2. Construction

First, we recall the definition of strong Morita equivalence for inclusions of $C^{*}$-algebras.

Let $A\subset C$ and $B\subset D$ be inclusions of $C^{*}$-algebras with $\overline {AC}=C$ and $\overline {CD}=D$. Let ${}_A \mathbf {B}_A (C,\, A)$, $_{B} \mathbf {B}_B (D,\, B)$ be the spaces of all bounded $A-A$-bimodule linear maps and all bounded $B-B$-bimodule linear maps from $C$ and $D$ to $A$ and $B$, respectively. We suppose that $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$. We construct an isometric isomorphism of ${}_B \mathbf {B}_B (D,\, B)$ onto ${}_A \mathbf {B}_A (C,\, A)$. For any $\phi \in {}_B \mathbf {B}_B (D,\, B)$, we define the linear map $\tau$ from $Y$ to $X$ by

\[ \langle x , \, \tau(y) \rangle_B =\phi( \langle x, y \rangle_D ) \]

for any $x\in X$, $y\in Y$.

Proof. Except for the uniqueness of $\tau$, we can prove this lemma in the same way as in the proof of [Reference Kodaka and Teruya8, Lemma 3.4 and Remark 3.3 (ii)]. Indeed, the definition of $\tau$, $\tau$ satisfies Condition (3). For any $x,\, z\in X$, $d\in D$,

\[ \langle z, \tau(x\cdot d)\rangle_B =\phi(\langle z, x\cdot d \rangle_D )=\phi(\langle z, x \rangle_B \, d) =\langle z, x \rangle_B \, \phi(d) =\langle z, x\cdot \phi(d)\rangle_B . \]

Hence $\tau (x\cdot d)=x\cdot \phi (d)$ for any $x\in X$, $d\in D$. Also, for any $b\in B$, $z\in X$ $y\in Y$,

\[ \langle z, \tau(y\cdot b)\rangle_B =\phi(\langle z, y\cdot b \rangle_D \, )=\phi(\langle z, y \rangle_D \, b)=\phi(\langle z, y \rangle_D \, )b =\langle z, \tau(y)\cdot b \rangle_B . \]

Thus, $\tau (y\cdot b)=\tau (y)\cdot b$ for any $b\in B$, $y\in Y$. By Raeburn and Williams [Reference Raeburn and Williams10, the proof of Lemma 2.18], for any $y\in Y$

\begin{align*} ||\tau(y)|| & =\sup \{||\langle x, \tau(y) \rangle_B|| \ | \ ||x||\leq 1, \, x\in X \} \\ & =\sup \{||\phi(\langle x, y \rangle_D ) || \ | \ ||x||\leq 1, \, x\in X \} \\ & \leq \sup \{||\phi|| \, ||x|| \, ||y|| \, \, | \, \, ||x||\leq 1, \, x\in X \} \\ & \leq ||\phi||\, ||y||. \end{align*}

Thus, $\tau$ is bounded and $||\tau ||\leq ||\phi ||$. Furthermore, let $\tau '$ be a linear map from $Y$ to $X$ satisfying Condition (3). Then for any $x\in X$, $y\in Y$,

\[ \langle x, \tau(y)\rangle_B =\phi(\langle x, y \rangle_D \, )=\langle x, \tau' (y) \rangle_B . \]

Hence $\tau (y)=\tau ' (y)$ for any $y\in Y$. Therefore, $\tau$ is unique.

Proof. We can prove this lemma by routine computations. Indeed, for any $x,\, z\in X$, $y\in Y$,

\[ \tau({}_A \langle x, z \rangle\cdot y)=\tau( x\cdot \langle z , y \rangle_D )=x\cdot \phi(\langle z, y \rangle_D ) =x\cdot \langle z, \tau (y)\rangle_B ={}_A \langle x, z \rangle \cdot \tau(y) . \]

Since $\overline {{}_A \langle X,\, \, X \rangle }=A$ and $\tau$ is bounded, we obtain the conclusion.

Let $\psi$ be the linear map from $C$ to $A$ defined by

\[ \psi(c)\cdot x=\tau(c\cdot x) \]

for any $c\in C$, $x\in X$, where we identify $\mathbf {K}_B (X)$ with $A$ as $C^{*}$-algebras by the map $a\in A \mapsto T_a \in \mathbf {K}_B (X)$, which is defined by $T_a (x)=a\cdot x$ for any $x\in X$.

Proof. We can prove this lemma in the same way as in the proof of [Reference Kodaka and Teruya8, Proposition 3.6]. Indeed, by the definition of $\psi$, $\psi$ satisfies Condition (1). Also, for any $x,\, z\in X$, $y\in Y$,

\[ \psi({}_C \langle y, x \rangle )\cdot z =\tau(\, {}_C \langle y, x \rangle \cdot z ) =\tau(y\cdot \langle x, z \rangle_B \, ) =\tau(y)\cdot \langle x, z \rangle_B ={}_A \langle \tau(y) , \, x \rangle \cdot z \]

Hence $\psi (\, {}_C \langle y ,\, x \rangle )={}_A \langle \tau (y) ,\, \, x \rangle$ for any $x\in X$, $y\in Y$. For any $c\in C$,

\begin{align*} ||\psi(c)|| & =\sup \{||\psi(c)\cdot x || \ | \ ||x||\leq 1, \, x\in X \} \\ & =\sup \{||\tau(c\cdot x) || \ | \ ||x||\leq 1, \, x\in X \} \\ & \leq \sup \{||\tau|| \, ||c|| \, ||x|| \ | \ ||x||\leq 1, \, x\in X \} \\ & =||\tau||\, ||c|| . \end{align*}

Thus, $\psi$ is bounded and $||\psi ||\leq ||\tau ||$. Next, we show that $\psi$ is an $A-A$-bimodule map from $C$ to $A$. It suffices to show that

\[ \psi(ac)=a\psi(c), \quad \psi(ca)=\psi(c)a \]

for any $a\in A$, $c\in C$. For any $a\in A$, $c\in C$, $x\in X$,

\[ \psi(ac)\cdot x=\tau(ac\cdot x)=\tau(a\cdot (c\cdot x))=a\tau(c\cdot x)=a\psi(c)\cdot x \]

by Lemma 2.2. Hence $\psi (ac)=a\psi (c)$. Also,

\[ \psi(ca)\cdot x=\tau(ca\cdot x)=\tau(c\cdot (a\cdot x))=\psi(c)\cdot (a\cdot x)=\psi(c)a\cdot x \]

since $a\cdot x\in X$. Hence $\psi (ca)=\psi (c)a$. Let $\psi '$ be a linear map from $C$ to $A$ satisfying Condition (1). Then for any $x\in X$, $c\in C$,

\[ \psi(c)\cdot x=\tau(c\cdot x)=\psi' (c)\cdot x . \]

Thus, $\psi (c)=\psi ' (c)$ for any $c\in C$. Therefore, we obtain the conclusion.

By getting the similar lemmas to Lemmas 2.1, 2.2 and 2.3, we obtain the following proposition:

We denote by $f_{(X, Y)}$ the map

\[ \phi\in{}_B \mathbf{B}_B (D, B)\mapsto\psi\in{}_A \mathbf{B}_A (C, A) \]

as above. By the definition of $f_{(X, Y)}$ and Propositions 2.4 and 2.5, we can see that $f_{(X, Y)}$ is an isometric isomorphism of ${}_B \mathbf {B}_B (D,\, B)$ onto ${}_A \mathbf {B}_A (C,\, A)$.

Lemma 2.6 With the above notation, let $\phi$ be any element in ${}_B \mathbf {B}_B (D,\, B)$. Then $f_{(X, Y)}(\phi )$ is the unique linear map from $C$ to $A$ satisfying that

\[ \langle x , f_{(X, Y)}(\phi)(c)\cdot z \rangle_B =\phi(\langle x , \, c\cdot z \rangle_D ) \]

for any $c\in C,$ $x,\, z\in X$.

Proof. It is clear that $f_{(X, Y)}(\phi )$ satisfies the above equation by the definition of $f_{(X, Y)}(\phi )$ and Lemma 2.1. Let $\psi$ be another linear map from $C$ to $A$ satisfying the above equation. Then for any $c\in C$, $x,\, z\in X$,

\[ \langle x, \, f_{(X, Y)}(\phi)(c)\cdot z \rangle_B=\langle x, \, \psi(c)\cdot z \rangle_B . \]

Hence $f_{(X, Y)}(\phi )(c)=\psi (c)$ for any $c\in C$. Thus, $f_{(X, Y)}(\phi )=\psi$.

Let ${{\rm {Equi}}}(A,\, C,\, B,\, D)$ be the set of all pairs $(X,\, Y)$ such that $Y$ is a $C-D$-equivalence bimodule and $X$ is its closed subspace satisfying Conditions (1), (2) in Definition 2.1. We define an equivalence relation “$\sim$” in ${{\rm {Equi}}} (A,\, C,\, B,\, D)$ as follows: For any $(X,\, Y),\, (Z,\, W)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$, we say that $(X,\, Y)\sim (Z,\, W)$ in ${{\rm {Equi}}}(A,\, C,\, B,\, D)$ if there is a $C-D$- equivalence bimodule map $\Phi$ of $Y$ onto $W$ such that $\Phi |_X$ is a bijection of $X$ onto $Z$. Then $\Phi |_X$ is an $A-B$-equivalence bimodule isomorphism of $X$ onto $Z$ by [Reference Kodaka6, Lemma 3.2]. We denote by $[X,\, Y]$ the equivalence class of $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$ and we denote by ${{\rm {Equi}}}(A,\, C,\, B,\, D)/\!\sim$ the set of all equivalence classes $[X,\, Y]$ of $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$.

Proof. Let $\Phi$ be a $C-D$-equivalence bimodule isomorphism of $Y$ onto $W$ satisfying that $\Phi |_X$ is a bijection of $X$ onto $Z$. Let $\phi \in {}_B\mathbf {B}_B (D,\, B)$. Then for any $x_1 ,\, x_2 \in X$, $c\in C$,

\begin{align*} \langle \Phi(x_1 ) , \, f_{(Z, W)}(\phi)(c)\cdot \Phi(x_2 ) \rangle_B & = \phi(\langle \Phi(x_1 ) , \, c\cdot \Phi(x_2 ) \rangle_D ) \\ & =\phi(\langle \Phi(x_1 ) , \, \Phi(c\cdot x_2 ) \rangle_D ) \\ & =\phi(\langle x_1 , \, c\cdot x_2 \rangle_D ) \\ & =\langle x_1 , \, f_{(X, Y)}(\phi)(c)\cdot x_2 \rangle_B . \end{align*}

On the other hand,

\begin{align*} \langle \Phi(x_1 ), f_{(Z, W)}(\phi)(c)\cdot \Phi(x_2 ) \rangle_B & =\langle \Phi(x_1 ) , \Phi(f_{(Z, W)}(\phi)(c)\cdot x_2 ) \rangle_B \\ & =\langle x_1 , f_{(Z, W)}(\phi)(c)\cdot x_2 \rangle_B . \end{align*}

Hence we obtain that

\[ \langle x_1 , f_{(X, Y)}(\phi)(c)\cdot x_2 \rangle_B =\langle x_1 , f_{(Z, W)}(\phi)(c)\cdot x_2 \rangle_B. \]

for any $c\in C$, $x_1 ,\, x_2 \in X$. Therefore, we obtain the conclusion.

We denote by $f_{[X, Y]}$ the isometric isomorphism of ${}_B \mathbf {B}_B (D,\, B)$ into ${}_A \mathbf {B}_A (C,\, A)$ induced by the equivalence class $[X,\, Y]$ of $(X,\, Y)\in {{\rm {Equi}}} (A,\, C,\, B,\, D)$.

Let $L\subset M$ be an inclusion of $C^{*}$-algebras with $\overline {LM}=M$, which is strongly Morita equivalent to the inclusion $B\subset D$ with respect to a $D-M$-equivalence bimodule $W$ and its closed subspace $Z$. Then the inclusion $A\subset C$ is strongly Morita equivalent to the inclusion $L\subset M$ with respect to the $C-M$-equivalence bimodule $Y\otimes _D W$ and its closed subspace $X\otimes _B Z$.

Lemma 2.8 With the above notation,

\[ f_{[X\otimes_B Z , \, Y\otimes_D W]}=f_{[X, Y]}\circ f_{[Z, W]} . \]

Proof. Let $x_1 ,\, x_2 \in X$ and $z_1 ,\, z_2 \in Z$. Let $c\in C$ and $\phi \in {}_L\mathbf {B}_L (M,\, L)$. Then

\begin{align*} & \langle x_1 \otimes z_1 , \, (f_{[X, Y]}\circ f_{[Z, W]})(\phi)(c)\cdot x_2 \otimes z_2 \rangle_L \\ & \quad=\langle z_1 , \, \langle x_1 , \, (f_{[X, Y]}\circ f_{[Z, W]})(\phi)(c)\cdot x_2 \rangle_B \cdot z_2 \rangle_L \\ & \quad=\langle z_1 , \, f_{[Z, W]}(\phi)(\langle x_1 , \, c\cdot x_2 \rangle_D \, )\cdot z_2 \rangle_L \\ & \quad=\phi(\langle z_1 , \, \langle x_1 , \, c\cdot x_2 \rangle_D \cdot z_2 \rangle_M ) . \end{align*}

On the other hand,

\begin{align*} & \langle x_1 \otimes z_1 , \, (f_{[X\otimes_B Z, \, Y\otimes_D W]})(\phi)(c)\cdot x_2 \otimes z_2 \rangle_L \\ & \quad= \phi(\langle x_1 \otimes z_1 , \, c\cdot x_2 \otimes z_2 \rangle_L) \\ & \quad=\phi(\langle z_1 , \, \langle x_1 , \, c\cdot x_2 \rangle_D \cdot z_2 \rangle_M ) . \end{align*}

By Lemma 2.6,

\[ (f_{[X, Y]}\circ f_{[Z, W]})(\phi)=f_{[X\otimes_B Z, \, Y\otimes_D W]}(\phi) \]

for any $\phi \in {}_L \mathbf {B}_L (M,\, L)$. Therefore, we obtain the conclusion.

3. Strong Morita equivalence

Let $A\subset C$ and $B\subset D$ be inclusions of $C^{*}$-algebras with $\overline {AC}=C$ and $\overline {BD}=D$. Let $\psi \in {}_A \mathbf {B}_A (C,\, A)$ and $\phi \in {}_B \mathbf {B}_B (D,\, B)$.

Let $\psi \in {}_A \mathbf {B}_A (C,\, A)$ and $\phi \in {}_B \mathbf {B}_B (D,\, B)$. We suppose that $\psi$ and $\phi$ are strongly Morita equivalent with respect to $(X,\, Y)$ in ${{\rm {Equi}}}(A,\, C,\, B,\, D)$. Let $L_X$ and $L_Y$ be the linking $C^{*}$-algebras for $X$ and $Y$, respectively. Then in the same way as in [Reference Kodaka6, Section 3] or Brown et al. [Reference Brown, Green and Rieffel2, Theorem 1.1], $L_X$ is a $C^{*}$-subalgebra of $L_Y$ and by easy computations, $\overline {L_X L_Y}=L_Y$. Furthermore, there are full projections $p,\, q\in M(L_X )$ with $p+q=1_{M(L_X )}$ satisfying the following conditions:

\begin{align*} pL_X p & \cong A , \quad pL_Y p \cong C , \\ qL_X q & \cong B , \quad qL_Y q \cong D \end{align*}

as $C^{*}$-algebras. We note that $M(L_X )\subset M(L_Y )$ by Pedersen [Reference Pedersen9, Section 3.12.12] since $\overline {L_X L_Y}=L_Y$.

Let $\phi$, $\psi$ be as above. We suppose that $\phi$ and $\psi$ are selfadjoint. Let $\tau$ be the unique bounded linear map from $Y$ to $X$ satisfying Conditions (1)–(6) in Proposition 2.4. Let $\rho$ be the map from $L_Y$ to $L_X$ defined by

\[ \rho\left(\begin{bmatrix} c & y \\ \widetilde{z} & d \end{bmatrix}\right) =\begin{bmatrix} \psi(c) & \tau(y) \\ \widetilde{\tau(z)} & \phi(d) \end{bmatrix} \]

for any $c\in C$, $d\in D$, $y,\, z\in Y$. By routine computations $\rho$ is a selfadjoint element in ${}_{L_X} \mathbf {B}_{L_X} (L_Y ,\, L_X )$, where ${}_{L_X} \mathbf {B}_{L_X} (L_Y ,\, L_X )$ is the space of all bounded $L_X -L_X$-bimodule linear maps from $L_Y$ to $L_X$. Furthermore, $\rho |_{pL_Y p}=\psi$ and $\rho |_{qL_Y q}=\phi$, where we identify $A,\, C$ and $B,\, D$ with $pL_X p$, $pL_Y p$ and $qL_X q$, $qL_Y q$ in the usual way, respectively. Thus, we obtain the following lemma:

Lemma 3.2 With the above notation, let $\psi \in {}_A \mathbf {B}_A (C,\, A)$ and $\phi \in {}_B \mathbf {B}_B (D,\, B)$. We suppose that $\psi$ and $\phi$ are selfadjoint and strongly Morita equivalent with respect to $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$. Then there is a selfadjoint element $\rho \in {}_{L_X} \mathbf {B}_{L_X} (L_Y ,\, L_X )$ such that

\[ \rho|_{pL_Y p}=\psi , \quad \rho|_{qL_Y q}=\phi . \]

Also, we have the converse direction:

Lemma 3.3 Let $A\subset C$ and $B\subset D$ be as above and let $\psi \in {}_A \mathbf {B}_A (C,\, A)$ and $\phi \in {}_B \mathbf {B}_B (D,\, B)$ be selfadjoint elements. We suppose that there are an inclusion $K\subset L$ of $C^{*}$-algebras with $\overline {KL}=L$ and full projections $p,\, q\in M(K)$ with $p+q=1_{M(K)}$ such that

\[ A\cong pKp , \quad C\cong pLp, \ B\cong qKq , \ D\cong qLq , \]

as $C^{*}$-algebras. Also, we suppose that there is a selfadjoint element $\rho$ in ${}_K \mathbf {B}_K (L,\, K)$ such that

\[ \rho|_{pLp}=\psi , \quad \rho|_{qLq}=\phi . \]

Then $\phi$ and $\psi$ are strongly Morita equivalent, where we identify $pKp,$ $pLp$ and $qKq,$ $qLq$ with $A,\, C$ and $B,\, D,$ respectively.

Proof. We note that $(Kp,\, Lp )\in {{\rm {Equi}}}(K,\, L,\, A,\, C)$, where we identify $A$ and $C$ with $pKp$ and $pLp$, respectively. By routine computations, we can see that

\[ \langle kp , \rho(l)\cdot k_1 p \rangle_A =\psi(\langle kp , \, l\cdot k_1 p \rangle_C ) \]

for any $k,\, k_1 \in K$, $l\in L$. Thus, by Lemma 2.6, $f_{[Kp , \, Lp ]}(\psi )=\rho$. Similarly, $f_{[Kq , \, Lq]}(\phi )=\rho$. Since $f_{[Kq , \, Lq ]}^{-1}(\rho )=\phi$,

\[ (f_{[Kq ,\, Lq]}^{{-}1} \circ f_{[Kp , \, Lp ]})(\psi)=\phi . \]

Since $f_{[Kq ,\, Lq ]}^{-1}=f_{[qK , \, qL ]}$, by Lemma 2.8

\[ \phi=f_{[qK , \, qL][Kp , \, Lp]}(\rho)=f_{[qKp ,\, qLp]}(\psi) . \]

Therefore, we obtain the conclusion.

4. Stable $C^{*}$-algebras and matrix algebras

Let $A\subset C$ be an inclusion of $C^{*}$-algebras with $\overline {AC}=C$. Let $A^{s} =A\otimes \mathbf {K}$ and $C^{s} =C\otimes \mathbf {K}$. Let $\{e_{ij}\}_{i, j=1}^{\infty }$ be a system of matrix units of $\mathbf {K}$. Clearly $A^{s} \subset C^{s}$ and $A\subset C$ are strongly Morita equivalent with respect to the $C^{s} -C$-equivalence bimodule $C^{s} (1_{M(A)}\otimes e_{11})$ and its closed subspace $A^{s} (1_{M(A)}\otimes e_{11})$, where we identify $A$ and $C$ with $(1\otimes e_{11})A^{s} (1\otimes e_{11})$ and $(1\otimes e_{11})C^{s} (1\otimes e_{11})$, respectively.

Lemma 4.1 With the above notation, for any $\phi \in {}_A \mathbf {B}_A (C,\, A),$

\[ f_{[A^{s} (1\otimes e_{11}) , \ C^{s} (1\otimes e_{11})]}(\phi)=\phi\otimes{{\rm{id}}}_{\mathbf{K}}. \]

Proof. It suffices to show that

\[ \langle a(1\otimes e_{11}) , \ (\phi\otimes{{\rm{id}}}_{\mathbf{K}})(c)\cdot b(1\otimes e_{11}) \rangle_A =\phi(\langle a(1\otimes e_{11}) , \ c\cdot b(1\otimes e_{11}) \rangle_C ) \]

for any $a,\, b\in A^{s}$, $c\in C^{s}$ by Lemma 2.6. Indeed, for any $a,\, b\in A^{s}$, $c\in C^{s}$,

\begin{align*} \langle a(1\otimes e_{11}), \, (\phi\otimes{{\rm{id}}}_{\mathbf{K}})(c)\cdot b(1\otimes e_{11}) \rangle_A & = (1\otimes e_{11})a^{*} (\phi\otimes{{\rm{id}}}_{\mathbf{K}})(c)b(1\otimes e_{11}) \\ & =(\phi\otimes{{\rm{id}}}_{\mathbf{K}})((1\otimes e_{11})a^{*} cb(1\otimes e_{11})). \end{align*}

On the other hand,

\[ \phi(\langle a(1\otimes e_{11}) , \ c\cdot b(1\otimes e_{11}) \rangle_C ) =\phi((1\otimes e_{11})a^{*} cb(1\otimes e_{11})) . \]

Since we identify $C$ with $(1\otimes e_{11})C^{s} (1\otimes e_{11})$,

\[ \langle a(1\otimes e_{11}), \ (\phi\otimes{{\rm{id}}}_{\mathbf{K}})(c)\cdot b(1\otimes e_{11}) \rangle_A =\phi(\langle a(1\otimes e_{11}) , \, c\cdot b(1\otimes e_{11}) \rangle_C ) \]

for any $a,\, b\in A^{s}$, $c\in C^{s}$. Therefore, we obtain the conclusion.

Let $\psi \in {}_A \mathbf {B}_A (C,\, A)$. Let $\{u_{\lambda }\}_{\lambda \in \Lambda }$ be an approximate unit of $A^{s}$ with $||u_{\lambda }||\leq 1$ for any $\lambda \in \Lambda$. Since $\overline {AC}=C$, $\{u_{\lambda }\}_{\lambda \in \Lambda }$ is an approximate unit of $C^{s}$. Let $c$ be any element in $M(C)$. For any $a\in A$, $\{a\psi (cu_{\lambda })\}_{\lambda \in \Lambda }$ and $\{\psi (cu_{\lambda })a\}_{\lambda \in \Lambda }$ are Cauchy nets in $A$. Hence there is an element $x\in M(A)$ such that $\{\psi (cu_{\lambda })\}_{\lambda \in \Lambda }$ is strictly convergent to $x\in M(A)$. Let $\underline {\psi }$ be the map from $M(C)$ to $M(A)$ defined by $\underline {\psi }(c)=x$ for any $c\in C$. By routine computations $\underline {\psi }$ is a bounded $M(A)-M(A)$-bimodule linear map from $M(C)$ to $M(A)$ and $\psi =\underline {\psi }|_C$. We note that $\underline {\psi }$ is independent of the choice of approximate unit of $A^{s}$.

Let $q$ be a full projection in $M(A)$, that is, $\overline {AqA}=A$. Since $\overline {AC}=C$, $M(A)\subset M(C)$ by [Reference Pedersen9, Section 3.12.12]. Thus

\[ \overline{CqC}=\overline{CAqAC}=\overline{CAC}=C . \]

We regard $qC$ and $qA$ as a $qCq-C$-equivalence bimodule and a $qAq- A$-equivalence bimodule, respectively. Then $(qA,\, qC)\in {{\rm {Equi}}}(qAq,\, qCq,\, A,\, C)$.

Lemma 4.2 With the above notation, for any $\psi \in {}_A \mathbf {B}_A (C,\, A)$

\[ f_{[qA, qC]}(\psi)=\psi|_{qCq} . \]

Proof. By easy computations, we see that

\[ \langle qx , \ \psi|_{qCq}(c)\cdot qz \rangle_A =\psi(\langle qx , \ c\cdot qz \rangle_C ) \]

for any $x,\, z \in A$, $c\in C$ since $\underline {\psi }(q)=q$. Thus, we obtain the conclusion by Lemma 2.6.

Let $A\subset C$ and $B\subset D$ be inclusions of $C^{*}$-algebras such that $A$ and $B$ are $\sigma$-unital and $\overline {AC}=C$ and $\overline {BD}=D$. Let $B^{s} =B\otimes \mathbf {K}$ and $D^{s} =D\otimes \mathbf {K}$. We suppose that $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to $(X,\, Y)\in {{\rm {Equi}}} (A,\, C,\, B,\, D)$. Let $X^{s} =X\otimes \mathbf {K}$ and $Y^{s} =Y\otimes \mathbf {K}$, an $A^{s} -B^{s}$-equivalence bimodule and a $C^{s} -D^{s}$-equivalence bimodule, respectively. We note that $(X^{s} ,\, \, Y^{s} )\in {{\rm {Equi}}}(A^{s} ,\, C^{s} ,\, B^{s} ,\, D^{s} )$. Let $L_{X^{s}}$ and $L_{Y^{s}}$ be the linking $C^{*}$-algebras for $X^{s}$ and $Y^{s}$, respectively. Let

\[ p_1 =\begin{bmatrix} 1_{M(A^{s} )} & 0 \\ 0 & 0 \end{bmatrix} , \quad p_2 =\begin{bmatrix} 0 & 0 \\ 0 & 1_{M(B^{s} )} \end{bmatrix} . \]

Then $p_1$ and $p_2$ are full projections in $M(L_{X^{s}})$. By easy computations, we can see that $\overline {L_{X^{s}}L_{Y^{s}}}=L_{Y^{s}}$. Hence by [Reference Pedersen9, Section 3.12.12], $M(L_{X^{s}})\subset M(L_{Y^{s}})$. Since $p_1$ and $p_2$ are full projections in $M(L_X )$, by Brown [Reference Brown1, Lemma 2.5], there is a partial isometry $w\in M(L_{X^{s}} )$ such that $w^{*} w=p_1$, $ww^{*} =p_2$. We note that $w\in M(L_{Y^{s}} )$. Let $\varPsi$ be the map from $p_2 L_{Y^{s}}p_2$ to $p_1 L_{Y^{s}}p_1$ defined by

\[ \varPsi\left(\begin{bmatrix} 0 & 0 \\ 0 & d \end{bmatrix}\right)=w^{*} \begin{bmatrix} 0 & 0 \\ 0 & d \end{bmatrix}w \]

for any $d\in D^{s}$. In the same way as in the discussions of [Reference Brown, Green and Rieffel2], $\varPsi$ is an isomorphism of $p_2 L_{Y^{s}}p_2$ onto $p_1 L_{Y^{s}}p_1$ and $\varPsi |_{p_2 L_{X^{s}}p_2}$ is an isomorphism of $p_2 L_{X^{s}}p_2$ onto $p_1 L_{X^{s} }p_1$. Also, we note the following:

\begin{align*} & p_1 L_{Y^{s}}p_1 \cong C^{s}, \quad p_1 L_{X^{s}}p_1 \cong A^{s} \\ & p_2 L_{Y^{s}}p_2 \cong D^{s} , \quad p_2 L_{X^{s}}p_2 \cong B^{s} \end{align*}

as $C^{*}$-algebras. We identify $A^{s}$, $C^{s}$ and $B^{s}$, $D^{s}$ with $p_1 L_{X^{s}}p_1$, $p_1 L_{Y^{s}}p_1$ and $p_2 L_{X^{s}}p_2$, $p_2 L_{Y^{s}}p_2$, respectively. Also, we identify $X^{s}$, $Y^{s}$ with $p_1 L_{X^{s}}p_2$, $p_1 L_{Y^{s}}p_2$.

Let $A_{\varPsi }^{s}$ be the $A^{s} -B^{s}$-equivalence bimodule induced by $\varPsi |_{B^{s}}$, that is, $A_{\varPsi }^{s} =A^{s}$ as $\mathbf {C}$-vector spaces. The left $A^{s}$-action and the $A^{s}$-valued inner product on $A_{\varPsi }^{s}$ are defined in the usual way. The right $B^{s}$-action and the right $B^{s}$-valued inner product on $A_{\varPsi }^{s}$ are defined as follows: For any $x,\, y\in A_{\varPsi }^{s}$, $b\in B^{s}$,

\[ x\cdot b= x\varPsi(b), \quad \langle x, y \rangle_{B^{s}}=\varPsi^{{-}1}(x^{*} y) . \]

Similarly, we define the $C^{s} -D^{s}$-equivalence bimodule $C_{\varPsi }^{s}$ induced by $\varPsi$. We note that $A_{\varPsi }^{s}$ is a closed subspace of $C_{\varPsi }^{s}$ and $(A_{\varPsi }^{s} ,\, \, C_{\varPsi }^{s} ) \in {{\rm {Equi}}}(A^{s} ,\, C^{s} ,\, B^{s} ,\, D^{s} )$.

Proof. We can prove this lemma in the same way as in the proof of [Reference Brown, Green and Rieffel2, Lemma 3.3]. Indeed, let $\pi$ be the map from $Y^{s}$ to $C_{\varPsi }^{s}$ defined by

\[ \pi(y)=\begin{bmatrix} 0 & y \\0 & 0 \end{bmatrix}w \]

for any $y\in Y^{s}$. By routine computations, $\pi$ is a $C^{s}- D^{s}$-equivalence bimodule isomorphism of $Y^{s}$ onto $C_{\varPsi }^{s}$ and $\pi |_{X^{s}}$ is a bijection from $X^{s}$ onto $A^{s}$. Hence by [Reference Kodaka6, Lemma 3.2], we obtain the conclusion.

Lemma 4.4 With the above notation, for any $\phi \in {}_{B^{s}} \mathbf {B}_{B^{s}} (D^{s} ,\, B^{s} ),$

\[ f_{[X^{s} , Y^{s} ]}(\phi)=\varPsi\circ\phi\circ\varPsi^{{-}1} \in {}_{A^{s}}\mathbf{B}_{A^{s}}(C^{s} , A^{s} ). \]

Proof. We claim that

\[ \langle x , (\varPsi\circ\phi\circ\varPsi^{{-}1})(d)\cdot z \rangle_{B^{s}}=\phi(\langle x, d\cdot z \rangle_{D^{s}}) \]

for any $\phi \in {}_{B^{s}} \mathbf {B}_{B^{s}}(D^{s} ,\, \, B^{s} )$, $x,\, z \in A_{\varPsi }^{s}$, $d\in D^{s}$. Indeed,

\begin{align*} \langle x , (\varPsi\circ\phi\circ\varPsi^{{-}1})(d)\cdot z \rangle_{B^{s}} & =\varPsi^{{-}1}(x^{*}(\varPsi\circ\phi\circ\varPsi^{{-}1})(d)z ) \\ & =\varPsi^{{-}1}(x^{*} )(\phi\circ\varPsi^{{-}1})(d)\varPsi^{{-}1}(z) . \end{align*}

On the other hand,

\begin{align*} \phi(\langle x, d\cdot z \rangle_{D^{s}}) & =\phi(\varPsi^{{-}1}(x^{*} dz))=\phi(\varPsi^{{-}1}(x^{*} )\varPsi^{{-}1}(d)\varPsi^{{-}1}(z)) \\ & =\varPsi^{{-}1}(x^{*} )(\phi\circ\varPsi^{{-}1})(d)\varPsi^{{-}1}(z) \end{align*}

since $\varPsi ^{-1}(x^{*} )$, $\varPsi ^{-1}(z)\in B^{s}$. Thus

\[ \langle x , (\varPsi\circ\phi\circ\varPsi^{{-}1})(d)\cdot z \rangle_{B^{s}} =\phi(\langle x, d\cdot z \rangle_{D^{s}}) \]

for any $\phi \in {}_{B^{s}} \mathbf {B}_{B^{s}}(D^{s} ,\, B^{s} )$, $x,\, z \in A_{\varPsi }^{s}$, $d\in D^{s}$. Hence by Lemma 2.6, $f_{[A_{\varPsi }^{s} ,\, C_{\varPsi }^{s} ]}(\phi )=\varPsi \circ \phi \circ \varPsi ^{-1}$ for any $\phi \in {}_{B^{s}} \mathbf {B}_{B^{s}}(D^{s} ,\,B^{s} )$. Therefore, $f_{[X^{s} , Y^{s} ]}(\phi )=\varPsi \circ \phi \circ \varPsi ^{-1}$ by Lemmas 2.7 and 4.3.

Let $\underline {\varPsi }$ be the strictly continuous isomorphism of $M(D^{s} )$ onto $M(C^{s} )$ extending $\varPsi$ to $M(D^{s} )$, which is defined in Jensen and Thomsen [Reference Jensen and Thomsen5, Corollary 1.1.15]. Then $\underline {\varPsi }|_{M(B^{s} )}$ is an isomorphism of $M(B^{s} )$ onto $M(A^{s} )$. Let $q=\underline {\varPsi }(1\otimes e_{11})$. Then $q$ is a full projection in $M(A^{s} )$ with $\overline {C^{s} q C^{s} }=C^{s}$ and $qA^{s} q \cong A$, $qC^{s} q \cong C$ as $C^{*}$-algebras. We identify with $qA^{s} q$ and $qC^{s} q$ with $A$ and $C$, respectively. Then we obtain the following proposition:

Proposition 4.5 Let $A\subset C$ and $B\subset D$ be inclusions of $C^{*}$-algebras such that $A$ and $B$ are $\sigma$-unital and $\overline {AC}=C$ and $\overline {BD}=D$. Let $\varPsi$ be the isomorphism of $D^{s}$ onto $C^{s}$ defined before Lemma 4.3 and let $q=\varPsi (1\otimes e_{11})$. Let $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$. For any $\phi \in {}_B \mathbf {B}_B (D,\, B),$

\[ f_{[X, Y]}(\phi)=(\varPsi\circ (\phi\otimes{{\rm{id}}}_{\mathbf{K}})\circ\varPsi^{{-}1})|_{qC^{s} q} , \]

where we identify $qA^{s} q$ and $qC^{s} q$ with $A$ and $C,$ respectively.

Proof. We note that $(1\otimes e_{11})B^{s} (1\otimes e_{11})$ and $(1\otimes e_{11})D^{s} (1\otimes e_{11})$ are identified with $B$ and $D$, respectively. Also, we identify $qA^{s} q$ and $qC^{s} q$ with $A$ and $C$, respectively. Thus, we see that

\[ [qA^{s} \otimes_{A^{s}}X^{s} \otimes_{B^{s}}B^{s} (1\otimes e_{11}) , \ qC^{s} \otimes_{C^{s}}Y^{s} \otimes_{D^{s}}D^{s} (1\otimes e_{11})]=[X, Y] \]

in ${{\rm {Equi}}}(A,\, C,\, B,\, D)/\!\sim$. Hence by Lemma 2.8,

\[ f_{[X, Y]}(\phi)=(f_{[qA^{s} , \, qC^{s} ]}\circ f_{[X^{s} , \, Y^{s}]}\circ f_{[B^{s} (1\otimes e_{11}) , \, D^{s} (1\otimes e_{11})]})(\phi) . \]

Therefore, by Lemmas 4.1, 4.2 and 4.4,

\[ f_{[X, Y]}(\phi)=(\varPsi\circ (\phi\otimes{{\rm{id}}}_{\mathbf{K}})\circ\varPsi^{{-}1})|_{qC^{s} q} . \]

Next, we consider the case that $C^{*}$-algebras are unital $C^{*}$-algebras. Let $A\subset C$ and $B\subset D$ be unital inclusions of unital $C^{*}$-algebras, which are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$. By [Reference Kodaka and Teruya8, Section 2], there are a positive integer $n$ and a full projection $p\in M_n (B)$ such that

\[ A\cong pM_n (B)p , \quad C\cong pM_n (D)p \]

as $C^{*}$-algebras and such that

\[ X\cong pM_n (B)(1\otimes e), \quad Y\cong pM_n (D)(1\otimes e ) \]

as $A-B$-equivalence bimodules and $C-D$-equivalence bimodules, respectively, where

\[ e=\begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix}\in M_n (\mathbf{C}) \]

and we identify $A$, $C$ and $B$, $D$ with $pM_n (B)p$, $pM_n (D)p$ and $(1\otimes e)M_n (B)(1\otimes e)$, $(1\otimes e)M_n (D)(1\otimes e)$, respectively. We denote the above isomorphisms by

\begin{align*} \varPsi_A & : A\longrightarrow pM_n (B)p , \\ \varPsi_C & : C\longrightarrow pM_n (D)p , \\ \varPsi_X & : X\longrightarrow pM_n (B)(1\otimes e) , \\ \varPsi_Y & : Y\longrightarrow pM_n (D)(1\otimes e), \end{align*}

respectively. Then we have the same results as in [Reference Kodaka and Teruya8, Lemma 2.6 and Corollary 2.7]. First, we construct a map from ${}_B \mathbf {B}_B (D,\, B)$ to the space of all $pM_n (B)p-pM_n (B)$-bimodule maps, ${}_{pM_n (B)p}\mathbf {B}_{pM_n (B)p}(pM_n(D)p ,\, \, pM_n (B)p)$. Let $\phi \in {}_B \mathbf {B}_B (D,\, B)$. Let $\psi$ be the map from $M_n (D)$ to $M_n (B)$ defined by

\[ \psi(x)=(\phi\otimes{{\rm{id}}}_{M_n (\mathbf{C})})(x) \]

for any $x\in M_n (D)$. Since $\psi (p)=p$, by easy computations, $\psi$ can be regarded as an element in ${}_{pM_n (B)p}\mathbf {B}_{pM_n (B)p}(pM_n(D)p ,\, \, pM_n (B)p)$. We denote by $F$ the map

\[ \phi\in{}_B \mathbf{B}_B (D, B)\mapsto \psi\in{}_{pM_n (B)p}\mathbf{B}_{pM_n (B)p}(pM_n (D)p, \, pM_n (B)p) \]

as above.

Lemma 4.7 With the above notation, let $\phi \in {}_B \mathbf {B}_B (D,\, B)$. Then for any $d\in M_n (D),$ $x,\, y\in M_n (B),$

\[ \langle px(1\otimes e) , \ F(\phi)(pdp)\cdot pz(1\otimes e) \rangle_B =\phi(\langle px(1\otimes e) , \ pdp\cdot pz(1\otimes e) \rangle_D ). \]

Proof. This can be proved by routine computations. Indeed, for any $d\in M_n (D)$, $x,\, y\in M_n (B)$,

\[ \langle px(1\otimes e) , \ F(\phi)(pdp)\cdot pz(1\otimes e) \rangle_B =(1\otimes e)x^{*} p(\phi\otimes{{\rm{id}}})(d)pz(1\otimes e) \]

On the other hand,

\[ \phi(\langle px(1\otimes e) , \ pdp\cdot pz(1\otimes e)\rangle_D ) =\phi((1\otimes e)x^{*} pdpz(1\otimes e)) . \]

Since we identify $D$ with $(1\otimes e)M_n (D)(1\otimes e)$,

\[ (1\otimes e)x^{*} p(\phi\otimes{{\rm{id}}})(d)pz(1\otimes e)=\phi((1\otimes e)x^{*} pdpz(1\otimes e)) . \]

Thus, we obtain the conclusion.

Lemma 4.8 With the above notation, for any $\phi \in {}_B \mathbf {B}_B (D,\, B),$

\[ f_{[X, Y]}(\phi)=\varPsi_A^{{-}1}\circ F(\phi)\circ\varPsi_C . \]

Proof. By Lemma 2.6, it suffices to show that

\[ \langle x , (\varPsi_A^{{-}1}\circ F(\phi)\circ\varPsi_C )(c)\cdot z \rangle_B =\phi(\langle x , c\cdot z \rangle_D) \]

for any $c\in C$, $x,\, z\in X$. Indeed, for any $c\in C$, $x,\, z\in X$,

\begin{align*} \langle x , (\varPsi_A^{{-}1}\circ F(\phi)\circ\varPsi_C )(c)\cdot z \rangle_B & =\langle \varPsi_X (x) , \varPsi_X ((\varPsi_A^{{-}1}\circ F(\phi)\circ\varPsi_C )(c)\cdot z) \rangle_B \\ & \quad(\text{by the same result as [8, Lemma 2.6 (3)]}) \\ & =\langle \varPsi_X (x) , \ (F(\phi)\circ\varPsi_C )(c)\cdot \varPsi_X (z) \rangle_B \\ & \quad(\text{by the same result as [8, Lemma 2.6 (2)]}) \\ & =\phi(\langle \varPsi_X (x) , \, \varPsi_C (c)\cdot \varPsi_X (z) \rangle_D ) \quad (\text{by Lemma 4.7}) \\ & =\phi(\langle \varPsi_X (x) , \, \varPsi_X (c\cdot z)\rangle_D \, ) \\ & \quad(\text{by the same results as [8, Corollary 2.7 (2), (5)]}) \\ & =\phi(\langle x , \, c\cdot z \rangle_D \, ) \\ & \quad(\text{by the same result as [8, Corollary 2.7 (3)]}) . \end{align*}

Therefore, we obtain the conclusion.

5. Basic properties

Let $A\subset C$ and $B\subset D$ be inclusions of $C^{*}$-algebras with $\overline {AC}=C$ and $\overline {BD}=D$. We suppose that they are strongly Morita equivalent with respect to $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$. Let ${}_A \mathbf {B}_A (C,\, A)$ and ${}_B \mathbf {B}_B (D,\, B)$ be as above and let $f_{[X, Y]}$ be the isometric isomorphism of ${}_B \mathbf {B}_B (D,\, B)$ onto ${}_A \mathbf {B}_A (C,\, A)$ induced by $(X,\, Y)\in {{\rm {Equi}}}(A,\, C,\, B,\, D)$ which is defined in § 2. We note that for any $\phi \in {}_B \mathbf {B}_B (D,\, B)$, we can regard $f_{[X, Y]}(\phi )$ as a linear map from $C$ to $A$ defined by

\[ \langle x , f_{[X, Y]}(\phi)(c)\cdot z \rangle_B =\phi(\langle x , c\cdot z \rangle_D ) \]

for any $c\in C$, $x,\, z\in X$ by Lemma 2.6. In this section, we give basic properties about $f_{[X, Y]}$.

Proof. (1) Let $\phi$ be any selfadjoint linear map in ${}_B \mathbf {B}_B (D,\, B)$ and let $c\in C$, $x,\, z\in X$. By lemma 2.6,

\begin{align*} \langle x , \, f_{[X, Y]}(\phi)(c^{*} )\cdot z \rangle_B & =\phi(\langle x , \ c^{*} \cdot z \rangle_D ) =\phi(\langle c\cdot x , \, z \rangle_D ) \\ & =\phi(\langle z, \, c\cdot x \rangle_D)^{*} =\langle z , \, f_{[X, Y]}(\phi)(c)\cdot x \rangle_B^{*} \\ & =\langle f_{[X, Y]}(\phi)(c)\cdot x , \, z \rangle_B = \langle x , \, f_{[X, Y]}(\phi)(c)^{*} \cdot z \rangle_B . \end{align*}

Hence $f_{[X, Y]}(\phi )(c^{*} )=f_{[X, Y]}(\phi )(c )^{*}$ for any $c\in C$.

(2) Let $\phi$ be any positive linear map in ${}_B \mathbf {B}_B (D,\, B)$ and let $c$ be any positive element in $C$. Then $\langle x ,\, c\cdot x \rangle _D \geq 0$ for any $x\in X$ by Raeburn and Williams [Reference Raeburn and Williams10, Lemma 2.28]. Hence $\phi (\langle x ,\, c\cdot x \rangle _D )\geq 0$ for any $x\in X$. That is, $\langle x,\, \, f_{[X, Y]}(\phi )(c)\cdot x \rangle _B \geq 0$ for any $x\in X$. Thus, $f_{[X, Y]}(\phi )(c)\geq 0$ by [Reference Raeburn and Williams10, Lemma 2.28]. Therefore, we obtain the conclusion.

Proof. Since $\phi (b)=b$ for any $b\in B$, for any $a\in A$, $x,\, z\in X$,

\[ \langle x, f_{[X, Y]}(\phi)(a)\cdot z \rangle_B =\phi(\langle x , \, a\cdot z \rangle_B ) =\langle x , \, a\cdot z \rangle_B \]

by Lemma 2.6. Thus, $f_{[X, Y]}(\phi )(a)=a$ for any $a\in A$. By Proposition 2.4 and Lemma 5.1, we obtain the conclusion.

Since $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to $(X,\, Y)\in {{\rm {Equi}}} (A,\, C,\, B,\, D)$, $A^{s} \subset C^{s}$ and $B^{s} \subset D^{s}$ are strongly Morita equivalent with respect to $(X^{s} ,\, Y^{s} )\in {{\rm {Equi}}} (A^{s} ,\, C^{s} ,\, B^{s} ,\, D^{s} )$. Let $\phi$ be any element in ${}_B \mathbf {B}_B (D,\, B)$. Then

\[ \phi\otimes{{\rm{id}}}_{\mathbf{K}} \in{}_{B^{s}} \mathbf{B}_{B^{s}} (D^{s} , B^{s} ) . \]

Lemma 5.3 With the above notation, for any $\phi \in {}_B \mathbf {B}_B (D,\, B)$

\[ f_{[X^{s} , Y^{s} ]}(\phi\otimes{{\rm{id}}}_{\mathbf{K}})=f_{[X, Y]}(\phi)\otimes{{\rm{id}}}_{\mathbf{K}} . \]

Proof. This can be proved by routine computations. Indeed, for any $c\in C$, $x,\, z\in X$, $k_1,\, k_2 ,\, k_3 \in \mathbf {K}$,

\begin{align*} & \langle x\otimes k_1 , \ f_{[X^{s} , Y^{s}]}(\phi\otimes{{\rm{id}}})(c\otimes k_2 )\cdot z\otimes k_3 \rangle_{B^{s}} \\ & \quad= (\phi\otimes{{\rm{id}}})(\langle x\otimes k_1, \ c\otimes k_2 \cdot z\otimes k_3 \rangle_{B^{s}} ) \\ & \quad = (\phi\otimes{{\rm{id}}})(\langle x\otimes k_1, c\cdot z \otimes k_2 k_3 \rangle_{D^{s}} ) \\ & \quad = (\phi\otimes{{\rm{id}}})(\langle x , \, c\cdot z \rangle_D \otimes k_1^{*} k_2 k_3 ) \\ & \quad =\langle x , \, f_{[X, Y]}(\phi)(c)\cdot z \rangle_B \otimes k_1^{*} k_2 k_3 \\ & \quad=\langle x\otimes k_1 , \, f_{[X, Y]}(\phi)(c)\otimes k_2 \cdot z\otimes k_3 \rangle_{B^{s}} \end{align*}

by Lemma 2.6. Therefore, we obtain the conclusion by Lemma 2.6.

Corollary 5.4 With the above notation, let $n\in \mathbf {N}$. Then for any $\phi \in {}_B \mathbf {B}_B (D,\, B),$

\[ f_{[X\otimes M_n (\mathbf{C}) , \, Y\otimes M_n (\mathbf{C})]}(\phi\otimes{{\rm{id}}})=f_{[X, Y]}(\phi)\otimes{{\rm{id}}}_{M_n (\mathbf{C})} . \]

Again, we consider the case that $C^{*}$-algebras are unital $C^{*}$-algebras. Let $A\subset C$ and $B\subset D$ be unital inclusions of unital $C^{*}$-algebras, which are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$. As mentioned in § 4, by [Reference Kodaka and Teruya8, Section 2], there are a positive integer $n$ and a full projection $p\in M_n (B)$ such that

\[ A\cong pM_n (B)p, \quad C\cong pM_n (D)p \]

as $C^{*}$-algebras, respectively. We denote the above isomorphisms by

\[ \varPsi_A : A\to pM_n (B)p , \quad \varPsi_C : C\to pM_n (D)p , \]

respectively. We note that $\varPsi _A =\varPsi _C |_A$.

Proposition 5.6 Let $\phi$ be a conditional expectation from $D$ onto $B$. We suppose that there is a positive number $t$ such that

\[ \phi(d)\geq td \]

for any positive element $d\in D$. Then there is a positive number $s$ such that

\[ f_{[X, Y]}(\phi)(c)\geq sc \]

for any positive element $c\in C$.

Proof. We recall the discussions after Proposition 4.5. That is, by Lemma 4.8, $f_{[X, Y]}(\phi )=\varPsi _A^{-1}\circ F(\phi )\circ \varPsi _C$ for any $\phi \in {}_B \mathbf {B}_B(D,\, B)$, where $F$ is the isometric isomorphism of ${}_B \mathbf {B}_B (D,\, B)$ onto ${}_{pM_n (B)p}\mathbf {B}_{pM_n (B)p}(pM_n (D)p,\, \, pM_n (B)p)$ defined before Remark 4.6. Also, $\varPsi _A$ and $\varPsi _C$ are isomorphisms of $A$ and $C$ onto $pM_n (B)p$ and $pM_n (D)p$ defined after Proposition 4.5, respectively. We note that $\varPsi _C |_A =\varPsi _A$. Since there is a positive number $t$ such that $\phi (d)\geq td$ for any positive element $d\in D$, by Frank and Kirchberg [Reference Frank and Kirchberg3, Theorem 1], there is a positive number $s$ such that $(\phi \otimes {{\rm {id}}}_{M_n (\mathbf {C})})(d)\geq sd$ for any positive element $d\in M_n (D)$. Thus, for any $c\in C$,

\begin{align*} f_{[X, Y]}(\phi)(c^{*} c) & =(\varPsi_A^{{-}1}\circ F(\phi)\circ\varPsi_C )(c^{*} c) \\ & =\varPsi_A^{{-}1}((\phi\otimes{{\rm{id}}}_{M_n (\mathbf{C})})(\varPsi_C (c)^{*} \varPsi_C (c)) \\ & \geq \varPsi_A^{{-}1}(s\varPsi_C (c)^{*} \varPsi_C (c)) \\ & = sc^{*} c \end{align*}

since $F(\phi )=\phi \otimes {{\rm {id}}}_{M_n(\mathbf {C})}$ and $\varPsi _C |_A =\varPsi _A$. Therefore, we obtain the conclusion.

Following Watatani [Reference Watatani13, Definition 1.11.1], we give the following definition.

Definition 5.1 Let $\phi \in {}_B \mathbf {B}_B (D,\, B)$. Then a finite set $\{(u_i ,\, v_i )\}_{i=1}^{m} \subset D\times D$ is called a quasi-basis for $\phi$ if it satisfies that

\[ d=\sum_{i=1}^{m} u_i \phi(v_i d)=\sum_{i=1}^{m} \phi(du_i )v_i \]

for any $d\in D$.

Lemma 5.7 With the above notation, let $\phi \in {}_B \mathbf {B}_B (D,\, B)$ with a quasi-basis $\{(u_i ,\, v_i )\}_{i=1}^{m}$. Then the finite set of $D\times D$

\[ \{(p(u_i \otimes I_n )a_j p , \, p b_j (v_i \otimes I_n )p )\}_{i=1,2,\ldots, m, j=1,2,\ldots, K} \]

is a quasi-basis for $F(\phi ),$ where $a_1 ,\, a_2,\, \ldots,\, a_K,\, b_1,\, b_2,\, \ldots,\, b_K$ are elements in $M_n (B)$ with

\[ \sum_{j=1}^{K} a_j p b_j =1_{M_n(B)} . \]

Proof. This lemma can be proved in the same way as in [Reference Kodaka and Teruya8, Section 2]. Indeed, $F(\phi )=(\phi \otimes {{\rm {id}}}_{M_n (\mathbf {C})})|_{pM_n (D)p}$ and since $p$ is a full projection in $M_n (B)$, there are elements $a_1,\, a_2,\, \ldots,\, a_K,\, b_1,\, b_2,\, \ldots,\, b_K$ in $M_n (B)$ such that $\sum \nolimits _{j=1}^{K} a_j pb_j =1_{M_n (B)}$. Then the finite set

\[ \{(p(u_i \otimes I_n )a_j p , \, p b_j (v_i \otimes I_n )p )\}_{i=1,2,\ldots, m, \, j=1,2,\ldots, K} \]

of $D\times D$ is a quasi-basis for $F(\phi )$ by easy computations.

Remark 5.10 Let $E^{B}$ be as in Corollary 5.9. Then $E^{B}$ satisfies the assumption of Proposition 5.6, that is, there is a positive number $t$ such that

\[ E^{B} (d)\geq td \]

for any positive element $d\in D$ by [Reference Watatani13, Proposition 2.1.5].

6. The Picard groups

First, we recall the definition of the Picard group of an inclusion of $C^{*}$-algebras. Let $A\subset C$ be an inclusion of $C^{*}$-algebras with $\overline {AC}=C$. Let ${{\rm {Equi}}} (A,\, C)$ be the set of all pairs $(X,\, Y)$ such that $Y$ is a $C-C$-equivalence bimodule and $X$ is its closed subspace satisfying Conditions (1), (2) in Definition 2.1, that is, let ${{\rm {Equi}}} (A,\,C)={{\rm {Equi}}}(A,\, C,\, A,\, C)$. We define an equivalence relation $`` \sim "$ as follows: For $(X,\, Y)$, $(Z,\, W)\in {{\rm {Equi}}}(A,\, C)$, $(X,\, Y)\sim (Z,\, W)$ in ${{\rm {Equi}}} (A,\, C)$ if and only if there is a $C-C$-equivalence bimodule isomorphism $\Phi$ of $Y$ onto $W$ such that the restriction of $\Phi$ to $X$, $\Phi |_X$ is an $A-A$-equivalence bimodule isomorphism of $X$ onto $Z$. We denote by $[X,\, Y]$, the equivalence class of $(X,\, Y)$ in ${{\rm {Equi}}}(A,\, C)$. Let ${{\rm {Pic}}}(A,\, C)={{\rm {Equi}}} (A,\, C)/\!\sim$. We define the product in ${{\rm {Pic}}}(A,\, C)$ as follows: For $(X,\, Y)$, $(Z,\, W)\in {{\rm {Equi}}}(A,\, C)$

\[ [X, Y][Z, W]=[X\otimes_A Z , \, Y\otimes_C W], \]

where the $A-A$-equivalence bimodule $X\otimes _A Z$ is identified with the closed subspace of $Y\otimes _C W$ spanned by the set

\[ \{ x\otimes z\in Y\otimes_D W \, | \, x\in X , \, z\in Z \} . \]

We note that $Y\otimes _C W$ and its closed subspace $X\otimes _A Z$ satisfy Conditions (1), (2) in Definition 2.1. By easy computations, we can see that ${{\rm {Pic}}}(A,\, C)$ is a group. Indeed, we regard $(A,\, C)$ as an element in ${{\rm {Equi}}}(A,\, C)$ in the evident way. Then $[A,\, C]$ is the unit element in ${{\rm {Pic}}}(A,\, C)$. For any element $(X,\, Y)\in {{\rm {Equi}}}(A,\, C)$, $(\widetilde {X},\, \widetilde {Y})\in {{\rm {Equi}}}(A,\, C)$ and $[\widetilde {X},\, \widetilde {Y}]$ is the inverse element of $[X,\, Y]$ in ${{\rm {Pic}}}(A,\, C)$, where $\widetilde {X}$ and $\widetilde {Y}$ are the dual $A-A$-equivalence bimodule of $X$ and the dual $C-C$-equivalence bimodule of $Y$, respectively. We note that $\widetilde {X}$ can be a closed subspace of $\widetilde {Y}$. We call the group ${{\rm {Pic}}}(A,\, C)$ defined in the above, the Picard group of the inclusion of $C^{*}$-algebras $A\subset C$ with $\overline {AC}=C$. We refer to [Reference Kodaka6] for more details about the Picard group of an inclusion of $C^{*}$-algebras

Let $A\subset C$ be an inclusion of $C^{*}$-algebras with $\overline {AC}=C$ and ${{\rm {Pic}}}(A,\, C)$ the Picard group of the inclusion $A\subset C$. Let ${}_A \mathbf {B}_A ( C,\, A)$ be the space of all bounded $A-A$-bimodule linear maps from $C$ to $A$.

Definition 6.1 Let $\phi \in {}_A \mathbf {B}_A (C,\, A)$. We define ${{\rm {Pic}}}(\phi )$ by

\[ {{\rm{Pic}}} (\phi)=\{[X, Y]\in{{\rm{Pic}}}(A, C) \, | \, f_{[X, Y]}(\phi)=\phi \} . \]

We call ${{\rm {Pic}}}(\phi )$ the Picard group of $\phi$.

Let $B\subset D$ be an inclusion of $C^{*}$-algebras with $\overline {BD}=D$. Let $\phi \in {}_B \mathbf {B}_B (D,\, B)$ and $\psi \in {}_A \mathbf {B}_A (C,\, A)$.

Proof. Let $g$ be the map from ${{\rm {Pic}}}(\phi )$ to ${{\rm {Pic}}}(A,\, C)$ defined by

\[ g([X, Y])=[Z\otimes_B X \otimes_B \widetilde{Z}, \, W\otimes_D Y\otimes_D \widetilde{W} ] \]

for any $[X,\, Y]\in {{\rm {Pic}}} (\phi )$. Then since $f_{[Z, W]}(\phi )=\psi$, by Lemma 2.8

\begin{align*} f_{[Z\otimes_B X \otimes_B \widetilde{Z}, \, W\otimes_D Y\otimes_D \widetilde{W} ]}(\psi) & =(f_{[Z, W]}\circ f_{[X, Y]} \circ f_{[\widetilde{Z}, \widetilde{W}]})(\psi) \\ & =(f_{[Z, W]}\circ f_{[X, Y]} \circ f_{[Z, W]}^{{-}1})(\psi)=\psi . \end{align*}

Hence $[Z\otimes _B X \otimes _B \widetilde {Z},\, \, W\otimes _D Y\otimes _D \widetilde {W} ]\in {{\rm {Pic}}}(\psi )$ and by easy computations, we can see that $g$ is an isomorphism of ${{\rm {Pic}}}(\phi )$ onto ${{\rm {Pic}}}(\psi )$.

Let $\phi \in {}_A \mathbf {B}_A (C,\, A)$. Let $\alpha$ be an automorphism of $C$ such that the restriction of $\alpha$ to $A$, $\alpha |_A$, is an automorphism of $A$. Let ${{\rm {Aut}}} (A,\, C)$ be the group of all such automorphisms and let

\[ {{\rm{Aut}}} (A, C, \phi)=\{\alpha\in{{\rm{Aut}}}(A, C) \, | \, \alpha\circ\phi\circ\alpha^{{-}1}=\phi \} . \]

Then ${{\rm {Aut}}} (A,\, C,\, \phi )$ is a subgroup of ${{\rm {Aut}}} (A,\, C)$. Let $\pi$ be the homomorphism of ${{\rm {Aut}}}(A,\, C)$ to ${{\rm {Pic}}}(A,\, C)$ defined by

\[ \pi(\alpha)=[X_{\alpha}, Y_{\alpha}] \]

for any $\alpha \in {{\rm {Aut}}} (A,\, C)$, where $(X_{\alpha },\, Y_{\alpha })$ is an element in ${{\rm {Equi}}} (A,\, C)$ induced by $\alpha$, which is defined in [Reference Kodaka6, Section 3], where ${{\rm {Equi}}} (A,\, C)={{\rm {Equi}}}(A,\, C,\, A,\, C)$. Let $u$ be a unitary element in $M(A)$. Then $u\in M(C)$ and ${{\rm {Ad}}}(u)\in {{\rm {Aut}}}(A,\, C)$ since $\overline {AC}=C$. Let ${{\rm {Int}}} (A,\, C)$ be the group of all such automorphisms in ${{\rm {Aut}}} (A,\, C)$. We note that ${{\rm {Int}}} (A,\, C)={{\rm {Int}}} (A)$, the subgroup of ${{\rm {Aut}}}(A)$ of all generalized inner automorphisms of $A$. Let $\imath$ be the inclusion map of ${{\rm {Int}}} (A,\, C)$ to ${{\rm {Aut}}} (A,\, C)$.

Proof. (1) Let $\alpha \in {{\rm {Aut}}}(A,\, C)$. Then for any $c\in C$, $x,\, z\in X_{\alpha }$,

\begin{align*} \langle x , \, (\alpha\circ\phi\circ\alpha^{{-}1})(c)\cdot z \rangle_A & =\langle x , \, (\alpha\circ\phi\circ\alpha^{{-}1})(c)z \rangle_A \\ & =\alpha^{{-}1}(x^{*} (\alpha\circ\phi\circ\alpha^{{-}1})(c)z) \\ & =\alpha^{{-}1}(x^{*} )(\phi\circ\alpha^{{-}1})(c)\alpha^{{-}1}(z) . \end{align*}

On the other hand,

\begin{align*} \phi(\langle x , \, c\cdot z \rangle_C ) & =\phi(\alpha^{{-}1}(x^{*} cz))=\phi(\alpha^{{-}1}(x^{*} )\alpha^{{-}1}(c)\alpha^{{-}1}(z)) \\ & =\alpha^{{-}1}(x^{*} )(\phi\circ\alpha^{{-}1})(c)\alpha^{{-}1}(z) . \end{align*}

Thus, by Lemma 2.6, $f_{[X_{\alpha },\, Y_{\alpha }]}(\phi )=\alpha \circ \phi \circ \alpha ^{-1}$.

(2) Let $\alpha$ be any element in ${{\rm {Aut}}}(A,\, C,\, \phi )$. Then by (1), $f_{[X_{\alpha },\, Y_{\alpha }]}(\phi )=\alpha \circ \phi \circ \alpha ^{-1}=\phi$. Hence $[X_{\alpha },\, Y_{\alpha }]\in {{\rm {Pic}}}(\phi )$.

(3) Let ${{\rm {Ad}}}(u)\in {{\rm {Int}}} (A,\, C)$. Then $u\in M(A)\subset M(C)$. For any $c\in C$,

\[ ({{\rm{Ad}}}(u)\circ\phi\circ{{\rm{Ad}}}(u^{*}))(c) =u\phi(u^{*} cu)u^{*}=uu^{*} \phi(c)uu^{*} =\phi(c) \]

since $\underline {\phi }(u)=u$. Thus, ${{\rm {Int}}}(A,\, C)\subset {{\rm {Aut}}}(A,\, C,\, \phi )$. It is clear by [Reference Kodaka6, Lemma 3.4] that the sequence

\[ 1\longrightarrow {{\rm{Int}}}(A, C)\overset{\imath}{\longrightarrow}{{\rm{Aut}}}(A, C, \phi)\overset{\pi}{\longrightarrow}{{\rm{Pic}}}(\phi) \]

is exact.

Proposition 6.3 Let $A\subset C$ be an inclusion of $C^{*}$-algebras with $\overline {AC}=C$ and we suppose that $A$ is $\sigma$-unital. Let $\phi \in {}_{A^{s}} \mathbf {B}_{A^{s}}(C^{s},\, A^{s} )$. Then the sequence

\[ 1\longrightarrow {{\rm{Int}}}(A^{s} , C^{s} )\overset{\imath}{\longrightarrow}{{\rm{Aut}}}(A^{s} , C^{s} , \phi) \overset{\pi}{\longrightarrow}{{\rm{Pic}}}(\phi)\longrightarrow 1 \]

is exact.

Proof. It suffices to show that $\pi$ is surjective by Lemma 6.2 (3). Let $[X,\, Y]$ be any element in ${{\rm {Pic}}}(\phi )$. Then by [Reference Kodaka6, Proposition 3.5], there is an element $\alpha \in {{\rm {Aut}}} (A^{s} ,\, C^{s} )$ such that

\[ \pi(\alpha)=[X, Y] \]

in ${{\rm {Pic}}}(A,\, C)$. Since $[X,\, Y]\in {{\rm {Pic}}}(\phi )$, $f_{[X, Y]}(\phi )=\phi$. Also, by Lemma 2.7, $f_{[X, Y]}=f_{[X_{\alpha },\, Y_{\alpha }]}$, where $[X_{\alpha },\, Y_{\alpha }]$ is the element in ${{\rm {Pic}}}(A,\, C)$ induced by $\alpha$. Hence

\[ f_{[X_{\alpha}, Y_{\alpha}]}(\phi)=f_{[X, Y]}(\phi)=\phi . \]

Since $f_{[X_{\alpha },\, Y_{\alpha }]}(\phi )=\alpha \circ \phi \circ \alpha ^{-1}$ by Lemma 6.2(1), $\phi =\alpha \circ \phi \circ \alpha ^{-1}$. Hence $\alpha \in {{\rm {Aut}}} (A^{s} ,\, C^{s} ,\, \phi )$.

7. The $C^{*}$-basic construction

The proofs of the results in this section are simple analogues of the arguments in [Reference Kodaka6, Reference Kodaka and Teruya8]. Also, the obtained results will not be used later except in a few of the examples in § 9. Thus, in this section, we essentially only present outlines of proofs but give precise references to the necessary arguments which would have to be applied.

Let $A\subset C$ be a unital inclusion of unital $C^{*}$-algebras and let $E^{A}$ be a conditional expectation of Watatani index-finite type from $C$ onto $A$. Let $e_A$ be the Jones’ projection for $E^{A}$ and $C_1$ the $C^{*}$-basic construction for $E^{A}$. Let $E^{C}$ be its dual conditional expectation from $C_1$ onto $C$. Let $e_C$ be the Jones’ projection for $E^{C}$ and $C_2$ the $C^{*}$-basic construction for $E^{C}$ Let $E^{C_1}$ be the dual conditional expectation of $E^{C}$ from $C_2$ onto $C_1$. Since $E^{A}$ and $E^{C}$ are of Watatani index-finite type, $C$ and $C_1$ can be regarded as a $C_1 -A$-equivalence bimodule and a $C_2 -C$-equivalence bimodule induced by $E^{A}$ and $E^{C}$, respectively. We suppose that the Watatani index of $E^{A}$, ${{\rm {Ind}}}_W (E^{A} )\in A$. Then by [Reference Kodaka and Teruya8, Examples], inclusions $A\subset C$ and $C_1 \subset C_2$ are strongly Morita equivalent with respect to the $C_2 -C$ equivalence bimodule $C_1$ and its closed subspace $C$, where we regard $C$ as a closed subspace of $C_1$ by the map

\[ \theta_C (x)={{\rm{Ind}}}_W (E^{A} )^{\frac{1}{2}}xe_A \]

for any $x\in C$ (See [Reference Kodaka and Teruya8, Examples]).

Proof. By [Reference Kodaka and Teruya8, Lemma 4.2], $A\subset C$ and $C_1 \subset C_2$ are strongly Morita equivalent with respect to $(C,\, C_1 )\in {{\rm {Equi}}} (C_1 ,\, C_2 ,\, A,\, C)$. Since we regard $C$ as a closed subspace of $C_1$ by the linear map $\theta _C$, we can obtain that

\[ \langle x , \, E^{C_1} (c_1 e_A c_2 e_C d_1 e_A d_2 )\cdot z \rangle_A =E^{A} (\langle \theta_C (x) , \, c_1 e_A c_2 e_C d_1 e_A d_2 \cdot \theta_C (z) \rangle_C ) \]

for any $c_1,\, c_2 ,\, d_1 ,\, d_2 \in C$, $x,\, z\in C$ by routine computations. Indeed,

\begin{align*} \langle x , E^{C_1} (c_1 e_A c_2 e_C d_1 e_A d_2 )\cdot z \rangle_A & =\langle x , \, {{\rm{Ind}}}_W (E^{A} )^{{-}1}c_1 e_A c_2 d_1 e_A d_2 \cdot z \rangle_A \\ & ={{\rm{Ind}}}_W (E^{A} )^{{-}1} \langle x , \, c_1 E^{A} ( c_2 d_1 ) E^{A} (d_2 z) \rangle_A \\ & ={{\rm{Ind}}}_W (E^{A} )^{{-}1}E^{A} (x^{*} c_1 )E^{A} (c_2 d_1 )E^{A} (d_2 z) . \end{align*}

for any $c_1,\, c_2 ,\, d_1 ,\, d_2 \in C$, $x,\, z\in C$. On the other hand,

\begin{align*} & E^{A} (\langle \theta_C (x) , c_1 e_A c_2 e_C d_1 e_A d_2 \cdot \theta_C (z) \rangle_C ) \\ & \quad={{\rm{Ind}}}_W (E^{A} )E^{A} (\langle xe_A , \, c_1 e_A c_2 E^{C} (d_1 e_A d_2 ze_A ) \rangle_C ) \\ & \quad=E^{A} (\langle xe_A , \, c_1 e_A c_2 d_1 E^{A} (d_2 z) \rangle_C ) \\ & \quad=E^{A} (E^{C} (e_A x^{*} c_1 e_A c_2 d_1 ))E^{A} (d_2 z) \\ & \quad=E^{A} (x^{*} c_1 )E^{A} (E^{C} (e_A c_2 d_1 ))E^{A} (d_2 z) \\ & \quad={{\rm{Ind}}}_W (E^{A} )^{{-}1}E^{A} (x^{*} c_1 )E^{A} (c_2 d_1 )E^{A} (d_2 z) . \end{align*}

Hence

\[ \langle x , \, E^{C_1} (c_1 e_A c_2 e_C d_1 e_A d_2 )\cdot z \rangle_A =E^{A} (\langle \theta_C (x) , \, c_1 e_A c_2 e_C d_1 e_A d_2 \cdot \theta_C (z) \rangle_C ) \]

for any $c_1,\, c_2 ,\, d_1 ,\, d_2 \in C$, $x,\, z\in C$. Thus, by Lemma 2.6, $f_{[C, C_1 ]}(E^{A} )=E^{C_1}$. Therefore, we obtain the conclusion.

Let $B\subset D$ be another unital inclusion of unital $C^{*}$-algebras and let $E^{B}$ be a conditional expectation of Watatani index-finite type from $D$ onto $B$. Let $e_B,\, D_1 ,\, E^{D}$, $e_D ,\, D_2 ,\, E^{D_1}$ be as above.

Proof. By Lemma 7.2, there is an element $(Z,\, Z_1 )\in {{\rm {Equi}}}(C_1 ,\, C_2 ,\, D_1 ,\, D_2 )$ such that $f_{[Z, Z_1 ]}(E^{D_1})=E^{C_1}$. Since ${{\rm {Ind}}}_W (E^{B} )\in B$ by [Reference Kodaka and Teruya8, Lemma 6.7],

\[ f_{[C, C_1 ]}(E^{A} )=E^{C_1} , \quad f_{[D, D_1 ]}(E^{B} )=E^{D_1} \]

by Lemma 7.1. Thus

\[ [\widetilde{C}\otimes_{C_1} Z\otimes_{D_1} D , \, \widetilde{C_1}\otimes_{C_2}Z_1 \otimes_{D_2}D_1 ]\in{{\rm{Equi}}}(A, C, B, D) \]

and

\[ f_{[\widetilde{C}\otimes_{C_1} Z\otimes_{D_1} D , \, \widetilde{C_1}\otimes_{C_2}Z_1 \otimes_{D_2}D_1 ]}(E^{B} ) = (f_{[C, C_1 ]}^{{-}1}\circ f_{[Z, Z_1 ]}\circ f_{[D, D_1 ]})(E^{B} ) =E^{A} \]

by Lemma 2.8. Therefore, we obtain the conclusion.

Let $A\subset C$ and $C_1$, $C_2$ be as above. Let $E^{A}$, $E^{C}$ and $E^{C_1}$ be also as above. We suppose that ${{\rm {Ind}}}_W (E^{A} )\in A$. We consider the Picard groups ${{\rm {Pic}}}(E^{A} )$ and ${{\rm {Pic}}}(E^{C} )$ of $E^{A}$ and $E^{C}$, respectively. For any $[X,\, Y]\in {{\rm {Pic}}}(E^{A} )$, there is the unique conditional expectation $E^{X}$ from $Y$ onto $X$ satisfying Conditions (1)–(6) in [Reference Kodaka and Teruya8, Definition 2.4] since $f_{[X, Y]}(E^{A} )=E^{A}$. Let $F$ be the map from ${{\rm {Pic}}}(E^{A} )$ to ${{\rm {Pic}}}(E^{C} )$ defined by

\[ F([X, Y])=[Y, Y_1 ] \]

for any $[X,\, Y]\in {{\rm {Pic}}}(E^{A} )$, where $Y_1$ is the upward basic construction for $E^{X}$ and by Proposition 7.4, $[Y,\, Y_1 ]\in {{\rm {Pic}}}(E^{C} )$. Since $E^{X}$ is the unique conditional expectation from $Y$ onto $X$ satisfying Conditions (1)–(6) in [Reference Kodaka and Teruya8, Definition 2.4] we can see that the same results as [Reference Kodaka6, Lemmas 4.3-4.5] hold. Hence in the same way as in the proof of [Reference Kodaka6, Lemma 5.1], we obtain that $F$ is a homomorphism of ${{\rm {Pic}}}(E^{A} )$ to ${{\rm {Pic}}}(E^{C} )$. Let $G$ be the map from ${{\rm {Pic}}}(E^{A} )$ to ${{\rm {Pic}}}(E^{C_1} )$ defined by for any $[X,\, Y]\in {{\rm {Pic}}}(E^{A} )$

\[ G([X, Y])=[C\otimes_A X\otimes_A \widetilde{C} , \, C_1 \otimes_C Y \otimes_C \widetilde{C_1}] , \]

where $(C,\, C_1 )$ is regarded as an element in ${{\rm {Equi}}}(C_1 ,\, C_2 ,\, A,\, C)$. By the proof of Lemma 6.1, $G$ is an isomorphism of ${{\rm {Pic}}}(E^{A} )$ onto ${{\rm {Pic}}}(E^{C_1} )$. Let $F_1$ be the homomorphism of ${{\rm {Pic}}}(E^{C} )$ to ${{\rm {Pic}}}(E^{C_1})$ defined as above. Then in the same way as in the proof of [Reference Kodaka6, Lemma 5.2], $F_1 \circ F=G$ on ${{\rm {Pic}}}(E^{A} )$. Furthermore, in the same way as in the proofs of [Reference Kodaka6, Lemmas 5.3 and 5.4], we obtain that $F \circ G^{-1}\circ F_1 ={{\rm {id}}}$ on ${{\rm {Pic}}}(E^{C} )$. Therefore, we obtain the same result as [Reference Kodaka6, Theorem 5.5].

8. Modular automorphisms and relative commutants

Following Watatani [Reference Watatani13, Section 1.11], we give the definitions of the modular condition and modular automorphisms.

Let $A\subset C$ be a unital inclusion of unital $C^{*}$-algebras. Let $\theta$ be an automorphism of $A' \cap C$ and $\phi$ an element in ${}_A \mathbf {B}_A (C,\, A)$.

Definition 8.1 Let $\theta$ and $\phi$ be as above. $\phi$ is said to satisfy the modular condition for $\theta$ if the following condition holds:

\[ \phi(xy)=\phi(y\theta(x)) \]

for any $x\in A' \cap C$, $y\in C$.

We have the following theorem which was proved by Watatani in [Reference Watatani13]:

Remark 8.2 Following the proof of [Reference Watatani13, Theorem 1.11.3], we give how to construct the modular automorphism $\theta ^{\phi }$. Let $\{(u_i ,\, v_i )\}_{i=1}^{m}$ be a quasi-basis for $\phi$. Put

\[ \theta^{\phi}(c)=\sum_{i=1}^{m} u_i \phi(cv_i ) \]

for any $c\in A' \cap C$. Then $\theta ^{\phi }$ is the unique automorphism of $A' \cap C$ satisfying the modular condition by the proof of [Reference Watatani13, Theorem 1.11.3].

Let $A\subset C$ and $B\subset D$ be unital inclusions of unital $C^{*}$-algebras, which are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$. Then by the discussions after Proposition 4.5