \colon { \mathsf {Enchilada}} \to { \mathsf {Quesadilla}}$ is the forgetful semi-functor given by considering nondegenerate $*$ -homomorphisms as strict completely positive maps, then

$$ \begin{align*} {\mathrm{KSGNS}}{} \circ U = {\mathrm{Id}}_{ \mathsf{Enchilada}} \quad \text{and} \quad U \circ {\mathrm{KSGNS}}{}~\text{is idempotent}. \end{align*} $$

In a general semi-category, we cannot talk about invertible morphisms, as we lack identities. In ${ \mathsf {Quesadilla}}$ , we may talk about invertible morphisms since we have right identities.

Definition 3.20. A morphism $(\rho ,{}_AX_B)$ in ${ \mathsf {Quesadilla}}$ is invertible if there exists a morphism $(\sigma ,{}_BY_A)$ such that

$$ \begin{align*} (\rho,{}_AX_B)\otimes(\sigma,{}_BY_A)=(\mathrm{Id},{}_AA_A) \quad\text{and}\quad (\sigma,{}_BY_A)\otimes(\rho,{}_AX_B)=(\mathrm{Id},{}_BB_B). \end{align*} $$

Here, $(\mathrm {Id},{}_AA_A)$ and $(\mathrm {Id},{}_BB_B)$ are the right identity morphisms for A and B.

The lemma below should be compared with [Reference Echterhoff, Kalisewski, Quigg and RaeburnEKQR06, Lemma 2.4], which says that the invertible morphisms in ${ \mathsf {Enchilada}}$ are the Morita equivalence bimodules.

Proof. Morita equivalence bimodules are invertible with inverse given by the conjugate module. Conversely, suppose that $(\rho , {}_A X_B)\colon A \to B$ and $(\sigma , {}_BY_A) \colon B \to A$ represent mutually inverse morphisms in ${ \mathsf {Quesadilla}}$ . Then,

$$ \begin{align*} (\rho, {}_A X_B) &\cong (\rho, {}_A X_B) \otimes ({\mathrm{Id}}_B,{}_B B_B) \cong (\rho, {}_A X_B) \otimes (\sigma, {}_B Y_A) \otimes (\rho, {}_A X_B)\\ &\cong ({\mathrm{Id}}_A, {}_A A_A) \otimes (\rho, {}_A X_B) \cong(\pi_\rho,A \otimes_\rho X). \end{align*} $$

Uniqueness of the KSGNS construction implies that $\rho $ must be a $*$ -homomorphism so $(\rho , {}_A X_B)$ is a $C^*$ -correspondence. A similar argument shows that $({\sigma },{}_BY_A)$ is also a $C^*$ -correspondence. By [Reference Echterhoff, Kalisewski, Quigg and RaeburnEKQR06, Lemma 2.4], the invertible morphisms in ${ \mathsf {Enchilada}}$ are imprimitivity bimodules, so $(\rho , {}_A X_B)$ is an imprimitivity bimodule, and the result follows.

Under certain conditions, the composition in ${ \mathsf {Quesadilla}}$ simplifies significantly.

Lemma 3.22. Let $\rho \colon A \to B$ and ${\sigma } \colon B \to C$ be strict completely positive maps between $C^*$ -algebras and assume that $\rho $ is a conditional expectation. There is an inclusion of C-modules

$$ \begin{align*} \psi \colon (A \otimes_\rho B) \otimes_B (B \otimes_{\sigma} C) \to A \otimes_{{\sigma}\circ \rho} C \end{align*} $$

satisfying $\psi ((a \otimes b_1) \otimes (b_2 \otimes c)) = ab_1 b_2 \otimes c$ for all $a\in A$ , $b_1, b_2\in B$ and $c\in C$ . If B contains an approximate unit for A, then $\psi $ is an isomorphism.

Proof. Fix $a \in A$ , $b_1,b_2 \in B$ and $c \in C$ , and observe that

$$ \begin{align*} &\langle{(a \otimes b_1) \otimes (b_2 \otimes c) \mid (a \otimes b_1) \otimes (b_2 \otimes c)}\rangle_C\\ &\quad = c^*{\sigma}(b_2^*b_1^*\rho(a^*a)b_1b_2)c = c^*{\sigma}\circ \rho((a b_1 b_2)^* a b_1 b_2)c \\ &\quad = \langle ab_1b_2 \otimes c \mid ab_1b_2 \otimes c\rangle_C, \end{align*} $$

where the first inner product is on $(A \otimes _\rho B) \otimes _B (B \otimes _{\sigma } C)$ and the second is on $A \otimes _{{\sigma }\circ \rho } C$ . It follows that there is a well-defined isometric linear map $\psi \colon (A \otimes _\rho B) \otimes _B (B \otimes _{\sigma } C)\to A \otimes _{{\sigma }\circ \rho } C$ satisfying $ \psi ((a \otimes b_1) \otimes (b_2 \otimes c)) = ab_1b_2 \otimes c.$ It is straightforward to see that $\psi $ respects the left action of A and right action of C.

For the second statement, choose an approximate unit $(b_i)_i$ for A contained in B. Then for fixed $a\in A$ and $c\in C$ ,

$$ \begin{align*} \lim_i \psi((a \otimes b_i^{1/2}) \otimes (b_i^{1/2} \otimes c)) = \lim_i a b_i \otimes c = a \otimes c, \end{align*} $$

and it follows that $\psi $ is surjective.

Example 3.23. We give two examples of expectations arising from the Cuntz– Pimsner algebra of a $C^*$ -correspondence $(\phi _X,{}_AX_A)$ . Let $\gamma \colon \mathbb {T} \to {\mathrm {Aut}}(\mathcal {O}_X)$ denote the gauge action and let $\mathcal {O}_X^{\gamma }$ denote the associated fixed-point algebra. There is a conditional expectation $\Phi \colon \mathcal {O}_X\to \mathcal {O}_X^\gamma $ given by averaging over $\gamma $ . Applying the $ {\mathrm {KSGNS}}$ construction yields the correspondence $(\pi _\Phi ,{}_{\mathcal {O}_X}L^2_{\mathcal {O}_X^\gamma }(\mathcal {O}_X,\Phi ))$ underlying the Kasparov module encoding the gauge action, [Reference Carey, Neshveyev, Nest and RennieCNNR11, Reference Pask and RenniePR06].

If X is a bi-Hilbertian bimodule of finite right Watatani index (see Section 5), there is another expectation $\Phi _\infty :\mathcal {O}_X\to A$ . Applying the $ {\mathrm {KSGNS}}$ functor yields the correspondence $(\pi _{\Phi _\infty }, {}_{\mathcal {O}_X}\Xi _A)$ underlying the Kasparov module representing the class of the defining extension (2-1) [Reference Goffeng, Mesland and RennieGMR18, Reference Rennie, Robertson and SimsRRS17]. We note that both A and $\mathcal {O}_X^\gamma $ contain an approximate unit for $\mathcal {O}_X$ .

Given a densely defined norm lower semi-continuous trace $\tau $ on $\mathcal {O}_X^\gamma $ or on A, we can then compose with the GNS representations $(\pi _\tau ,{}_{\mathcal {O}_X^\gamma }L^2(\mathcal {O}_X^\gamma ,\tau ))$ or $(\pi _\tau ,{}_AL^2(A,\tau ))$ using Lemma 3.22, and apply the KSGNS construction to obtain

$$ \begin{align*} (\pi_{\tau\circ\Phi},{}_{\mathcal{O}_X}L^2(\mathcal{O}_X,\tau\circ\Phi))\quad\text{or}\quad (\pi_{\tau\circ\Phi_\infty},{}_AL^2(A,\tau\circ\Phi_\infty)). \end{align*} $$

The process of identifying KMS states of quasi-free real actions on $\mathcal {O}_X$ outlined by Laca and Neshveyev is compatible with these decompositions [Reference Goffeng, Rennie and UsachevGRU19, Reference Laca and NeshveyevLN04, Reference Rennie, Robertson and SimsRRS17].

To understand the composition of morphisms in ${ \mathsf {Quesadilla}}$ to and from Cuntz–Pimsner algebras, we need to understand how composition in ${ \mathsf {Quesadilla}}$ interacts with quotient maps. Recall that for a quotient map $q \colon A \to A/I$ with linear splitting $s \colon A/I \to A$ , we have $s(ab) - s(a)s(b) \in I$ and $s(a^*) - s(a)^* \in I$ for all $a,b\in A/I$ .

Lemma 3.24. Suppose that we have a commuting diagram of $C^*$ -algebras

with exact rows of $*$ -homomorphisms. Suppose that the top row has a completely positive splitting and that $\rho $ is a strict completely positive map satisfying $\rho (I)\subseteq J$ . Then, the map $\widetilde {\rho } \colon A/I \to B/J$ defined by $\widetilde {\rho }(a+I)=\rho (a)+J$ is completely positive and strict.

In the case where $\rho \colon A \to B$ is a conditional expectation, the KSGNS space of the induced completely positive map $\widetilde {\rho } \colon A/I \to B/J$ enjoys a construction similar to that of Example 3.9.

Lemma 3.25. Suppose that we have the commuting diagram

as in Lemma 3.24. Suppose further that $\rho \colon A \to B$ is a conditional expectation with nondegenerate inclusion $\alpha \colon B \to A$ . Let $(\mathrm {Id}_{A/I},L^2_{B/J}(A/I,\widetilde {\rho }))$ be the completion (after quotienting by zero vectors) of $A/I$ in the norm coming from the semi-inner product

(3-5) $$ \begin{align} \langle a_1+I\mid a_2+I\rangle_{B/J}=\rho(a_1^*a_2)+J,\quad a_1,a_2\in A. \end{align} $$

Then, $ {\mathrm {KSGNS}}(\widetilde {\rho },{}_{A/I}(B/J)_{B/J}) \cong (\mathrm {Id}_{A/I},L^2_{B/J}(A/I,\widetilde {\rho })). $

Proof. Apart from linearity, it is easy to check that (3-5) defines an inner product. Fix a completely positive splitting $s_A$ of $q_A$ so that $\widetilde {\rho } = q_B \circ \rho \circ s_A$ . For the $B/J$ -linearity, let $a+I\in A/I$ and $b+J\in B/J$ , and compute

$$ \begin{align*} \widetilde{\rho}(a+I)(b+J)&=\rho\circ s_A(a+I)b+J =\rho(a+i)b+J\quad \text{for some }i\in I\\ &=\rho(a\alpha(b)+i\alpha(b))+J =\rho(a\alpha(b))+J. \end{align*} $$

A straightforward computation shows that

$$ \begin{align*} \langle a_1+I\mid a_2+I\rangle_{B/J}(b+J)=\langle a_1+I\mid a_2\alpha(b)+I\rangle_{B/J}. \end{align*} $$

For $a\in A$ and $b\in B$ , we have the equality

$$ \begin{align*} \langle [a\alpha(b)+I]\mid[a\alpha(b)+I]\rangle_{B/J} &=\rho(\alpha(b^*)a^*a\alpha(b))+J =(b^*+J)(\rho(a^*a)+J)(b+J)\\ &=\langle b+J\mid \widetilde{\rho}(a^*a+I)(b+J)\rangle_{B/J}\\ &=\langle b+J\mid \widetilde{\rho}(\langle a+I\mid a+I\rangle_{A/I})(b+J)\rangle_{B/J}\\ &=\langle (a+I)\otimes(b+J)\mid (a+I)\otimes(b+J)\rangle_{B/J}, \end{align*} $$

so there is a well-defined isometric linear map $ \Lambda \colon A/I\otimes _{\widetilde {\Psi }}B/J\to L^2_{B/J}(A/I,\widetilde {\rho })$ satisfying . The right $B/J$ -linearity of $\Lambda $ is routine, and the nondegeneracy of $\alpha $ gives the surjectivity of $\Lambda $ .

The following result is analogous to Lemma 3.22 once we pass to quotients. The result is needed in Section 4.1.

Lemma 3.26. Suppose that

is a commuting diagram of $C^*$ -algebras with exact rows of $*$ -homomorphisms that have completely positive splittings $s_A \colon A/I \to A$ and $s_B \colon B/J \to B$ . Let $\rho \colon A \to B$ be a conditional expectation with corresponding inclusion $\alpha \colon B \to A$ and let $\sigma \colon B \to C$ be a strict completely positive map. Further suppose that $\rho (I) \subseteq J$ and $\sigma (J) \subseteq K$ , and that the maps $\widetilde {\rho }$ and $\widetilde {\sigma }$ are the strict completely positive maps induced by Lemma 3.24. Then there is an isometric inclusion of correspondences

$$ \begin{align*} \psi \colon (A/I \otimes_{\widetilde{\rho}} B/J) \otimes_{B/J} (B/J \otimes_{\widetilde{\sigma}} C/K) \to A/I \otimes_{\widetilde{\sigma}\circ\widetilde{\rho}} C/K \end{align*} $$

satisfying

(3-6) $$ \begin{align} \psi((a \otimes b_1) \otimes (b_2 \otimes c)) = a q_A( \alpha (s_B(b_1b_2))) \otimes c \end{align} $$

for $a\in A/I$ , $b_1,b_2\in B/J$ and $c\in C/K$ . If $q_A \circ \alpha \circ s_B(B/J)$ contains an approximate unit for $A/I$ , then $\psi $ is also surjective.

Proof. We first show that (3-6) yields a well-defined map. Let $s_A \colon A/I \to A$ and $s_B \colon B/J \to J$ be the completely positive splittings of $q_A$ and $q_B$ , respectively. For $a \in A/I$ , $b_1,b_2 \in B/J$ and $c \in C/K$ , a computation similar to that of Lemma 3.22 shows that

$$ \begin{align*} \langle(a \otimes b_1) \otimes (b_2 \otimes c) \mid (a \otimes b_1) \otimes (b_2 \otimes c)\rangle_{C/K} & = c^* \widetilde{\sigma} (b_2^*b_1^* \widetilde{\rho}(a^*a) b_1b_2)c. \end{align*} $$

Recall from Lemma 3.24 that $\widetilde {\rho } = q_B \circ \rho \circ s_A$ and $\widetilde {\sigma } = q_C \circ \sigma \circ s_B$ . Since for all $a\in A/I$ $s_A(a^*)s_A(a) - s_A(a^*a) \in I$ and $\rho (I) \subseteq J$ , it follows that

$$ \begin{align*} b_2^*b_1^* \widetilde{\rho}(a^*a) b_1b_2 & = q_B ( s_B(b_2^*b_1^*) \rho(s_A(a^*)s_A(a)) s_B(b_1b_2) ) \\ & = q_B \circ \rho ( \alpha( s_B(b_1b_2))^*s_A(a)^*s_A(a) \alpha(s_B(b_1b_2)) ). \end{align*} $$

Define and note that

$$ \begin{align*} s_A(d^*d) - \alpha(s_B(b_1b_2))^*s_A(a)^*s_A(a) \alpha(s_B(b_1b_2)) \in I. \end{align*} $$

Consequently,

$$ \begin{align*} \langle(a \otimes b_1) \otimes (b_2 \otimes c) \mid (a \otimes b_1) \otimes (b_2 \otimes c)\rangle_C & = c^* \widetilde{\sigma} ( q_B \circ \rho \circ s_A(d^*d)) c = c^* \widetilde{\sigma} \widetilde{\rho} (d^*d) c \\ & = \langle d \otimes c \mid d \otimes c \rangle_{C/K}, \end{align*} $$

where the latter inner product is on $A/I \otimes _{\widetilde {\sigma }\circ \widetilde {\rho }} C/K$ . This shows that $\psi $ is a well-defined isometric linear map. It is straightforward to see that $\psi $ respects the left action of $A/I$ and the right action of $C/K$ .

For the final statement, suppose $(b_i)_i$ is a net in $B/J$ such that defines an approximate unit for $A/I$ . Then,

$$ \begin{align*} \lim_i \psi( (a \otimes b_{i}^{1/2}) \otimes (b_i^{1/2} \otimes c)) = \lim_i a a_i \otimes c = a \otimes c, \end{align*} $$

and it follows that $\psi $ is surjective.

4.1 Morphisms from correspondence projections

A complemented sub- correspondence is a $C^*$ -correspondence that appears as a direct summand of a $C^*$ -correspondence. This is stronger than being complemented as a right $C^*$ -module since the left action must also be respected. Not every sub-correspondence is complemented. A sub-correspondence is complemented precisely if it is the image of a correspondence projection.

If P is a correspondence projection on $(\phi ,{}_AX_B)$ , then $1-P$ is a correspondence projection, and $(P\phi , {}_A PX_B)$ is a complemented sub-A–B-correspondence determining the direct sum decomposition $X_B \cong PX_B \oplus (1-P)X_B$ . Conversely, every complemented sub-A–B-correspondence is the image of a correspondence projection. If $(\phi _Y,{}_{A} Y_{A})$ is the image of a correspondence projection P on $(\phi _X,{}_{A} X_{A})$ , we write $P \colon (\phi _X,{}_{A} X_{A}) \to (\phi _Y,{}_{A} Y_{A})$ , where $Y_A = PX_A$ . We will also write .

If $Y = PX$ , then the left action $\phi _X$ acts diagonally on X with respect to the direct sum decomposition. This follows from the computation

$$ \begin{align*} \phi_X(a) x & = P\phi_X(a) P x + (1-P) \phi_X(a) Px + P \phi_X(a) (1-P)x + (1-P) \phi_X(a) (1-P)x \\ & =\phi_Y(a)Px + \phi_{Y^{\perp}}(a) (1-P)x, \end{align*} $$

which holds for all $a\in A$ and $x\in X$ . In particular, $\phi _X(a) \in {\mathrm {End}}^0_{A}(X)$ if and only if $\phi _Y(a) \in {\mathrm {End}}^0_{A}(Y) $ and $\phi _{Y^{\perp }}(a) \in {\mathrm {End}}^0_{A}(Y^{\perp })$ .

Also observe that $\ker (\phi _X) = \ker (\phi _Y) \cap \ker (\phi _{Y^{\perp }})$ . Since $\ker (\phi _Y)^{\perp } \cap \ker (\phi _{Y^\perp })^{\perp }$ is a subset of $(\ker (\phi _Y) \cap \ker (\phi _{Y^{\perp }}))^{\perp }$ , the covariance ideals satisfy $J_Y \cap J_{Y^{\perp }} \subseteq J_X$ . In general, it is not true that $J_Y \subseteq J_X$ nor that $J_X \subseteq J_Y$ . However, in the case that $(\phi _{Y^{\perp }}, Y_A^{\perp })$ is regular, we have the following immediate result.

Let $(\phi _Y,{}_{A} Y_{A})$ be a complemented sub-correspondence of $(\phi _X,{}_{A} X_{A})$ and let $\iota \colon Y \to X$ denote the inclusion. Then $( {\mathrm {Id}}_A,\iota )$ is a correspondence morphism, so the universal property of the Toeplitz algebra $\mathcal {T}_Y$ yields an injective $*$ -homomorphism

satisfying $\alpha (T_\xi T_\eta ^*) = T_{\iota (\xi )}T_{\iota (\eta )}^*$ for all $\xi , \eta \in Y$ . The correspondence morphism $( {\mathrm {Id}}_A,\iota )$ is typically not covariant as the following example shows, so $\alpha \colon \mathcal {T}_Y \to \mathcal {T}_X$ does not typically descend to a $*$ -homomorphism between Cuntz–Pimsner algebras.

Aside from $\alpha $ , there is a second representation of $\mathcal {T}_Y$ in $ {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ arising from the correspondence projection P that we will take a moment to explain. First, observe that if $(\phi _Y,{}_{A} Y_{A})$ is a sub-A–A-correspondence of $(\phi _X,{}_{A} X_{A})$ , then the Fock module $ {\mathrm {Fock}}_Y$ may be identified as a sub-A–A-correspondence of $ {\mathrm {Fock}}_X$ . The following lemma shows that complementability is also preserved.

Lemma 4.4. Suppose that $P_0 \colon (\phi _X,{}_{A} X_{A}) \to (\phi _Y,{}_{A} Y_{A})$ is a correspondence projection. Then $P_0$ extends to a correspondence projection $P_n \colon (\phi _X,{}_{A} X^{\otimes n}_{A}) \to (\phi _Y,{}_{A} Y^{\otimes n}_{A})$ by

(4-1)

The sum $\sum _{n \ge 0} P_n$ converges strictly to a projection $P \colon (\phi _X, {\mathrm {Fock}}_X) \to (\phi _Y, {\mathrm {Fock}}_Y)$ , so $ {\mathrm {Fock}}_Y$ is complemented in $ {\mathrm {Fock}}_X$ .

If $ {\mathrm {Fock}}_Y$ is complemented in $ {\mathrm {Fock}}_X$ , then there is an inclusion $\overline {\iota } \colon {\mathrm {End}}_A( {\mathrm {Fock}}_Y) \to {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ defined by $\overline {\iota }(T)(x \oplus x^{\perp }) = Tx$ for $x\in {\mathrm {Fock}}_Y$ and $x^{\perp } \in {\mathrm {Fock}}_Y^{\perp }$ . Since $\mathcal {T}_Y$ is a subalgebra of $ {\mathrm {End}}_A( {\mathrm {Fock}}_Y)$ , the map $\overline {\iota }$ restricts to an injective $*$ -homomorphism

$$ \begin{align*} \beta \colon \mathcal{T}_Y \to {\mathrm{End}}_A({\mathrm{Fock}}_X). \end{align*} $$

The representations $\alpha $ and $\beta $ of $\mathcal {T}_Y$ in $ {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ do not usually coincide. Indeed, if A were unital, then $\alpha (1_A) = {\mathrm {Id}}_{ {\mathrm {Fock}}_X}$ , while $\beta (1_A) = \overline {\iota }( {\mathrm {Id}}_{ {\mathrm {Fock}}_Y})$ is typically a proper subprojection of $ {\mathrm {Id}}_{ {\mathrm {Fock}}_X}$ . Moreover, if $T_y\in \mathcal {T}_Y$ is a creation operator on $ {\mathrm {Fock}}_Y$ for some $y \in Y_A$ , then $\beta (T_y)$ is typically not a creation operator on $ {\mathrm {Fock}}_X$ since it is zero on $ {\mathrm {Fock}}_Y^{\perp }$ .

Despite neither $\alpha $ nor $\beta $ inducing a $*$ -homomorphism from $\mathcal {O}_Y$ to $\mathcal {O}_X$ , we can use the projection P of Lemma 4.4 to build a conditional expectation from $\mathcal {T}_X$ to $\mathcal {T}_Y$ that respects both inclusions $\alpha $ and $\beta $ . By then applying Lemma 3.25, we will construct a positive map from $\mathcal {O}_X$ to $\mathcal {O}_Y$ that gives morphisms in both ${ \mathsf {Quesadilla}}$ and ${ \mathsf {Enchilada}}$ .

Lemma 4.5. Let $P \colon (\phi _X, {\mathrm {Fock}}_X) \to (\phi _Y, {\mathrm {Fock}}_Y)$ be the correspondence projection of Lemma 4.4. The map $\Psi _P \colon {\mathrm {End}}_A( {\mathrm {Fock}}_X) \to {\mathrm {End}}_A( {\mathrm {Fock}}_Y)$ defined by

(4-2) $$ \begin{align} \Psi_P(T) = PTP \end{align} $$

is a conditional expectation. It restricts to a conditional expectation $\Psi _P \colon \mathcal {T}_X \to \mathcal {T}_Y$ such that

(4-3) $$ \begin{align} \Psi_P (\alpha(b_1)a \alpha(b_2)) = b_1\Psi_P( a ) b_2 = \Psi_P (\beta(b_1)a \beta(b_2)) \end{align} $$

for all $a \in \mathcal {T}_X$ and $b_1,\,b_2 \in \mathcal {T}_Y$ .

Proof. There are multiple Fock spaces and Toeplitz algebras around, so if $T \in \mathcal {T}_X$ , then we write $T^X$ for the corresponding operator on $ {\mathrm {Fock}}_X$ , and if $T \in \mathcal {T}_Y$ , we write $T^Y$ for the corresponding operator on $ {\mathrm {Fock}}_Y$ . We also use $\iota \colon {\mathrm {Fock}}_Y \to {\mathrm {Fock}}_X$ to identify $ {\mathrm {Fock}}_Y$ as a submodule of $ {\mathrm {Fock}}_X$ . In particular, if $\xi \in Y$ , then $\alpha (T_\xi ^Y) = T^X_{\xi }$ .

That (4-2) defines an expectation for the inclusion $\overline {\iota } \colon {\mathrm {End}}_A( {\mathrm {Fock}}_Y) \to {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ is clear. It remains to show that $\Psi _P$ restricts to Toeplitz algebras. Fix elementary tensors $y = y_1 \otimes \cdots \otimes y_k \in {\mathrm {Fock}}_Y \subseteq {\mathrm {Fock}}_X$ and $\xi , \eta \in {\mathrm {Fock}}_X$ . It follows from the A-bilinearity of P that for $k \ge 1$ ,

$$ \begin{align*} \Psi_P(T^X_\xi T^{X*}_\eta) y &= PT^X_\xi T^{X*}_\eta P y\\ & = \begin{cases} P(\xi \cdot\langle \eta \mid Py_1 \otimes \cdots \otimes y_{|\eta|} \rangle_A \otimes y_{|\eta|+1} \otimes \cdots \otimes y_{k}) & \text{if } k \geq |\eta|, \\ 0 & \text{if } k < |\eta|, \end{cases} \\ & = \begin{cases} (P\xi )\cdot \langle P \eta \mid y_1 \otimes \cdots \otimes y_{|\eta|} \rangle_A \otimes y_{|\eta|+1} \otimes \cdots \otimes y_{k} & \text{if } k \geq |\eta|, \\ 0 & \text{if } k < |\eta|, \end{cases}\\ &= T_{P\xi}^Y T_{P\eta}^{Y*} y. \end{align*} $$

Similarly, if $y \in A$ , then $\Psi _P(T^X_\xi T^{X*}_\eta )y = T_{P\xi }^Y T_{P\eta }^{Y*} y$ , so $\Psi _P (\mathcal {T}_X) = \mathcal {T}_Y$ .

Recall that for elementary tensors $\xi ,\eta \in {\mathrm {Fock}}_Y$ , we have $\alpha (T^Y_\xi T_\eta ^{Y*}) = T_{\xi }^X T_{\eta }^{X*}$ . The previous calculation shows that $ \Psi _P ( \alpha (T^Y_\xi T_\eta ^{Y*})) = T^Y_\xi T_\eta ^{Y*}, $ so $\Psi _P \circ \alpha = {\mathrm {Id}}_{\mathcal {T}_Y}$ . Since $\Psi _P\circ \overline {\iota }=\Psi _P\circ \beta = {\mathrm {Id}}_{\mathcal {T}_Y}$ , we also have $\Psi _P \circ \beta =\Psi _P \circ \alpha $ .

For the first equality of (4-3), fix elementary tensors $y \in Y^{\otimes n}_A$ and $\xi \in X^{\otimes m}_A$ . Then,

$$ \begin{align*} T^{X*}_{y} T^X_\xi & = \begin{cases} \langle y \mid \xi_1 \otimes \cdots \otimes \xi_n \rangle _A T^X_{\xi_{n+1} \otimes \cdots \otimes \xi_{m}} & \text{if } m \ge n, \\ T^{X*}_{{y_{m+1} \otimes \cdots \otimes y_n}} \langle y_1 \otimes \cdots \otimes y_{m} \mid \xi\rangle_A & \text{if } n> m, \end{cases} \\ & = \begin{cases} T^X_{ \phi_X(\langle y \mid P (\xi_1 \otimes \cdots \otimes \xi_n) \rangle_A) \xi_{n+1} \otimes \cdots \otimes \xi_{m}} & \text{if } m \ge n, \\ T^{X*}_{ \phi_X(\langle P\xi \mid y_1 \otimes \cdots \otimes y_{m}\rangle_A) {y_{m+1} \otimes \cdots \otimes y_n}} & \text{if } n > m. \end{cases} \end{align*} $$

So, for elementary tensors $x \in {\mathrm {Fock}}_Y$ and $\eta \in {\mathrm {Fock}}_X$ ,

$$ \begin{align*} \Psi_P(\alpha(T_x^Y T_y^{Y*})T_\xi^X T_\eta^{X*}) & = P T_{x}^X T_{y}^{X*} T_\xi^X T_\eta^{X*} P \\ & =\begin{cases} P T^X_{x \otimes \phi_X(\langle y \mid P_0 \xi_1 \otimes \cdots \otimes P_0\xi_n \rangle_A) \xi_{n+1} \otimes \cdots \otimes \xi_{m}} T_\eta^{X*} P & \text{if } m \ge n, \\ P T_{x}^X T_{\eta \otimes \phi_X(\langle P\xi \mid y_1 \otimes \cdots \otimes y_{m}\rangle_A) {y_{m+1} \otimes \cdots \otimes y_n}}^{X*} P & \text{if } n> m, \end{cases}\\ & =\begin{cases} T^Y_{x \otimes \phi_Y(\langle y \mid P_0 \xi_1 \otimes \cdots \otimes P_0\xi_n \rangle_A) P_0 \xi_{n+1} \otimes \cdots \otimes P_0 \xi_{m}} T_\eta^{Y*} & \text{if } m \ge n, \\ T_{x}^Y T_{P\eta \otimes \phi_Y(\langle P\xi \mid y_1 \otimes \cdots \otimes y_{m}\rangle_A) {y_{m+1} \otimes \cdots \otimes y_n}}^{Y*} & \text{if } n > m, \end{cases} \\ & = T_x^Y T_y^{Y*} T_{P\xi}^Y T_{P\eta}^{Y*} = T_x^Y T_y^{Y*} \Psi_P(T_{\xi}^X T_{\eta}^{X*}). \end{align*} $$

Similarly, $\Psi _P(T_\xi ^X T_\eta ^{X*}\alpha (T_x^X T_y^{X*})) = \Psi _P(T_{\xi }^X T_{\eta }^{X*}) T_x^Y T_y^{Y*}$ .

Since $\Psi _P\colon {\mathrm {End}}_A( {\mathrm {Fock}}_X) \to {\mathrm {End}}_A( {\mathrm {Fock}}_Y)$ is an expectation for the inclusion $\overline {\iota }$ , it follows that $\Psi _P$ descends to an expectation $\Psi _P \colon \mathcal {T}_X\to \mathcal {T}_Y$ for the inclusion $\alpha $ . The second equality of (4-3) follows because $\Psi _P$ is an expectation for $\overline {\iota }$ and $\beta =\overline {\iota }|_{\mathcal {T}_Y}$ .

With the expectation $\Psi _P \colon \mathcal {T}_X \to \mathcal {T}_Y$ in hand, we aim to appeal to Lemma 3.24 to induce a strict completely positive map between the corresponding Cuntz–Pimsner algebras. Since $\mathcal {O}_X$ is the quotient $\mathcal {T}_X / {\mathrm {End}}_A^0( {\mathrm {Fock}}_X \cdot J_X)$ , we need to show that $\Psi _P$ takes $ {\mathrm {End}}_A^0( {\mathrm {Fock}}_X \cdot J_X)$ to $ {\mathrm {End}}_A^0( {\mathrm {Fock}}_Y \cdot J_Y)$ .

To use Lemma 3.24, we need the existence of a completely positive splitting $\mathcal {O}_X \to \mathcal {T}_X$ . By the Choi–Effros lifting theorem [Reference Brown and OzawaBO08, Theorem C.3], such a splitting exists whenever $\mathcal {O}_X$ is nuclear. A sufficient condition for nuclearity of $\mathcal {O}_X$ is that the coefficient algebra A is nuclear [Reference KatsuraKat04a, Corollary 7.4]. The resulting strict completely positive map $\widetilde {\Psi }_P$ is typically not a conditional expectation itself since there is no obvious choice for an inclusion of $\mathcal {O}_Y$ as a subalgebra of $\mathcal {O}_X$ .

By applying Lemma 3.24, the map $\Psi _P \colon \mathcal {T}_X\to \mathcal {T}_Y$ passes to the Cuntz–Pimsner quotients.

A direct application of Lemma 3.26 gives the following description of composition.

Corollary 4.9. Suppose that $(\phi _X, {}_A X_A) \overset {P_0}{\to } (\phi _Y,{}_A Y_A) \overset {Q_0}{\to } (\phi _Z,{}_A Z_A)$ are correspondence projections that induce correspondence projections

$$ \begin{align*}(\phi _X, {\mathrm {Fock}}_X) \overset {P}{\to } (\phi _Y, {\mathrm {Fock}}_Y) \overset {Q}{\to } (\phi _Z, {\mathrm {Fock}}_Z).\end{align*} $$

Suppose that $J_Z = J_Y = J_Z$ and that there are completely positive splittings $\mathcal {O}_X \to \mathcal {T}_X$ and $\mathcal {O}_Y \to \mathcal {T}_Y$ . Then there is an isometric inclusion of correspondences

$$ \begin{align*} \psi \colon (\mathcal{O}_X \otimes_{\widetilde{\Psi}_P} \mathcal{O}_Y) \otimes ( \mathcal{O}_Y \otimes_{\widetilde{\Psi}_Q} \mathcal{O}_Z ) \to \mathcal{O}_X \otimes_{\widetilde{\Psi}_{QP}} \mathcal{O}_Z. \end{align*} $$

Moreover, if there is a completely positive splitting $s_{Y} \colon \mathcal {O}_Y \to \mathcal {T}_Y$ with the additional property that the subspace $\alpha \circ s_Y(\mathcal {O}_Y) + {\mathrm {End}}_A^{0}( {\mathrm {Fock}}_X \cdot J_X) $ contains an approximate unit for $\mathcal {O}_X$ , then $\psi $ is also surjective.

Proof. It follows from Theorem 4.7 and Lemma 3.26 that there is an isometric embedding

$$ \begin{align*} \psi\colon (\mathcal{O}_X \otimes_{\widetilde{\Psi}_P} \mathcal{O}_Y) \otimes ( \mathcal{O}_Y \otimes_{\widetilde{\Psi}_Q} \mathcal{O}_Z ) \to \mathcal{O}_X \otimes_{\widetilde{\Psi}_{PQ}} \mathcal{O}_Z. \end{align*} $$

If $s\colon \mathcal {O}_X \to T_X$ is a nondegenerate splitting of the quotient map $\mathcal {T}_X \to \mathcal {O}_X$ , then there is an approximate unit $(e_i)_i$ in $\mathcal {O}_X$ such that $(s(e_i))_i$ converges strictly to the identity in $ {\mathrm {Mult}}(T_Y)$ . Then, $(s(e_i))_i$ is an approximate unit for $\mathcal {T}_X$ and it follows from the last part of Lemma 3.26 that $\psi $ is an isomorphism.

Since the KSGNS construction is fairly explicit, it is often possible to get one’s hands on the module $\mathcal {O}_X \otimes _{\widetilde {\Psi }_P} \mathcal {O}_Y$ in examples. One class of examples arises from graph $C^*$ -algebras.

Example 4.11. Let $E = (E^0,E^1,r,s)$ be a directed graph. We refer to [Reference RaeburnRae05, Ch. 8] for the description of the graph $C^*$ -algebra $C^*(E)$ as the Cuntz–Pimsner $\mathcal {O}_{X(E)}$ of the graph correspondence $X(E)$ .

Suppose that $F = (F^0,F^1,r,s)$ and are subgraphs of E such that $E^0 = F^0$ . Further, suppose that $E \setminus F$ is a regular subgraph in the sense that it contains no sources or infinite receivers. In particular, $E \setminus F$ must be a ‘large’ subgraph of E since isolated vertices are sources. Then, the graph correspondence $X(E)$ splits into a direct sum $X(E) = X(F) \oplus X(E \setminus F)$ of correspondences over $C_0(E^0)$ . Since $X(E\setminus F)$ is regular, Theorem 4.7 provides a $C^*(E)$ – $C^*(F)$ -correspondence.

To be more explicit, let $\{S_\mu \colon \mu \in E^*\}$ be the generating partial isometries of $C^*(E)$ and let $\{W_\mu \colon \mu \in F^*\}$ be the generating partial isometries of $C^*(F)$ . For $\mu ,\nu \in E^*$ ,

$$ \begin{align*} \widetilde{\Psi}_P (S_{\mu}S_{\nu}^*) = \begin{cases} W_{\mu}W_{\nu}^* &\text{if } \mu,\nu \in F^*,\\ 0 &\text{otherwise}. \end{cases} \end{align*} $$

In the KSGNS module $C^*(E) \otimes _{\widetilde {\Psi }_P} C^*(F)$ , the inner product satisfies

$$ \begin{align*} \langle S_\mu S_\nu^* \otimes W_\xi W_\eta^* \mid S_{\alpha} S_{\beta}^* \otimes W_{\rho} W_{\sigma}^* \rangle_{C^*(F)} &= W_\eta W_\xi^* \widetilde{\Psi}_P(S_\nu S_\mu^* S_\alpha S_\beta^*) W_{\rho} W_{\sigma}^*. \end{align*} $$

In particular, if $\mu = \alpha $ , $\nu = \beta $ , $\xi = \rho $ and $\eta = \sigma $ , then

$$ \begin{align*} \langle S_\mu S_\nu^* \otimes W_\xi W_\eta^* \mid S_{\mu} S_{\nu}^* \otimes W_\xi W_\eta^*\rangle_{C^*(F)} &= W_\eta W_\xi^* \widetilde{\Psi}_P(S_\nu S_\mu^* S_\mu S_\nu^*) W_{\xi} W_{\eta}^* \\ &= \begin{cases} W_\eta W_\xi^* W_{\nu} W_{\nu}^* W_\xi W_\eta^* & \text{if } \nu \in F^*, \\ 0 & \text{otherwise}, \end{cases} \\ &= \begin{cases} W_{\eta\nu'} W_{\eta \nu'}^* & \text{if } \nu = \xi \nu' \text{ and } \nu \in F^* \\ W_{\eta}W_{\eta}^* & \text{if } \xi = \nu \xi' \text{ and } \nu \in F^*,\\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Therefore, a necessary condition for $S_\mu S_\nu ^* \otimes W_\xi W_\eta ^* \in C^*(E) \otimes _{\widetilde {\Psi }_P} C^*(F)$ to be nonzero is that $\nu \in F^*$ and $\xi $ extends $\nu $ (or vice versa).

We claim that $C^*(E) \otimes _{\widetilde {\Psi }_P} C^*(F)$ is isomorphic to $ {\mathrm {Fock}}_{X(E)}\otimes _{C_0(E^0)} C^*(F)$ as right $C^*(F)$ -modules. Recall that $E^k$ denotes the paths of length k in E, and for each $k \ge 0$ , we can identify $X(E)^{\otimes k}$ with a completion of $C_c(E^k)$ . With this identification, for each $\mu \in E^*$ , we let $\delta _\mu \in {\mathrm {Fock}}_{X(E)}$ denote the point mass at $\mu $ in the completion. In $ {\mathrm {Fock}}_{X(E)}\otimes _{C_0(E^0)} C^*(F)$ , we calculate the inner product

$$ \begin{align*} \langle \delta_\mu \otimes W_\nu^*W_\xi W_\eta^* \mid \delta_\mu \otimes W_\nu^*W_\xi W_\eta^* \rangle_{C^*(F)} &= \langle W_\nu^*W_\xi W_\eta^* \mid P_{s(\mu)} W_\nu^*W_\xi W_\eta^* \rangle_{C^*(F)} \\ &= \begin{cases} W_\eta W^*_\xi W_\nu W_\nu^*W_\xi W_\eta^* & \text{if } s(\mu) = s(\nu),\\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

This yields an isometric linear map $\kappa \colon C^*(E)\otimes _{\tilde {\Psi }} C^*(F) \to {\mathrm {Fock}}_{X(E)}\otimes _{C_0(F^0)} C^*(F)$ given by $\kappa (S_\mu S_\nu \otimes W_\xi W_\eta ^*) = \delta _\mu \otimes W_\nu ^* W_\xi W_\eta ^*$ for every $\mu \in E^*$ and $\nu , \xi , \eta \in F^*$ . The target $ {\mathrm {Fock}}_{X(E)}\otimes _{C_0(F^0)} C^*(F)$ is generated by elements of the form $\delta _\mu \otimes W_\xi W_\eta ^*$ , and this is nonzero only if $r(\xi ) = s(\mu )$ . Choosing $\nu = s(\mu )$ so that $S_\nu = P_{s(\mu )}$ , we see that $\kappa (S_\mu S_\nu \otimes W_\xi W_\eta ^*) = \delta _\mu \otimes W_\xi W_\eta ^*$ . This shows that $\kappa $ is surjective.

Routine computations show that $\kappa$ intertwines the right actions of $C^*(F)$ and inner products. Define $\rho \colon C^*(E) \to {\mathrm {End}}_{C^*(F)} ( {\mathrm {Fock}}_{X(E)}\otimes _{C_0(F^0)} C^*(F))$ by

$$ \begin{align*} \rho(S_{\alpha}S_{\beta}^*) ( \delta_\mu \otimes W_\nu^*W_\xi W_\eta^*) = \begin{cases} \delta_{\alpha \mu'} \otimes W_\nu^*W_\xi W_\eta^* & \text{if } \mu = \beta \mu',\\ \delta_{\alpha} \otimes W_{\nu \beta'}^* W_{\xi} W_{\eta}^* & \text{if } \beta = \mu \beta' \text{ and } \beta' \in F^*, \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Comparing with the (positive) left action on the KSGNS module completes the isomorphism of correspondences.

For a particular instance, suppose that $E^{0} = \{v\}$ and that there are n edges (which must have source and range v). Let F be the subgraph consisting of $m \le n$ edges. Then, $E \setminus F$ is regular, $X(E) \cong \mathbb {C}^n$ and $X(F) \cong \mathbb {C}^n$ . The graph algebra $C^*(E)$ is isomorphic to the Cuntz algebra $\mathcal {O}_n$ , and $C^*(F)\cong \mathcal {O}_m$ . Our construction provides a positive $\mathcal {O}_n$ – $\mathcal {O}_m$ -correspondence with underlying $\mathcal {O}_m$ -module isomorphic to $ {\mathrm {Fock}}_{\mathbb {C}^n} \otimes _{\mathbb {C}} \mathcal {O}_m$ . Note that $\mathcal {O}_n$ typically does not act by adjointable operators on $ {\mathrm {Fock}}_{\mathbb {C}^n}$ , but it does act on the module $ {\mathrm {Fock}}_{\mathbb {C}^n} \otimes _{\mathbb {C}} \mathcal {O}_m$ .

4.2 Fock projections and subproduct systems

The previous subsection started from a correspondence projection (see Definition 4.1), then built a correspondence projection on the associated Fock module and finally constructed a positive map. In certain circumstances, we can start directly with a projection on the Fock module and obtain sensible morphisms on the associated Cuntz–Pimsner algebras.

If we restrict $\Psi _P$ to a subalgebra of $ {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ , such as $\mathcal {T}_X$ , we get a completely positive map onto some subalgebra of $ {\mathrm {End}}_A(P {\mathrm {Fock}}_X)$ .

Definition 4.15. Let $(\phi ,{}_A X_A)$ be a correspondence and let $P\in {\mathrm {End}}_A( {\mathrm {Fock}}_X)$ be a Fock projection. For $\xi \in P {\mathrm {Fock}}_X$ , we define a projected creation operator , where $T_\xi \in \mathcal {T}_X$ . Then, $T^P_\xi $ is adjointable with adjoint $(T^P_\xi )^*=T^*_\xi P$ .

We define the projected Toeplitz algebra $\mathcal {T}^P_X$ of $P {\mathrm {Fock}}_X$ to be the $C^*$ -algebra generated by $\{T^P_\xi \mid \xi \in P {\mathrm {Fock}}_X\}$ . We define the projected Cuntz–Pimsner algebra $\mathcal {O}^P_X$ of $P {\mathrm {Fock}}_X$ to be the quotient of $\mathcal {T}^P_X$ by the ideal

$$ \begin{align*} {\mathrm{End}}^0_A(P{\mathrm{Fock}}_X\cdot J_X)\cap \mathcal{T}^P_X. \end{align*} $$

By construction, $\Psi _P \colon \mathcal {T}_X \to \mathcal {T}_X^P$ is a conditional expectation. Building on similar ideas to the previous section, we use Lemma 3.24 to construct a strict completely positive map between a Cuntz–Pimsner algebra and its projected relative.

By applying Lemma 3.24, the map $\Psi _P\colon \mathcal {T}_X\to \mathcal {T}_X^P$ passes to the Cuntz–Pimsner quotients.

The construction of $\mathcal {O}_X^P$ is closely related to the construction of subproduct systems over $\mathbb {N}$ . The various definitions found in [Reference Arici and KaadAK21, Reference Dor-On and MarkiewiczDM14, Reference Shalit and SolelSS09, Reference ViselterVis12] (for instance) can all be seen to be equivalent using [Reference Shalit and SolelSS09, Lemma 6.1].

One can now deduce the existence of projections $P_k\colon X_1^{\otimes _Ak}\to X_k$ , see [Reference Shalit and SolelSS09, Lemma 6.1]. Setting $P_0=\mathrm {Id}_A$ , the strict sum

(4-4)

is a Fock projection on $ {\mathrm {Fock}}_{X_1}$ .

Given a subproduct system, we can construct the associated Toeplitz and Cuntz–Pimsner algebras. We use the definition of [Reference Arici and KaadAK21], which is derived from [Reference ViselterVis12].

Given $X=(X_m)_m$ , the distinction between $\mathcal {O}_{X_1}^P$ and $\mathbb {O}_X$ is the distinction between the ideals $ {\mathrm {End}}^0_A(P {\mathrm {Fock}}_{X_1}\cdot J_{X_1})$ and $\mathbb {I}$ ; see [Reference ViselterVis12].

5.1 Bi-Hilbertian bimodules

Amongst nondegenerate $C^*$ -correspondences, there is a useful subclass more general than Morita equivalences, but much tamer than general $C^*$ -correspondences. The properties of these bi-Hilbertian bimodules were developed by Watatani and coauthors (we refer to [Reference Kajiwara, Pinzari and WatataniKPW04] for most facts) as a $C^*$ -algebraic version of the setting of Jones index theory, and have since arisen in work of [Reference Clare, Crisp and HigsonCCH18] on representation theory, [Reference Arici and RennieAR19, Reference Goffeng, Mesland and RennieGMR18, Reference Rennie, Robertson and SimsRRS17, Reference Rennie, Robertson and SimsRRS19] on the Kasparov theory of Cuntz–Pimsner algebras and in the characterisation of ‘noncommutative vector bundles’ [Reference Rennie and SimsRS17].

In this section, we assume that all $C^*$ -modules are countably generated.

We usually do not refer to the left and right action $*$ -homomorphisms and simply write ${}_A F_B$ for a bi-Hilbertian A–B-bimodule since the definition is symmetric, unlike that of a $C^*$ -correspondence. By [Reference Kajiwara, Pinzari and WatataniKPW04, Proposition 2.16], the actions on ${}_A F_B$ are automatically nondegenerate. Many examples can be found in [Reference Rennie, Robertson and SimsRRS17, Section 2.1].

Before we examine bi-Hilbertian bimodules, we address the question of when a correspondence has a compatible left inner product. A less general version of the following lemma appears in [Reference Lord, Rennie and VárillyLRV12, Lemma 3.23].

Lemma 5.2 (See [Reference Kajiwara, Pinzari and WatataniKPW04, Lemma 2.6])

Let $F_B$ be a countably generated right B-module and let $A\subseteq {\mathrm {End}}_B(F)$ be a $C^*$ -subalgebra. Suppose that ${}_A\langle \cdot \mid \cdot \rangle $ is a left A-valued inner product on $F_B$ for which the right action of B is adjointable. Then there is an A-bilinear faithful positive map $\Upsilon \colon {\mathrm {End}}_B^{00}(F) \to A$ such that for all $e,f \in F_B$ . For any frame $(f_i)$ for $F_B$ ,

$$ \begin{align*} \Upsilon(T) = \sum_i {}_A\langle T f_i\mid f_i\rangle\quad\text{for all } T\! \in {\mathrm{End}}_B^{00}(F). \end{align*} $$

Conversely, if there exists an A-bilinear faithful positive map $\Upsilon \colon {\mathrm {End}}_B^0(F) \to A$ , then defines a left A-valued inner product on $F_B$ for which the right action of B is adjointable.

Definition 5.3 [Reference Kajiwara, Pinzari and WatataniKPW04, Definition 2.8]

Let ${}_A F_B$ be a bi-Hilbertian A–B-bimodule. We say that ${}_A F_B$ has finite right numerical index if there exists $\lambda> 0$ such that

(5-1) $$ \begin{align} \bigg\|\sum_{i=1}^n {}_A \langle f_i \mid f_i\rangle\bigg\| \le \lambda \bigg\|\sum_{i=1}^n \Theta_{f_i, f_i}\bigg\| \quad\text{for all } n\in\mathbb{N} \text{ and all } f_1, \ldots , f_n \in F. \end{align} $$

The right numerical index of ${}_A F_B$ is the infimum of the numbers $\lambda $ satisfying (5-1).

Remark 5.4. Following [Reference Kajiwara, Pinzari and WatataniKPW04, Corollary 2.11], if ${}_A F_B$ has finite right numerical index, then the map $\Upsilon \colon {\mathrm {End}}_B^{00}(F) \to A$ of Lemma 5.2 extends to an A-bilinear $*$ -homomorphism $\Upsilon \colon {\mathrm {End}}_B^{0}(F) \to A$ satisfying

$$ \begin{align*} \Upsilon(\Theta_{e,f}) = {}_A \langle e \mid f \rangle \quad \text{for all } e \in E,\, f \in F. \end{align*} $$

If $T\in {\mathrm {End}}_B^{0}(F)$ commutes with all $a\in A$ , then $ a\Upsilon (T)=\Upsilon (aT)=\Upsilon (Ta)=\Upsilon (T)a, $ so $\Upsilon (T)$ is in the centre of A. In particular, $\Upsilon (\mathrm {Id}_F)$ is central in A.

In [Reference Kajiwara, Pinzari and WatataniKPW04, Proposition 2.13], it is shown that tensor products of bi-Hilbertian bimodules with finite left and right numerical indices are again bi-Hilbertian bimodules.

Let ${}_A F_B$ be a bi-Hilbertian bimodule, countably generated as a right module and with finite right numerical index. By [Reference Kajiwara, Pinzari and WatataniKPW04, Theorem 2.22], the left action of A on ${}_A F_B$ is by compact endomorphisms with respect to $\langle \cdot \mid \cdot \rangle _B$ if and only if, for every frame $(f_j)$ for the right B-module $F_B$ , the series

$$ \begin{align*} \sum_{j\geq 1}{}_A\langle f_j \mid f_j\rangle \end{align*} $$

converges strictly in $ {\mathrm {Mult}}(A)$ . In this case, we denote the strict limit by $r\text {-}\operatorname {Ind}(F)$ . We note that $r\text {-}\operatorname {Ind} (F)$ is independent of the frame $(f_j)$ [Reference Kajiwara, Pinzari and WatataniKPW04, Theorem 2.22].

We record the following facts about $r\text {-}\operatorname {Ind} (F)$ .

Lemma 5.8 motivates the following definition.

If ${}_AF_B$ is a regular bi-Hilbertian bimodule, then the left and right Watatani indices are central, positive and invertible elements of $ {\mathrm {Mult}}(A)$ and $ {\mathrm {Mult}}(B)$ , respectively. This allows us to write $e^{\beta } \in {\mathrm {Mult}}(A)$ for the right Watatani index, where $\beta = \beta _R$ is central and self-adjoint. The notation $e^{\beta }$ will always mean the right Watatani index unless stated otherwise. When needed, the left Watatani index is denoted $e^{\beta _L}$ .

The next result gives useful relationships between frames for the different module structures on a bi-Hilbertian bimodule, its conjugate module and its algebra of compact operators. Recall from Notation 2.5 that if ${}_A F_B$ is a bi-Hilbertian bimodule, then $ {\mathrm {End}}^0_B(F)$ is identified with $F \otimes _B F^*$ . Since ${}_A F_B$ comes equipped with a left A-valued inner product, $ {\mathrm {End}}_B^0(F)$ inherits the structure of a bi-Hilbertian A-bimodule [Reference Kajiwara, Pinzari and WatataniKPW04, Corollary 2.29].

The following property was discovered and described in [Reference Kajiwara, Pinzari and WatataniKPW04, Section 4.2] using the language of intertwiners.

Lemma 5.11. Let ${}_A F_B$ be a regular bi-Hilbertian bimodule. Then the $A$ -bilinear map $P_0\colon F\otimes _B F^* \to F\otimes _B F^*$ defined by

$$ \begin{align*} P_0 (T) = e^{-\beta}\Upsilon(T)\, \mathrm{Id}_F \end{align*} $$

is an adjointable projection. In particular, $P(F \otimes _B F^*) = A {\mathrm {Id}}_F$ is a complemented bi-Hilbertian sub-bimodule of ${}_A F \otimes _B F^*_A$ , which is isomorphic to ${}_A A_A$ .

Proof. Recall that if $e^{\beta } \in {\mathrm {Mult}}(A)$ has finite index, then $\Upsilon (T) \in A \subseteq F \otimes _B F^*$ for all $T\in F\otimes _B F^*$ . Since $F\otimes _B F^*$ is an ideal in $ {\mathrm {End}}_B(F)$ , we have $P_0(T) \in F \otimes _B F^*$ . The A-bilinearity of $\Upsilon $ and centrality of $e^{\beta }$ imply that $P_0$ is A-bilinear. Fullness of the left inner product implies that $P_0$ surjects onto $A {\mathrm {Id}}_F$ .

To see that $P_0$ is idempotent, fix a right frame $(u_i)_i$ for $F_B$ . Then, for $a \in A$ ,

$$ \begin{align*} P_0(a {\mathrm{Id}}_F) = \lim_i \sum_i P_0(a u_i \otimes u_i^*) = a {\mathrm{Id}}_F. \end{align*} $$

Since $e^{-\beta }\Psi (T)\in A$ , it follows that $P^2(T) = P_0(e^{-\beta } \Upsilon (T) {\mathrm {Id}}_F) = e^{-\beta } \Upsilon (T) {\mathrm {Id}}_F$ .

For adjointability of $P_0$ , fix $e_1 \otimes e_2^*,\,f_1 \otimes f_2^* \in F \otimes _B F^*$ . We first observe that

$$ \begin{align*} \sum_l\langle u_l\otimes u_l^*\mid f_1\otimes f_2^*\rangle_A &=\sum_l\langle u_l^*\mid\langle u_l\mid f_1\rangle_A \cdot f_2^*\rangle_A\\ &=\sum_l\langle (\langle f_1\mid u_l\rangle_A \cdot u_l)^*\mid f_2^*\rangle_A ={}_A\langle f_1\mid f_2\rangle. \end{align*} $$

Then, using centrality of $e^{-\beta }$ at the last equality,

$$ \begin{align*} \langle P_0(e_1 \otimes e_2^*) \mid f_1 \otimes f_2^* \rangle_A &= \sum_k\langle e^{-\beta} {}_A \langle e_1 \mid e_2 \rangle u_k\otimes u_k^*\mid f_1\otimes f_2^*\rangle_A\\ &=\sum_k\langle u_k\otimes u_k^*|e^{-\beta} {}_A \langle e_2 \mid e_1 \rangle f_1\otimes f_2^*\rangle_A ={}_A\langle e^{-\beta}{}_A \langle e_2 \mid e_1 \rangle f_1\mid f_2\rangle \\ &=e^{-\beta}{}_A \langle e_2 \mid e_1 \rangle {}_A\langle f_1\mid f_2\rangle ={}_A \langle e_2 \mid e_1 \rangle e^{-\beta} {}_A\langle f_1\mid f_2\rangle. \end{align*} $$

A symmetric calculation shows $\langle e_1 \otimes e_2^* \mid P_0(f_1 \otimes f_2^*) \rangle _A ={}_A \langle e_2 \mid e_1 \rangle e^{-\beta } {}_A\langle f_1\mid f_2\rangle $ , from which it follows that $P_0^* = P_0$ .

A bi-Hilbertian bimodule ${}_AF_B$ does not typically give a covariant correspondence between a $C^*$ -correspondence ${}_AX_A$ and $F^*\otimes _AX\otimes _AF$ in the sense of [Reference Meyer and SehnemMS19, Definition 2.21]. This is because $X\otimes _A F \cong A \otimes _A X \otimes _A F \subseteq F\otimes _B F^* \otimes _A X \otimes _A F$ , where the inclusion is as a complemented submodule. The only time this inclusion is an isomorphism is when F is a Morita equivalence.

Although they do not typically induce covariant correspondences, in the next section, we show that conjugation of a $C^*$ -correspondence by a bi-Hilbertian bimodule yields a strict completely positive map between corresponding Cuntz–Pimsner algebras.

5.2 From bi-Hilbertian bimodules to positive maps

Suppose that ${}_{A} F_B$ is a bi-Hilbertian bimodule of finite index, and ${}_AX_A$ is a $C^*$ -correspondence. Lemma 5.11 provides natural projections on the Fock module of the B–B-correspondence . We use these projections to induce correspondences between their associated Cuntz–Pimsner algebras. We start on Fock space and build expectations on the associated Toeplitz algebras.

Let $P_0 \colon F \otimes _B F^* \to A {\mathrm {Id}}_F$ be the projection of Lemma 5.11, so

$$ \begin{align*} P_0(f \otimes g^*) = e^{-\beta} {}_A\langle f \mid g\rangle {\mathrm{Id}}_F, \end{align*} $$

where $e^{\beta }$ is the right Watatani index of $F_B$ . Consider the element

Since F is finite index, A acts compactly and so $aZ\in {\mathrm {End}}_B^0(F)=F\otimes _B F^*$ for all $a\in A$ . Then, for each $a \in A$ , and right frame $(u_i)$ ,

$$ \begin{align*} P_0(aZ) = e^{-\beta/2}\sum_i P_0(a u_i \otimes u_i^*) = e^{-\beta/2} \sum_i e^{-\beta} a{}_A \langle u_i \mid u_i \rangle {\mathrm{Id}}_F = aZ. \end{align*} $$

Since $e^{\beta } \in {\mathrm {Mult}}(A)$ commutes with elements of A, Z commutes with elements of $A \subseteq {\mathrm {End}}_B^0(F)$ . We record how inner products with $aZ$ work in $ {\mathrm {End}}_B^0(F)$ .

Lemma 5.12. Let ${}_A F_B$ be a regular bi-Hilbertian bimodule. Then, for a simple tensor $f_1 \otimes f_2^* \in F \otimes _B F^*$ and $a \in A$ ,

$$ \begin{align*} \langle f_1 \otimes f_2^* \mid aZ \rangle_A = e^{-\beta/2} {}_A \langle f_1 \mid f_2 \rangle a. \end{align*} $$

In particular, for $a_1,\,a_2 \in A$ , we have $\langle a_1 Z \mid a_2 Z \rangle _A = a_1^* a_2$ .

Proof. We compute

$$ \begin{align*} \langle f_1 \otimes f_2^* \mid aZ \rangle_{A} &= \sum_i \langle f_2^* \mid \langle f_1 \mid u_i \rangle_B \cdot u_i^* \rangle_A e^{-\beta/2}a = \sum_i\langle (f_2 \cdot \langle f_1 \mid u_i \rangle_B)^* \mid u_i^* \rangle_A e^{-\beta/2}a\\ &= \sum_i {}_A \langle f_2 \mid u_i \cdot \langle u_i \mid f_1 \rangle_B \rangle e^{-\beta/2}a = e^{-\beta/2}{}_A \langle f_2 \mid f_1 \rangle a. \end{align*} $$

The second statement follows from the first.

Let ${}_A F_B$ be a regular bi-Hilbertian bimodule and let $(\phi ,{}_A X_A)$ be a $C^*$ -correspondence. Since $e^{-\beta }$ is positive and invertible, we have $P_0(F\otimes F^*) = A Z$ . Using A-bilinearity of $P_0$ , we can define a projection $P_n$ on $F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^*$ by

where $P_0$ appears $n+1$ times. Since the image of $P_0$ is isomorphic to ${}_A A_A$ , the image of $P_n$ can be identified with $X^{\otimes n}$ . We make this more precise.

Lemma 5.13. Let $(\phi ,{}_A X_A)$ be a $C^*$ -correspondence and let ${}_A F_B$ be a regular bi-Hilbertian bimodule. For all $n \in \mathbb {N}_0$ , there is an adjointable A-bilinear isometric inclusion $ W_n \colon X^{\otimes _A n} \to F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^* $ such that for $n \ge 1$ ,

$$ \begin{align*} W_n(x_1 \otimes \cdots \otimes x_n) = Z \otimes x_1 \otimes Z \otimes \cdots \otimes Z \otimes x_n \otimes Z \end{align*} $$

and $W_0(a) = aZ$ for $a \in A \cong X^{\otimes 0}$ . For $f,g \in F$ and $\bigotimes _{i=1}^n (f_i \otimes x_i \otimes g_i^*) \in {\mathrm {Fock}}_{F^*XF}$ , the adjoint satisfies

$$ \begin{align*} W_n^*\bigg(f \otimes \bigotimes_{i=1}^n (f_i^* \otimes x_i \otimes g_i) \otimes g^* \bigg) &= e^{-\beta/2} {}_A\langle f \mid f_1 \rangle \cdot x_1 \otimes e^{-\beta/2}\cdot {}_A \langle g_1 \otimes f_2^* \rangle \cdot x_2 \otimes \cdots\\ &\quad \otimes e^{-\beta/2} {}_A \langle g_{n-1} \mid f_n \rangle \cdot x_n \cdot e^{-\beta/2} {}_A \langle g_n \mid g \rangle. \end{align*} $$

Moreover, $ W_n(X^{\otimes _A n}) = P_n(F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^*) $ is a complemented sub- correspondence of $F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^*$ . If ${}_AF_B$ is a Morita equivalence bimodule, then conjugation by ${}_A F_B$ yields $X^{\otimes _A n}\cong F\otimes _B(F^* X F)^{\otimes _A n}\otimes _B F^*$ .

Proof. For each $x \in {}_A X_A$ , we can write $x = a_1 \cdot x' \cdot a_2$ for some $a_1,a_2 \in A$ and $x' \in {}_A X_A$ . So, $Z \otimes x \otimes Z = Za_1 \otimes x' \otimes a_2 Z$ belongs to $F \otimes _B F^* \otimes _A X \otimes _A F \otimes _B F^*$ . In particular, the formula defining $W_n$ makes sense.

The case where $n = 0$ follows from the fact that $P_0(F\otimes F^*)$ is the A-span of Z. For $n \ge 1$ , we begin by showing that $W_n$ preserves inner products and is therefore isometric. Let $(u_i)$ be a right frame for F and observe that for any $f \in F$ , we have $\sum _{i} \langle f\mid u_i\rangle _B \cdot u_i^* = f^*$ . Fix elementary tensors $x = x_1 \otimes \cdots \otimes x_n$ and $y = y_1 \otimes \cdots \otimes y_n$ in $X^{\otimes n}$ . Then,

$$ \begin{align*} \langle Wx \mid Wy\rangle_{A} &= \langle x_n \otimes Z \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \otimes Z\rangle_A, \end{align*} $$

where $x' = x_1 \otimes \cdots \otimes x_{n-1}$ and $y' = y_1 \otimes \cdots \otimes y_{n-1}$ . Using the frame relation at the fourth equality and the definition of $e^{\beta }$ at the fifth equality,

$$ \begin{align*} \langle W_nx \mid W_ny\rangle_{A} &= \sum_{i,j} \langle x_n \otimes e^{-\beta/2} \cdot u_i \otimes u_i^* \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \otimes e^{-\beta/2} \cdot u_j \otimes u_j^*\rangle_A\\ &= \sum_{i,j} \langle u_i^* \mid e^{-\beta/2} \cdot u_i \mid \langle x_n \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \rangle_A e^{-\beta/2} \cdot u_j\rangle_B \cdot u_j^* \rangle_A\\ &= \sum_{i,j} {}_A\langle u_i \cdot \langle e^{-\beta/2} \cdot u_i \mid \langle x_n \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \rangle_A e^{-\beta/2} \cdot u_j\rangle_B \mid u_j \rangle\\ &= \sum_{j} {}_A\langle e^{-\beta} \langle x_n \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \rangle_A \cdot u_j \mid u_j \rangle\\ &= \langle x_n \mid \langle W_{n-1}x' \mid W_{n-1} y' \rangle_A \cdot y_n \rangle_A = \langle W_{n-1}x' \otimes x_n \mid W_{n-1} y' \otimes y_n\rangle_A. \end{align*} $$

An inductive argument now shows that $\langle W_nx \mid W_ny\rangle _A = \langle x \mid y\rangle _A$ . The preceding argument extends to spans of elementary tensors, and so $W_n$ extends to an isometric linear map from $X^{\otimes _A n}$ to $ F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^*$ . The A-bilinearity of $W_n$ follows from the fact that Z commutes with elements of A.

The formula for $W_n^*$ follows by using Lemma 5.12 to check that $W_n^* W_n = {\mathrm {Id}}$ on elementary tensors. Additionally,

$$ \begin{align*} W_n(X^{\otimes_A n})= A Z \otimes_A X \otimes_A AZ \otimes \cdots \otimes A Z \otimes_A X \otimes A Z = P_n(F\otimes_B(F^*XF)^{\otimes_A n}\otimes_BF^* ), \end{align*} $$

which is complimented in $F\otimes _B(F^*XF)^{\otimes _A n}\otimes _BF^*$ .

Finally, observe that if ${}_AF_B$ is a Morita equivalence, then $F \otimes _B F^* \cong ~A$ .