# On possible values of the interior angle between intermediate subalgebras

Ved Prakash Gupta  and  Deepika Sharma  School of Physical Sciences,
Jawaharlal Nehru University, New Delhi, INDIA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We show that all values in the interval $[0,\frac{\pi}{2}]$ can be attained as
the interior angle between intermediate subalgebras (as introduced in [3]) of
a certain inclusion of simple unital $C^{*}$-algebras. We also calculate the
interior angle between intermediate crossed product subalgebras of any
inclusion of crossed product algebras corresponding to any action of a
countable discrete group and its subgroups on a unital $C^{*}$-algebra.

###### Key words and phrases:

Inclusions of $C^{*}$-algebras, intermediate subalgebras, finite-index
conditional expectations, $C^{*}$-basic construction, interior angle, crossed
products, group $C^{*}$-algebras

###### 2010 Mathematics Subject Classification:

46L05, 47L40

##  1\. Introduction

In any category, in order to classify its objects, the analysis of the
relative positions of the subobjects of an object has proved to be a very
rewarding approach. In the same vein, in the category of operator algebras, a
great deal of work has been done by some eminent mathematicians - see, for
instance, [1, 3, 6, 7, 8, 11] and the references therein. The theory of
subfactors and, more generally, the theory of inclusions of (simple)
$C^{*}$-algebras are two limelights of this aspect.

In this article, our focus lies only on unital $C^{*}$-algebras and their
subalgebras. Over the years, various significant tools and theories have been
developed to understand the relative positions of subalgebras of a given
unital $C^{*}$-algebra. Among them, Watatani’s notions of finite-index
conditional expectations and $C^{*}$-basic construction with respect to a
finite-index conditional expectation ([11]) have proved to be fundamental in
the development of the theory of inclusions of $C^{*}$-algebras - see [11, 9,
7, 6, 3]. Based on these two notions, and motivated by [1], very recently,
Bakshi and the first named author, in [3], introduced the notions of interior
and exterior angles between intermediate $C^{*}$-subalgebras of a given
inclusion $B\subset A$ of unital $C^{*}$-algebras with a finite-index
conditional expectation. As an application of the notion of interior angle,
the authors in [3] were able to improve Longo’s upper bound for the
cardinality of the lattice of intermediate $C^{*}$-subalgebras of any
irreducible inclusion of simple unital $C^{*}$-algebras.

Apart from the above mentioned quantitative application of the notion of
interior angle, we expect some significant qualitative consequences too to be
visible soon. In this direction, it is then quite natural to first ask whether
one can make some concrete calculations of these angles and the possible
values that they can attain. This article essentially answers these questions
to a certain level of satisfaction. Being precise, through some elementary
calculations, we are able to show that all values in the interval
$[0,\frac{\pi}{2}]$ are attained as the interior angles between intermediate
subalgebras of a certain inclusion of simple unital $C^{*}$-algebras. Further,
motivated by [2], we also calculate the interior angle between intermediate
crossed product subalgebras of any inclusion of crossed product algebras
corresponding to any action of a countable discrete group and its subgroups on
a given unital $C^{*}$-algebra.

The article is organized as follows:

After the introduction, we have a relatively longer section on preliminaries
wherein we recall and derive some basic nuances related to finite-index
conditional expectations and Watatani’s $C^{*}$-basic construction related to
inclusions of unital $C^{*}$-algebras. This discussion is fundamental to the
formalism of the interior and exterior angles, which we briefly recall in
Section 3; and, in the same section, we also derive some useful expressions
related to them. Then, in Section 4, we prove that for any
$t\in[0,\frac{\pi}{2}]$, there exists a $2\times 2$ unitary matrix $u$ such
that the interior angle $\alpha(\Delta,u\Delta u^{*})=t$ with respect to the
canonical conditional expectation from $M_{2}(\mathbb{C})$ onto $\mathbb{C}$,
where $\Delta$ denotes the diagonal subalgebra of $M_{2}(\mathbb{C})$;
thereby, establishing that all values in the interval $[0,\frac{\pi}{2}]$ are
attained as the interior angles between intermediate subalgebras. Finally, in
Section 5, as an application of some expressions derived in Section 3, given
any quadruple of countable discrete groups $H\subsetneq K,L\subsetneq G$ with
$[G:H]<\infty$ and with an action $\alpha$ of $G$ on a unital $C^{*}$-algebra
$P$, we derive an expression for the interior angle between the (reduced as
well as universal) intermediate crossed product subalgebras $P\rtimes K$ and
$P\rtimes L$ of the inclusion $P\rtimes H\subset P\rtimes G$.

##  2\. Preliminaries

###  2.1. Watatani’s index and basic construction

In this subsection, we first recall Watatani’s notions of finite-index
conditional expectations and the $C^{*}$-basic constructions with respect to
such conditional expectations; and then, we touch upon some generalities
related to intermediate $C^{*}$-subalgebras.

####  2.1.1. Finite-index conditional expectations

Recall that, for an inclusion $B\subset A$ of unital $C^{*}$-algebras, a
conditional expectation $E:A\to B$ is said to have finite index if there
exists a finite set $\\{\lambda_{1},\ldots,\lambda_{n}\\}\subset A$ such that

$x=\sum_{i=1}^{n}E(x\lambda_{i})\lambda^{*}_{i}=\sum_{i=1}^{n}\lambda_{i}E(\lambda^{*}_{i}x)$

for every $x\in A$ \- see [11, 6, 7]. Such a set
$\\{\lambda_{1},\ldots,\lambda_{n}\\}$ is called a quasi-basis for $E$ and the
Watatani index of $E$ is defined as

$\mathrm{Ind}(E)=\sum_{i=1}^{n}\lambda_{i}\lambda^{*}_{i}.$

It is known that $\mathrm{Ind}(E)$ is a positive invertible element of
$\mathcal{Z}(A)$ and is independent of the quasi-basis $\\{\lambda_{i}\\}$ \-
see [11, $\S 2$]. Also, $E$ is faithful, $E(1_{A})=1_{B}$ and
$\mathrm{Ind}(E)\geq 1$.

######  Remark 2.1.

  1. (1)

If $B\subset C\subset A$ are inclusions of unital $C^{*}$-algebras with
$1_{A}\in B$ and $E:A\to B$, $F:A\to C$ and $G:C\to B$ are faithful
conditional expectations satisfying $E=G\circ F$, then $E$ has finite index if
and only if $F$ and $G$ have finite index - see [9, Proposition 3.5].

  2. (2)

For an inclusion $B\subset A$, in general, if $E,E^{\prime}:A\to B$ are two
conditional expectations, one may be of finite index and the other may fail -
see [11, Example 2.10.1].

Interestingly, if there exists a finite index conditional expectation from $A$
onto $B$, then all faithful conditional expectations from $A$ onto $B$ are of
finite index if the centralizer of $B$ in $A$, i.e.,
$\mathcal{C}_{A}(B):=\\{x\in A:xb=bx\ \text{for all}\ b\in B\\}$, is finite
dimensional - see [11, Propositions 2.10.2].

Thus, when $\mathcal{C}_{A}(B)$ is finite dimensional, one can roughly say
that the property of ‘finite index’ is an intrinsic property of the inclusion
$B\subset A$ and not of a conditional expectation from $A$ onto $B$.

  3. (3)

There exist finite-index conditional expectations even when the corresponding
centralizers are not finite dimensional. For instance, see [11, Example
2.6.7].

Let $A=C(X)$ and $B:=A^{\alpha}$, where $X$ is an infinite compact Hausdorff
space and $\alpha$ is a free action of a finite group $G$ on $A$. Define
$E:A\to B$ by

$E(f)=\frac{\sum_{g}\alpha_{g}(f)}{|G|},\ f\in A.$

Then, $E$ has finite-index and $\mathrm{Ind}(E)=|G|$ \- see [11, Proposition
2.8.1] \- whereas $\mathcal{C}_{A}(B)$ is infinite dimensional as $A=C(X)$ is
a commutative $C^{*}$-algebra.

####  2.1.2. Watatani’s $C^{*}$-basic construction

Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with common unit
and suppose $E:A\to B$ is a faithful conditional expectation. Let $A_{1}$
denote the Watatani’s $C^{*}$-basic construction of the inclusion $B\subset A$
with respect to the conditional expectation $E$, i.e., in short, one
essentially shows the following:

  1. (1)

$A$ is a pre-Hilbert $B$-module with respect to the $B$-valued inner product
given by

$\langle a,a^{\prime}\rangle_{B}:=E(a^{*}a^{\prime})\text{ for
}a,a^{\prime}\in A;$

and, if $\mathfrak{A}$ denotes the Hilbert $B$-module completion of $A$, then

  2. (2)

the space of adjointable maps on $\mathfrak{A}$, denoted by
$\mathcal{L}_{B}(\mathfrak{A})$, is a unital $C^{*}$-algebra (with the usual
operator norm) and $A$ embeds in it as a unital $C^{*}$-subalgebra (and, by a
slight abuse of notation, we identify $A$ with its image in
$\mathcal{L}_{B}(\mathfrak{A})$);

  3. (3)

there exists a projection $e_{B}\in\mathcal{L}_{B}(\mathfrak{A})$ (called the
Jones projection associated to $E$) such that $e_{B}ae_{B}=E(a)e_{B}$ for all
$a\in A$ (it is standard to denote $e_{B}$ by $e_{1}$ as well); and

  4. (4)

one considers $A_{1}:=\overline{\mathrm{span}}\\{xe_{B}y:x,y\in
A\\}\subseteq\mathcal{L}_{B}(\mathfrak{A})$, which turns out to be a
$C^{*}$-algebra (not always unital) and is called the $C^{*}$-basic
construction of the inclusion $B\subset A$.

The system $(A,B,E,e_{B},A_{1})$ has the following natural universal property.

######  Theorem 2.2.

[11, Proposition 2.2.11] Let $B\subset A$ be an inclusion of unital
$C^{*}$-algebras with a faithful conditional expectation $E:A\to B$. Suppose
that $A$ acts faithfully on some Hilbert space $H$ and $e$ is a projection on
$H$ satisfying $eae=E(a)e$ for all $a\in A$. If the linear map $B\ni b\mapsto
be\in B(H)$ is injective, then there is a $*$-isomorphism
$\theta:A_{1}\to\overline{AeA}\subset B(H)$ such that $\theta(xe_{B}y)=xey$
for all $x,y\in A$.

######  Remark 2.3.

If $E:A\to B$ has finite index with a quasi-basis $\\{\lambda_{i}\\}$, then

  1. (1)

the two norms $\|\cdot\|_{A}$ and $\|\cdot\|$ are equivalent on $A$ (where
$\|x\|_{A}:=\|E_{B}(x^{*}x)\|^{1/2}$)- see [11] or the proof of [3, Lemma
2.11]; in particular, $A$ itself is a Hilbert $B$-module;

  2. (2)

$A_{1}$ is unital and is equal to $C^{*}(A,e_{B})$ \- see [11, Proposition
.1.5];

  3. (3)

there exists a finite-index conditional expectation $E_{1}:A_{1}\to A$ (called
the dual conditional expectation) with a quasi-basis
$\\{\lambda_{i}e_{B}(\mathrm{Ind}(E))^{1/2}\\}$ which satisfies the equation

(2.1) $E_{1}(xe_{B}y)=\mathrm{Ind}(E)^{-1}xy$

for all $x,y\in A$ and
$\mathrm{Ind}(E_{1})=\sum_{i}\lambda_{i}E(\mathrm{Ind}(E))e_{B}\lambda_{i}^{*}$;
moreover, if $\mathrm{Ind}(E)\in B$, then
$\mathrm{Ind}(E_{1})=\mathrm{Ind}(E)$ \- see [11, Propositions 2.3.2 $\&$
2.3.4]; and,

  4. (4)

if $F:A\to B$ is another finite-index conditional expectation and
$C^{*}(A,f_{B})$ denotes the corresponding $C^{*}$-basic construction, then
there exists a $*$-isomorphism $\theta:A_{1}\to C^{*}(A,f_{B})$ such that
$\theta(e_{B})=f_{B}$ and $\theta(a)=a$ for all $a\in A$ \- [11, Proposition
2.10.11]; and,

  5. (5)

$A_{1}=\mathrm{span}\\{xe_{B}y:x,y\in A\\}=:Ae_{B}A$ \- see [11, Lemma 2.2.2].

###  2.2. Intermediate $C^{*}$-subalgebras

Throughout this subsection, we let $B\subset A$ be an inclusion of unital
$C^{*}$-algebras, $E:A\to B$ be a finite-index conditional expectation with a
quasi-basis $\\{\lambda_{i}:1\leq i\leq n\\}$, $A_{1}:=Ae_{B}A\
(=C^{*}(A,e_{B}))$ denote the $C^{*}$-basic construction of $B\subset A$ with
respect to $E$ and $E_{1}:A_{1}\to A$ denote the dual conditional expectation.

As in [7], let $\mathrm{IMS}(B,A,E)$ denote the set of intermediate
$C^{*}$-subalgebras $C$ between $B$ and $A$ with a conditional expectation
$F:A\to C$ satisfying the compatibility condition $E=E_{\restriction_{C}}\circ
F$.

######  Remark 2.4.

  1. (1)

If $C\in\mathrm{IMS}(B,A,E)$ with respect to two compatible conditional
expectations $F,F^{\prime}:A\to C$, then $F=F^{\prime}$ \- see [7, Page 3].

  2. (2)

If $C\in\mathrm{IMS}(B,A,E)$ with respect to the compatible conditional
expectation $F:A\to C$, then $F$ is faithful (since $E$ is so) and, therefore,
by Remark 2.1(1), $F$ has finite index.

  3. (3)

It must be mentioned here that it was presumed (without mention) in [3] that
the compatible conditional expectation has finite index and was implicitly
used while defining the notions of interior and exterior angles between
intermediate subalgebras of an inclusion of unital $C^{*}$-algebras.

  4. (4)

For $C\in\mathrm{IMS}(B,A,E)$ with respect to the compatible conditional
expectation $F:A\to C$, we observe that $A$ is a Hilbert $C$-module (Remark
2.3(2)); we let $e_{C}$ denote the corresponding Jones projection in
$\mathcal{L}_{C}(A)$ and $C_{1}$ denote the Watatani basic construction of the
inclusion $C\subset A$; thus,
$C_{1}=C^{*}(A,e_{C})\subseteq\mathcal{L}_{C}(A)$.

######  Remark 2.5.

In general, if $Q\subset P$ is an inclusion of unital $C^{*}$-algebras with a
finite-index conditional expectation $G:P\to Q$, then not every intermediate
$C^{*}$-subalgebra $R$ of $Q\subset P$ belongs to $\mathrm{IMS}(Q,P,G)$ \- see
[7, Example 2.5]. In fact, the example given in [7] illustrates that there
need not exist even a single conditional expectation from $P$ onto $R$.

Izumi observed that the intermediate subalgebras of an inclusion of simple
$C^{*}$-algebras have certain specific structures.

######  Proposition 2.6.

[6] Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with a finite-
index conditional expectation $E:A\to B$. If either $A$ or $B$ is simple then,
every $C$ in $\mathrm{IMS}(B,A,E)$ is a finite direct sum of simple closed
two-sided ideals.

###### Proof.

Let $C\in\mathrm{IMS(B,A,E)}$ with respect to the compatible conditional
expectation $F:A\to C$. Then, by Remark 2.4(2) and [11, Proposition 2.1.5],
$F$ and $E_{\restriction_{C}}$ satisfy the Pimsner-Popa inequality. Further,
since $A$ or $B$ is simple and unital, it then follows from [6, Theorem 3.3]
that $C$ is a finite direct sum of simple closed two-sided ideals ∎

The following useful observations will be needed ahead when we recall and
derive some generalities related to the notions of interior and exterior
angles.

######  Proposition 2.7.

Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras, $E:A\to B$ be a
finite-index conditional expectation with a quasi-basis $\\{\lambda_{i}:1\leq
i\leq n\\}$, $A_{1}$ denote the $C^{*}$-basic construction of $B\subset A$
with respect to $E$, $E_{1}:A_{1}\to A$ denote the dual conditional
expectation and $C\in\mathrm{IMS}(B,A,E)$ with respect to the compatible
finite-index conditional expectation $F:A\to C$. Then,

  1. (1)

$\mathcal{L}_{C}(A)\subset\mathcal{L}_{B}(A)$;

  2. (2)

$C_{1}\subset A_{1}$, so that $e_{C}\in A_{1}$;

  3. (3)

$e_{C}e_{B}=e_{B}=e_{B}e_{C}$;

  4. (4)

$E_{\restriction_{C}}$ has finite index with a quasi-basis
$\\{F(\lambda_{i})\\}$ and $e_{C}=\sum{\mu_{j}e_{B}\mu_{j}^{*}}$ for any
quasi-basis $\\{\mu_{j}\\}$ of the conditional expectation
$E_{\restriction_{C}}$;

  5. (5)

$E_{1}(e_{B})=\mathrm{Ind}(E)^{-1}\in\mathcal{Z}(A)$;

  6. (6)

$E_{1}(e_{C})=\mathrm{Ind}(E)^{-1}\mathrm{Ind}(E_{\restriction_{C}})\in\mathcal{Z}(C)$;
and,

  7. (7)

in addition, if $\mathrm{Ind}(E_{\restriction_{C}})\in\mathcal{Z}(A)$, then

    1. (a)

$\mathrm{Ind}(E)=\mathrm{Ind}(F)\mathrm{Ind}(E_{\restriction_{C}})$;

    2. (b)

${E_{1}}_{\restriction_{C_{1}}}=F_{1}$, where $F_{1}$ denotes the dual
conditional expectation of $F$; and

    3. (c)

$C_{1}\in\mathrm{IMS}(A,A_{1},E_{1})$ with respect to the conditional
expectation $G:A_{1}\to C_{1}$ satisfying
$G(xe_{B}y)=\mathrm{Ind}(E_{\restriction_{C}})^{-1}xe_{C}y$ for all $x,y\in A$
and has a quasi-basis
$\\{\lambda_{i}e_{B}\mathrm{Ind}(E_{\restriction_{C}})^{1/2}:1\leq i\leq
n\\}$.

In particular, we then have $E_{1}(e_{C})=\mathrm{Ind}(F)^{-1}.$

###### Proof.

(1): Let $T\in\mathcal{L}_{C}(A)$ and $T^{*}$ denote its adjoint in
$\mathcal{L}_{C}(A)$. Then, we see that

$\displaystyle{c}\langle
T(x),y\rangle_{B}=E(T(x)^{*}y)=(E_{\restriction_{C}}\circ
F)(T(x)^{*}y)=E_{\restriction_{C}}(\langle T(x),y\rangle_{C})$
$\displaystyle=E_{\restriction_{C}}(\langle
x,T^{*}(y)\rangle_{C})=(E_{\restriction_{C}}\circ F)(xT^{*}(y))=\langle
x,T^{*}(y)\rangle_{B}$

for all $x,y\in A$. Hence, $T\in\mathcal{L}_{B}(A)$.

Because of (1), (2) now follows on the lines [3, Lemma 4.2].

(3): Clearly, $e_{C}e_{B}=e_{B}$ (as $B\subset C$). Next, we observe that

$e_{B}e_{C}(a)=e_{B}(F(a))=E(F(a))=(E_{\restriction_{C}}\circ
F)(a)=E(a)=e_{B}(a)$

for all $a\in A$. Thus, $e_{B}e_{C}=e_{B}$.

(4): That $E_{\restriction_{C}}$ has finite-index with quasi-basis
$\\{F(\lambda_{i})\\}$ follows from [7, Page 3] (also see [11, Proposition
1.7.2]). Further, for any quasi-basis $\\{\mu_{j}\\}$ for
$E_{\restriction_{C}}$, we have

$\displaystyle(\sum\mu_{j}e_{B}\mu_{j}^{*})(a)$ $\displaystyle=$
$\displaystyle\sum\mu_{j}e_{B}\mu_{j}^{*}(a)$ $\displaystyle=$
$\displaystyle\sum\mu_{j}E(\mu_{j}^{*}(a))$ $\displaystyle=$
$\displaystyle\sum\mu_{j}(E_{\restriction_{C}}\circ F)(\mu_{j}^{*}(a))$
$\displaystyle=$
$\displaystyle\sum\mu_{j}E_{\restriction_{C}}(\mu_{j}^{*}F(a))$
$\displaystyle=$ $\displaystyle F(a)$ $\displaystyle=$ $\displaystyle
e_{C}(a)$

for all $a\in A$. Hence, $e_{C}=\sum\mu_{j}e_{B}\mu_{j}^{*}$.

(5): See [11, Proposition 2.3.2].

(6): For any quasi-basis $\\{\mu_{j}\\}$ for the conditional expectation
$E_{\restriction_{C}}$, we have $e_{C}=\sum\mu_{j}e_{B}\mu_{j}^{*}$. Hence,

$\displaystyle E_{1}(e_{C})$ $\displaystyle=$ $\displaystyle
E_{1}(\sum\mu_{j}e_{B}\mu_{j}^{*})$ $\displaystyle=$ $\displaystyle\sum
E_{1}(\mu_{j}e_{B}\mu_{j}^{*})$ $\displaystyle=$
$\displaystyle\mathrm{Ind}(E)^{-1}\sum\mu_{j}\mu_{j}^{*}$ $\displaystyle=$
$\displaystyle\mathrm{Ind}(E)^{-1}\mathrm{Ind}(E_{\restriction_{C}}).$

(7a): Let $\\{\mu_{1},\mu_{2},...,\mu_{n}\\}$ be a quasi-basis for
$E_{\restriction_{C}}$ and $\\{\gamma_{1},\gamma_{2},...,\gamma_{m}\\}$ be a
quasi-basis for $F$. Then, it is (known and can be) easily seen that
$\\{\gamma_{i}\mu_{j}:1\leq i\leq m,1\leq j\leq n\\}$ is a quasi-basis for $E$
\- see also [11, Proposition 1.7.1]. Thus,

$\displaystyle\mathrm{Ind}(E)$ $\displaystyle=$
$\displaystyle\sum_{i,j}(\gamma_{i}\mu_{j})(\gamma_{i}\mu_{j})^{*}$
$\displaystyle=$
$\displaystyle\sum_{i}\gamma_{i}(\sum_{j}\mu_{j}\mu_{j}^{*})\gamma_{i}^{*}$
$\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})\mathrm{Ind}(F).$

(7b): We have $C_{1}=\mathrm{span}\\{xe_{C}y:x,y\in A\\}$. Fix a quasi-basis
$\\{\mu_{j}\\}$ for $E_{\restriction_{C}}$. Then, for every pair $x,y\in C$,
we observe that

$\displaystyle E_{1}(xe_{C}y)$ $\displaystyle=$ $\displaystyle
E_{1}(x\sum_{j}\mu_{j}e_{B}\mu_{j}^{*}y)$ $\displaystyle=$
$\displaystyle\mathrm{Ind}(E)^{-1}\sum_{j}x\mu_{j}\mu_{j}^{*}y$
$\displaystyle=$
$\displaystyle\mathrm{Ind}(E)^{-1}x\,\mathrm{Ind}(E_{\restriction_{C}})y=\mathrm{Ind}(F)^{-1}xy$
$\displaystyle=$ $\displaystyle F_{1}(xe_{C}y),$

where the second last equality follows from (7a). Hence,
$(E_{1})_{\restriction_{C_{1}}}=F_{1}$.

(7c): We have $A_{1}=\mathrm{span}\\{xe_{B}y:x,y\in A\\}$. Consider the linear
map $G:A_{1}\to C_{1}$ given by

$G(\sum_{i}x_{i}e_{B}y_{i})=\mathrm{Ind}(E_{\restriction_{C}})^{-1}\sum_{i}x_{i}e_{C}y_{i}$

for $x_{i},y_{i}\in A$, $i=1,\ldots,n$.

We first assert that $G$ is a conditional expectation of finite-index.

Fix a quasi-basis $\\{\mu_{j}\\}$ for $E_{\restriction_{C}}$. Then, for any
$x,y\in A$, by Item (4), we have

$\displaystyle G(xe_{C}y)$ $\displaystyle=$ $\displaystyle
G(x\sum_{j}\mu_{j}e_{B}\mu_{j}^{*}y)$ $\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})^{-1}\sum_{j}x\mu_{j}e_{C}\mu_{j}^{*}y$
$\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})^{-1}\sum_{j}x\mu_{j}\mu_{j}^{*}e_{C}y\qquad(\text{since
}e_{C}\in C^{\prime}\cap C_{1})$ $\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})^{-1}x\mathrm{Ind}(E_{\restriction_{C}})e_{C}y$
$\displaystyle=$ $\displaystyle xe_{C}y.\qquad(\text{since
}\mathrm{Ind}(E_{\restriction_{C}})\in\mathcal{Z}(A))$

This implies that $G^{2}=G$. Further, for any $\sum_{i}x_{i}e_{B}y_{i}\in
A_{1}$, we have

$\displaystyle
G\big{(}(\sum_{i}x_{i}e_{B}y_{i})^{*}(\sum_{i}x_{i}e_{B}y_{i})\big{)}$
$\displaystyle=$ $\displaystyle
G\big{(}\sum_{i,j}y_{i}^{*}E(x_{i}^{*}x_{j})e_{B}y_{j}\big{)}$
$\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})^{-1}\sum_{i,j}y_{i}^{*}e_{C}E(x_{i}^{*}x_{j})e_{C}y_{j}.$

Then, taking $a_{i,j}:=e_{C}E(x_{i}^{*}x_{j})e_{C}\in C_{1}$,
$i,j=1,\ldots,n$, we have

$[a_{i,j}]=\mathrm{diag}(e_{C},\ldots,e_{C})[E(x_{i}^{*}x_{j})]\mathrm{diag}(e_{C},\dots,e_{C}).$

By [10, Lemma 3.1], $[x_{i}^{*}x_{j}]$ is positive in $M_{n}(A)$ and since
$E:A\to B$ is completely positive, it follows that $[E(x_{i}^{*}x_{j})]$ is
positive in $M_{n}(B)$. Hence, $[a_{i,j}]$ is positive in $M_{n}(C_{1})$.
Thus, by [10, Lemma 3.2], it follows that
$\sum_{i,j}y_{i}^{*}e_{C}E(x_{i}^{*}x_{j})e_{C}y_{j}\geq 0$ in $C_{1}$.
Further, since
${\mathrm{Ind}(E_{\restriction_{C}})}^{-1}\in\mathcal{Z}(A)\cap\mathcal{Z}(C)$
and $e_{C}\in C^{\prime}\cap C_{1}$, it follows that
${\mathrm{Ind}(E_{\restriction_{C}})}^{-1}$ commutes with
$\sum_{i,j}y_{i}^{*}e_{C}E(x_{i}^{*}x_{j})e_{C}y_{j}$ and hence

$G\big{(}(\sum_{i}x_{i}e_{B}y_{i})^{*}(\sum_{i}x_{i}e_{B}y_{i})\big{)}\geq 0.$

Thus, $G:A_{1}\to C_{1}$ is positive and, therefore, it is a conditional
expectation.

Further, $G:A_{1}\to C_{1}$ has finite index with quasi basis
$\\{\lambda_{i}e_{B}\mathrm{Ind}(E_{\restriction_{C}})^{1/2}:1\leq i\leq n\\}$
because, for any $x,y\in A$, we have

$\displaystyle\sum_{i}\lambda_{i}e_{B}\mathrm{Ind}(E_{\restriction_{C}})^{1/2}G\big{(}(\mathrm{Ind}(E_{\restriction_{C}})^{1/2}e_{B}\lambda_{i}^{*}xe_{B}y\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i}\lambda_{i}e_{B}(\mathrm{Ind}(E_{\restriction_{C}})^{1/2})G\big{(}(\mathrm{Ind}(E_{\restriction_{C}})^{1/2})E(\lambda_{i}^{*}x)e_{B}y\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i}\lambda_{i}e_{B}\mathrm{Ind}(E_{\restriction_{C}})^{1/2}\mathrm{Ind}(E_{\restriction_{C}})^{-1}\mathrm{Ind}(E_{\restriction_{C}})^{1/2}E(\lambda_{i}^{*}x)e_{C}y$
$\displaystyle=$ $\displaystyle xe_{B}y.\qquad\text{(since $e_{C}\in C_{1}\cap
B^{\prime}$ , $e_{B}e_{C}=e_{B}$ and $e_{B}\in A_{1}\cap B^{\prime}$)}$

Finally, since ${E_{1}}_{\restriction_{C_{1}}}=F_{1}$, we observe that

$\displaystyle({E_{1}}_{\restriction_{C_{1}}}\circ G)(xe_{B}y)$
$\displaystyle=$ $\displaystyle
E_{1}(\mathrm{Ind}(E_{\restriction_{C}})^{-1}xe_{C}y)$ $\displaystyle=$
$\displaystyle F_{1}(\mathrm{Ind}(E_{\restriction_{C}})^{-1}xe_{C}y)$
$\displaystyle=$
$\displaystyle\mathrm{Ind}(E_{\restriction_{C}})^{-1}\mathrm{Ind}({F})^{-1}xy$
$\displaystyle=$ $\displaystyle\mathrm{Ind}({E})^{-1}xy$ $\displaystyle=$
$\displaystyle E_{1}(xe_{B}y)$

for all $x,y\in A$, where the second last equality follows from Item (7a).
Hence, $({E_{1}}_{\restriction_{C_{1}}}\circ G)=E_{1}$.

These show that $C_{1}\in\mathrm{IMS}(A,A_{1},E_{1})$ with respect to the
finite-index conditional expectation $G:A_{1}\to C_{1}$. ∎

Recall that for an inclusion $B\subset A$ of unital $C^{*}$-algebras, the
normalizer of $B$ in $A$ is defined as

$\mathcal{N}_{A}(B):=\\{u\in\mathcal{U}(A):uBu^{*}=B\\},$

where $\mathcal{U}(A)$ denotes the group of unitaries in $A$; and, (as already
mentioned above) the centralizer of $B$ in $A$ is defined as

$\mathcal{C}_{A}(B):=\\{a\in A:ab=ba\text{ for all }b\in B\\}.$

Clearly, $\mathcal{U}(B)$ is a normal subgroup of $\mathcal{N}_{A}(B)$ and
$\mathcal{C}_{A}(B)$ is a unital $C^{*}$-subalgebra of $A$, which is also
denoted by $B^{\prime}\cap A$ and is called the relative commutant of $B$ in
$A$.

The following observation provides us with some easy examples of elements in
$\mathrm{IMS}(B,A,E)$.

######  Lemma 2.8.

Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with a finite-
index conditional expectation $E:A\to B$. Let $C\in\mathrm{IMS}(B,A,E)$ with
respect to the compatible finite-index conditional expectation $F:A\to C$ and
$u\in\mathcal{U}(A)$. Then,

  1. (1)

$F_{u}:A\to uCu^{*}$ given by $F_{u}=\mathrm{Ad}_{u}\circ
F\circ\mathrm{Ad}_{u^{*}}$, i.e., $F_{u}(a)=uF(u^{*}au)u^{*}\text{ for }a\in
A,$ is a finite-index conditional expectation with a quasi-basis
$\\{u\eta_{i}u^{*}:1\leq i\leq n\\}$, where $\\{\eta_{i}:1\leq i\leq n\\}$ is
a quasi basis for $F$; $\mathrm{Ind}(F_{u})=\mathrm{Ind}(F)$; and,

  2. (2)

in addition, if $u\in\mathcal{N}_{A}(B)$ and $E$ satisfies the tracial
property, i.e., $E(xy)=E(yx)$ for all $x,y\in A$, then
$uCu^{*}\in\mathrm{IMS}(B,A,E)$ with respect to $F_{u}$.

###### Proof.

(1) is a straight forward verification.

(2): Let $D:=uCu^{*}$. Since $u\in\mathcal{N}_{A}(B)$, $B=uBu^{*}\subset
uCu^{*}$ and we have

$\displaystyle E_{\restriction_{D}}\circ F_{u}(a)$ $\displaystyle=$
$\displaystyle E_{\restriction_{D}}(uF(u^{*}au)u^{*})$ $\displaystyle=$
$\displaystyle E(u^{*}uF(u^{*}au))\qquad(\text{by tracial property})$
$\displaystyle=$ $\displaystyle E_{\restriction_{C}}\circ F(u^{*}au)$
$\displaystyle=$ $\displaystyle E(u^{*}au)$ $\displaystyle=$ $\displaystyle
E(uu^{*}a)\qquad(\text{by tracial property again})$ $\displaystyle=$
$\displaystyle E(a)$

for all $a\in A$. Hence, $uCu^{*}\in\mathrm{IMS}(B,A,E)$ with respect to
$F_{u}$. ∎

######  Remark 2.9.

Note that $ue_{C}u^{*}$ is a projection in $C_{1}$ (as $u\in A\subset C_{1}$)
and, for each $x\in A$, we have

$(ue_{C}u^{*})x(ue_{C}u^{*})=uF(u^{*}xu)e_{C}u^{*}=F_{u}(x)ue_{C}u^{*}.$

So, it is quite tempting to think that maybe the basic construction of
$B\subset uCu^{*}$ is given by $(uCu^{*})_{1}=C_{1}$ (as the $C^{*}$-algebra)
with Jones projection $e_{uCu^{*}}=ue_{C}u^{*}$. However, this is not the
case.

For instance, if we let $A$, $B$, $C$, $E:A\to B$ and $F:A\to C$ be the same
as in Section 4, then taking
$u=\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}\\\
\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$, we observe that
$ue_{C}u^{*}=\begin{bmatrix}1/2&0&-i/2&0\\\ 0&1/2&0&i/2\\\ i/2&0&1/2&0\\\
0&-i/2&0&1/2\end{bmatrix}$ whereas $e_{uCu^{*}}=\begin{bmatrix}1/2&0&0&1/2\\\
0&1/2&-1/2&0\\\ 0&-1/2&1/2&0\\\ 1/2&0&0&1/2\end{bmatrix}$ (using values of
$e_{C}$ from Lemma 4.2 and $e_{uCu^{*}}$ from Lemma 4.3).

######  Remark 2.10.

  1. (1)

In general, the dual conditional expectation of a tracial conditional
expectation need not be tracial.

For instance, consider the inclusion
$B=\mathbb{C}\ni\lambda\hookrightarrow(\lambda,\lambda)\in
A=\mathbb{C}\oplus\mathbb{C}$ with respect to the conditional expectation
$E:A\to B$ given by $E((\lambda,\mu))=\frac{\lambda+\mu}{2}$. Clearly, $E$ is
a finite-index tracial conditional expectation and we see that one can
identify $A_{1}$ with $M_{2}(\mathbb{C})$ and then the dual conditional
expectation $E_{1}:A_{1}\to A$ is given by $E_{1}([a_{ij}])=(a_{11},a_{22})$.
Clearly, $E_{1}(AB)\neq E_{1}(BA)$ for $A=\begin{pmatrix}1&0\\\
1&1\end{pmatrix}$ and $B=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$.

  2. (2)

It is natural to wonder whether traciality of $E$ can be dropped or not while
showing that $uCu^{*}$ belongs $\mathrm{IMS(B,A,E)}$ with respect to $F_{u}$.
And, it turns out that it can’t be dropped always.

For instance, consider $A=M_{2}(\mathbb{C})$ and $B=\mathbb{C}I_{2}$ with the
conditional expectation $E:A\to B$ given by
$E([a_{ij}])=a_{11}t+a_{22}(1-t)\quad\text{where t}\neq 1/2$ is fixed. Let
$C=\\{\mathrm{diag}(\lambda,\mu):\lambda,\mu\in\mathbb{C}\\}$ and $F:A\to C$
be the conditional expectation defined by
$F([a_{ij}])=\mathrm{diag}(a_{11},a_{22})$. Clearly, $E$ and $F$ are finite-
index conditional expectations with quasi-bases
$\\{\sqrt{t}e_{11},\sqrt{1-t}e_{12},\sqrt{t}e_{21},\sqrt{1-t}e_{22}\\}$ and
$\\{e_{ij}:1\leq i,j\leq 2\\}$, respectively, and $E_{\restriction_{C}}\circ
F=E$. If $u=\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}\\\
\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$, then $u\in U(2)$ and

$F_{u}([a_{ij}])=\begin{bmatrix}(a_{11}+a_{22})/2&(a_{12}-a_{21})/2\\\
(a_{21}-a_{12})/2&(a_{11}+a_{22})/2\end{bmatrix}$

for all $[a_{ij}]\in A$. Thus, $E_{\restriction_{uCu^{*}}}\circ
F_{u}\big{(}[a_{ij}]\big{)}=\frac{a_{11}+a_{22}}{2}$ which is not equal to
$E\big{(}[a_{ij}]\big{)}$ (as $t\neq 1/2$).

##  3\. Interior and exterior angles

Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with a conditional
expectation $E:A\to B$. Then, for the $B$-valued inner product
$\langle\cdot,\cdot\rangle_{B}$ on $A$ given by $\langle
x,y\rangle_{B}=E(x^{*}y)$, one has the following well known analogue of the
Cauchy-Schwarz inequality

(3.1) $\|\langle x,y\rangle_{B}\|\leq\|x\|_{A}\|y\|_{A}\text{ for all }x,y\in
A,$

where $\|x\|_{A}:=\|E_{B}(x^{*}x)\|^{1/2}$. And, unlike for usual inner
products, equality in (3.1) does not imply that $\\{x,y\\}$ is linearly
dependent. For instance, consider
$B=\\{\mathrm{diag}(\lambda,\mu):\lambda,\mu\in\mathbb{C}\\}$ in
$A=M_{2}(\mathbb{C})$ with the natural finite-index conditional expectation
$E:A\to B$ given by $E([x_{ij}])=\mathrm{diag}(x_{11},x_{22})$. Then, for
$x=\mathrm{diag}(1,1)$ and $y=\mathrm{diag}(i,1)$ in $A$, one easily verifies
that $\|\langle x,y\rangle_{B}\|=\|x\|_{A}\|y\|_{A}$ whereas $\\{x,y\\}$ is
linearly independent.

Employing Inequality (3.1), motivated by [1], Bakshi and the first named
author introduced the following definitions of the interior and exterior
angles between intermediate $C^{*}$-subalgebras.

######  Definition 3.1.

[3] Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with a finite-
index conditional expectation $E:A\to B$. Then, for
$C,D\in\mathrm{IMS}(B,A,E)\setminus\\{B\\}$, the interior angle between $C$
and $D$ (with respect to $E$), denoted as $\alpha(C,D)$, is given by the
expression

(3.2) $\cos(\alpha(C,D))=\frac{\|\langle
e_{C}-e_{B},e_{D}-e_{B}\rangle_{A}\|}{\|e_{C}-e_{B}\|_{A_{1}}\|e_{D}-e_{B}\|_{A_{1}}};$

and, for $C,D\in\mathrm{IMS}(B,A,E)\setminus\\{A\\}$ with
$C_{1},D_{1}\in\mathrm{IMS}(A,A_{1},E_{1})$, the exterior angle between $C$
and $D$ is defined as

(3.3) $\beta(C,D)=\alpha(C_{1},D_{1}),$

where $\alpha(C_{1},D_{1})$ is defined with respect to the dual conditional
expectation $E_{1}:A_{1}\to A$.

By definition, both angles are allowed to take values only in the interval
$[0,\frac{\pi}{2}]$.

######  Remark 3.2.

  1. (1)

Note that if
$\mathrm{Ind}(E_{\restriction_{D}}),\mathrm{Ind}(E_{\restriction_{C}})\in\mathcal{Z}(A)$,
then by Proposition 2.7(7), $C_{1},D_{1}\in\mathrm{IMS}(A,A_{1},E_{1})$. Thus,
$\beta(C,D)$ is defined for such intermediate subalgebras.

  2. (2)

If $B\subset C,D\subset A$ is a quadruple of simple unital $C^{*}$-algebras,
then $\beta(C,D)$ is always defined.

We now derive some useful formulae for the interior and exterior angles in
terms of certain related quasi-bases.

######  Proposition 3.3.

Let $B\subset A$ be an inclusion of unital $C^{*}$-algebras with a finite
index conditional expectation $E:A\to B$ with quasi-basis
$\\{\lambda_{i}:1\leq i\leq p\\}$. Let
$C,D\in\mathrm{IMS}(B,A,E)\setminus\\{B\\}$ with respect to the conditional
expectations $F:A\to C$ and $F^{\prime}:A\to D$, respectively. Let
$\\{\mu_{j}:1\leq\mu_{j}\leq m\\}$ and $\\{\delta_{k}:1\leq\delta_{k}\leq
n\\}$ be quasi-bases for $E_{\restriction_{C}}$ and $E_{\restriction_{D}}$,
respectively. Then, we have the following:

  1. (1)

The interior angle between $C$ and $D$ is given by

(3.4)
$\cos(\alpha(C,D))=\frac{\|(\mathrm{Ind}(E))^{-1}\left(\sum_{j,k}\mu_{j}E(\mu_{j}^{*}\delta_{k})\delta_{k}^{*}-1\right)\|}{\|(\mathrm{Ind}(E))^{-1}(\mathrm{Ind}(E_{\restriction_{C}})-1)\|^{\frac{1}{2}}\|(\mathrm{Ind}(E))^{-1}(\mathrm{Ind}(E_{\restriction_{D}})-1)\|^{\frac{1}{2}}}.$

In particular, if $\mathrm{Ind}(E)$ is a scalar, then

(3.5)
$\cos(\alpha(C,D))=\frac{\|\sum_{j,k}\mu_{j}E(\mu_{j}^{*}\delta_{k})\delta_{k}^{*}-1\|}{\|{\mathrm{Ind}(E_{\restriction_{C}})-1}\|^{\frac{1}{2}}\|{\mathrm{Ind}(E_{\restriction_{D}})-1}\|^{\frac{1}{2}}}.$

  2. (2)

Whenever $\mathrm{Ind}(E_{\restriction_{C}})$ and
$\mathrm{Ind}(E_{\restriction_{D}})$ belong to $\mathcal{Z}(A)$, the exterior
angle between $C$ and $D$ can be derived from (3.3) using the following
expressions:

$\displaystyle\langle e_{C_{1}}-e_{2},e_{D_{1}}-e_{2}\rangle_{A_{1}}$
$\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\Big{[}(\mathrm{Ind}(E_{\restriction_{C}}))^{-2}(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}\sum_{i,i^{\prime}}\lambda_{i}e_{C}\mathrm{Ind}(F^{\prime})$
$\displaystyle\quad\times\,\sum_{j,k}\mu_{j}E(\mu_{j}^{*}\lambda_{i}^{*}\lambda_{i^{\prime}}\delta_{k})\delta_{k}^{*})e_{D}\lambda_{i^{\prime}}^{*}-1\Big{]},$
$\|e_{C_{1}}-e_{2}\|_{A_{2}}=\Big{\|}(\mathrm{Ind}(E_{1}))^{-1}\Big{[}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}\big{(}\sum_{i}\lambda_{i}F(\mathrm{Ind}(F))e_{C}\lambda_{i}^{*}\big{)}-1\Big{]}\Big{\|}^{\frac{1}{2}}$

and a similar expression for $\|e_{D_{1}}-e_{2}\|_{A_{2}}$.

###### Proof.

(1) follows immediately by substituting the expressions for $e_{C},e_{D}$, as
obtained in Proposition 2.7 (4),(5),(6), in the definition of interior angle
(3.2).

(2): Note that the dual conditional expectation $E_{1}:A_{1}\to A$ is of
finite index with a quasi-basis
$\\{\lambda_{i}e_{B}(\mathrm{Ind}(E))^{1/2}\\}$ \- see Remark 2.3(3). Further,
from Proposition 2.7(4), a quasi-basis for ${E_{1}}_{\restriction_{C_{1}}}$ is
given by $\\{w_{i}:=G(\lambda_{i}e_{B}(\mathrm{Ind}(E))^{1/2}):1\leq i\leq
n\\}$, where $G:A_{1}\to C_{1}$ is the conditional expectation as in the proof
of Proposition 2.7(7). Thus, by Proposition 2.7(4), we see that

$\displaystyle e_{C_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i}w_{i}e_{2}w_{i}^{*}$ $\displaystyle=$
$\displaystyle\sum_{i}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}(\mathrm{Ind}(E))^{1/2}e_{2}(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}e_{C}\lambda_{i}^{*},$

since $\mathrm{Ind}(E_{\restriction_{C}})\in\mathcal{Z}(A)\cap\mathcal{Z}(C)$
and $e_{C}\in C^{\prime}\cap C_{1}$. Thus,

$\displaystyle E_{2}(e_{C_{1}})-E_{2}(e_{2})$ $\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\left[\left(\sum_{i}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}e_{C}\lambda_{i}^{*}\right)-1\right]$
$\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\left[\left(\sum_{i}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-2}(\mathrm{Ind}(E))e_{C}\lambda_{i}^{*}\right)-1\right]$
$\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\left[\left(\sum_{i}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}(\mathrm{Ind}(F))e_{C}\lambda_{i}^{*}\right)-1\right]\quad\text{(by
Prop. \ref{eC-e1}(7a))}$ $\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\left[\left((\mathrm{Ind}(E_{\restriction_{C}}))^{-1}\sum_{i}\lambda_{i}F((\mathrm{Ind}(F))e_{C}\lambda_{i}^{*}\right)-1\right],$

which shows that

$\|e_{C_{1}}-e_{2}\|_{A_{2}}=\|(\mathrm{Ind}(E_{1}))^{-1}\Big{[}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}\big{(}\sum_{i}\lambda_{i}F(\mathrm{Ind}(F))e_{C}\lambda_{i}^{*}\big{)}-1\Big{]}\|^{\frac{1}{2}}.$

Further, as above, we have

$e_{D_{1}}=\sum_{i}\lambda_{i}e_{D}(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}(\mathrm{Ind}(E))^{1/2}e_{2}(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}e_{D}\lambda_{i}^{*};$

so that

$\displaystyle e_{C_{1}}e_{D_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i,i^{\prime}}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}(\mathrm{Ind}(E))^{1/2}E_{1}\left[(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}e_{C}\lambda_{i}^{*}\lambda_{i^{\prime}}e_{D}\right.$
$\displaystyle\qquad\times\left.(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}(\mathrm{Ind}(E))^{1/2}\right]e_{2}(\mathrm{Ind}(E))^{1/2}(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}e_{D}\lambda_{i^{\prime}}^{*}$
$\displaystyle=$
$\displaystyle\sum_{i,i^{\prime}}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}\,\mathrm{Ind}(F)\,E_{1}(e_{C}\lambda_{i}^{*}\lambda_{i^{\prime}}e_{D})(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}\mathrm{Ind}(F^{\prime})e_{2}e_{D}\lambda_{i^{\prime}}^{*},$

where the last equality holds because of Proposition 2.7 (7)(a). Then, we
obtain

$\displaystyle E_{2}(e_{C_{1}}e_{D_{1}})-E_{2}(e_{2})$ $\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\sum_{i,i^{\prime}}\lambda_{i}e_{C}(\mathrm{Ind}(E_{\restriction_{C}}))^{-1}(\mathrm{Ind}(F))E_{1}(e_{C}\lambda_{i}^{*}\lambda_{i^{\prime}}e_{D})(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}$
$\displaystyle\quad\times\,\mathrm{Ind}(F^{\prime})e_{D}\lambda_{i^{\prime}}^{*}-(\mathrm{Ind}(E_{1}))^{-1}$
$\displaystyle=$
$\displaystyle(\mathrm{Ind}(E_{1}))^{-1}\Big{[}(\mathrm{Ind}(E_{\restriction_{C}}))^{-2}(\mathrm{Ind}(E_{\restriction_{D}}))^{-1}\sum_{i,i^{\prime}}\lambda_{i}e_{C}\mathrm{Ind}(F^{\prime})$
$\displaystyle\quad\times\,\sum_{j,k}\mu_{j}E(\mu_{j}^{*}\lambda_{i}^{*}\lambda_{i^{\prime}}\delta_{k})\delta_{k}^{*})e_{D}\lambda_{i^{\prime}}^{*}-1\Big{]}.$

Since $\langle
e_{C_{1}}-e_{2},e_{D_{1}}-e_{2}\rangle_{A_{1}}=E_{2}(e_{C_{1}}e_{D_{1}})-E_{2}(e_{2}),$
we are done.∎

######  Remark 3.4.

A priori, for $C,D\in\mathrm{IMS}(B,A,E)\setminus\\{B\\}$, it is not clear
whether $\alpha(C,D)=0$ implies $C=D$ or not. Though, when $B\subset A$ is an
irreducible inclusion of simple unital $C^{*}$-algebras, then it is known to
be true - see [3, Proposition 5.10]. Also, this phenomenon holds for a certain
collection of intermediate subalgebras even in some non-irreducible set up as
we shall see ahead in Corollary 4.5.

##  4\. Possible values of the interior angle

Throughout this section, we let $A=M_{2}(\mathbb{C})$, $B=\mathbb{C}I_{2}$,
$\Delta=\\{\mathrm{diag}(\lambda,\mu):\lambda,\mu\in\mathbb{C}\\}$, $E:A\to B$
denote the canonical (tracial) conditional expectation given by

$E([a_{ij}])=\frac{(a_{11}+a_{22})}{2}I_{2}\text{ for }[a_{ij}]\in A$

and $F:A\to\Delta$ denote the conditional expectation given by
$F([a_{ij}])=\mathrm{diag}(a_{11},a_{22})$.

The following useful observations are standard - see [11, Example 2.4.5].

######  Lemma 4.1.

With running notations, the following hold:

  1. (1)

$E$ is a finite-index conditional expectation with a quasi-basis

$\\{\sqrt{2}e_{ij}:1\leq i,j\leq 2\\},$

where $\\{e_{ij}:1\leq i,j\leq 2\\}$ denotes the set of standard matrix units
of $M_{2}(\mathbb{C})$.

  2. (2)

$\mathrm{Ind}(E)=4$ and $E$ is the (unique) minimal conditional expectation
from $A$ onto $B$.

  3. (3)

The $C^{*}$-basic construction $A_{1}$ for $B\subset A$, with respect to the
conditional expectation $E$, can be identified with $M_{4}(\mathbb{C})$ and
the Jones projection $e_{1}$ corresponding to the conditional expectation $E$
is given by

$e_{1}=\frac{1}{2}\begin{bmatrix}1&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\
1&0&0&1\end{bmatrix}.$

  4. (4)

Identifying $M_{4}(\mathbb{C})$ with $M_{2}(\mathbb{C})\otimes
M_{2}(\mathbb{C})$, the dual conditional expectation $E_{1}:A_{1}\to A$ is
given by $E_{1}=\mathrm{id_{M_{2}}}\otimes E$; thus,

$E_{1}\left(X\right)=\begin{bmatrix}E(X_{(1,1)})&E(X_{(1,2)})\\\
E(X_{(2,1)})&E(X_{(2,2)})\end{bmatrix},X\in M_{4}(\mathbb{C}),$

where $X_{(i,j)}$ denotes the $(i,j)^{\text{th}}$ $2\times 2$ block of a
matrix $X\in M_{4}(\mathbb{C})$.

######  Lemma 4.2.

  1. (1)

$F$ has finite index with a quasi-basis $\\{e_{ij}:1\leq i,j\leq 2\\}$ and
scalar index equal to $2$.

  2. (2)

$\Delta\in\mathrm{IMS}(B,A,E)$ with respect to the conditional expectation $F$
and the corresponding Jones projection in $\Delta_{1}$ ($\subset
A_{1}=M_{4}(\mathbb{C})$) is given by

$e_{\Delta}=\begin{bmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&1\end{bmatrix}.$

######  Lemma 4.3.

For every unitary $u$ in $A$,

  1. (1)

the map $F_{u}:A\to u\Delta u^{*}$ given by $F_{u}=\mathrm{Ad}_{u}\circ
F\circ\mathrm{Ad}_{u^{*}}$ is a finite-index conditional expectation with a
quasi-basis $\\{e_{ij}u:1\leq i,j\leq 2\\}$ and $\mathrm{Ind}(F_{u})=2$;

  2. (2)

$D:=u\Delta u^{*}\in\mathrm{IMS}(B,A,E)$ with respect to the conditional
expectation $F_{u}$; and,

  3. (3)

if $u=[\lambda_{ij}]$, then the corresponding Jones projection in $D_{1}$
$(\subset A_{1}=M_{4}(\mathbb{C}))$ is given by $e_{D}=[x_{ij}]$, where

$x_{11}=|\lambda_{11}|^{4}+|\lambda_{12}|^{4},x_{12}=\lambda_{21}\bar{\lambda}_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2}),x_{14}=2|\lambda_{11}|^{2}|\lambda_{12}|^{2},$
$x_{22}=2|\lambda_{11}|^{2}|\lambda_{21}|^{2},x_{23}=2\bar{\lambda}_{21}^{2}\lambda_{11}^{2}$

and the remaining entries are given by
$x_{12}=\bar{x_{13}}=\bar{x_{21}}={x_{31}}=-\bar{x_{24}}=-x_{42}$,
$x_{41}=x_{14}$, $x_{33}=x_{22}$, $x_{32}=\bar{x_{23}}$ and $x_{44}=x_{11}$.

###### Proof.

(1): Clearly, the map $F_{u}:A\to D$ is a conditional expectation and we can
easily verify that

$x=\sum_{i,j}e_{ij}uF_{u}(u^{*}e_{ij}^{*}x)=\sum_{i,j}F_{u}(xe_{ij}u)u^{*}e_{ij}^{*}$

for all $x\in A$. Thus, $\\{e_{ij}u:1\leq i,j\leq 2\\}$ is a quasi-basis for
$F_{u}$ and $\mathrm{Ind}(F_{u})=2=\mathrm{Ind}(F)$.

Since $E$ satisfies the tracial property and $\mathcal{N}_{A}(B)=\mathcal{A}$,
(2) follows from Lemma 2.8.

(3): After some routine calculation, for any $\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in A$, we obtain

$F_{u}\left(\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\right)=\begin{bmatrix}x|\lambda_{11}|^{2}+y|\lambda_{12}|^{2}&x\bar{\lambda}_{21}\lambda_{11}+y\bar{\lambda}_{22}\lambda_{12}\\\
x\lambda_{21}\bar{\lambda}_{11}+y\lambda_{22}\bar{\lambda}_{12}&x|\lambda_{12}|^{2}+y|\lambda_{11}|^{2}\end{bmatrix},$

where
$x=a|\lambda_{11}|^{2}+d|\lambda_{12}|^{2}+b\lambda_{21}\bar{\lambda}_{11}+c\bar{\lambda}_{21}\lambda_{11}$
and
$y=a|\lambda_{12}|^{2}+d|\lambda_{11}|^{2}+b\lambda_{22}\bar{\lambda}_{12}+c\bar{\lambda}_{22}\lambda_{12}$.
Since $u$ is a unitary, we have
$\bar{\lambda}_{12}\lambda_{22}=-(\bar{\lambda}_{11}\lambda_{21})$,
$\lambda_{12}\bar{\lambda}_{22}=-(\lambda_{11}\bar{\lambda}_{21})$,
$|\lambda_{11}|^{2}=|\lambda_{22}|^{2}$ and
$|\lambda_{12}|^{2}=|\lambda_{21}|^{2}$; thus, we further deduce that

$\displaystyle x|\lambda_{11}|^{2}+y|\lambda_{12}|^{2}$ $\displaystyle=$
$\displaystyle
a(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})+2d|\lambda_{11}|^{2}|\lambda_{12}|^{2}+c\bar{\lambda}_{21}\lambda_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})$
$\displaystyle\qquad+b\lambda_{21}\bar{\lambda}_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2});\text{
and }$ $\displaystyle x|\lambda_{12}|^{2}+y|\lambda_{11}|^{2}$
$\displaystyle=$ $\displaystyle
2a|\lambda_{11}|^{2}|\lambda_{12}|^{2}+d(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})+b\lambda_{21}\bar{\lambda}_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})$
$\displaystyle\qquad+c\bar{\lambda}_{21}\lambda_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2}).$

Then, using the above expression for $F_{u}([a_{ij}])$ for $[a_{ij}]\in A$ and
the matrix $e_{D}$ as given in the statement, we can easily verify that

  1. (i)

$e_{D}xe_{D}=F_{u}(x)e_{D}$ and

  2. (ii)

$e_{D}(x)=F_{u}(x)$

for all $x\in A$. This completes the proof. ∎

We are now all set to derive a concrete expression for the interior angle
between $\Delta$ and its conjugate $u\Delta u^{*}$, in terms of the entries of
$u$.

######  Theorem 4.4.

If $u=[\lambda_{ij}]\in U(2)$, then

(4.1) $\cos(\alpha(\Delta,u\Delta
u^{*}))=\sqrt{1-(2|\lambda_{11}||\lambda_{12}|)^{4}}.$

###### Proof.

Let $D=u\Delta u^{*}$. From the matrix expressions of $e_{1}$, $e_{\Delta}$
and $e_{D}$ obtained above, we easily see that
$E_{1}(e_{1})=\frac{1}{4}I_{2}$, $E_{1}(e_{\Delta})=\frac{1}{2}I_{2}$,
$E_{1}(e_{D})=\frac{1}{2}I_{2}$ and

$E_{1}(e_{\Delta}e_{D})=\begin{bmatrix}{(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})}/{2}&\bar{\lambda}_{21}\lambda_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})/2\\\
\lambda_{21}\bar{\lambda}_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})/2&(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})/2\end{bmatrix}.$

Thus,

$\|e_{\Delta}-e_{1}\|_{A_{1}}=\|E(e_{\Delta}-e_{1})\|=\frac{1}{2}=\|E(e_{D}-e_{1})\|=\|e_{D}-e_{1}\|_{A_{1}}.$

Next, we calculate $\|E_{1}(e_{\Delta}e_{D}-e_{1})\|$. Let

$T=E_{1}(e_{\Delta}e_{D}-e_{1})=\begin{bmatrix}(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})/2-1/4&\bar{\lambda}_{21}\lambda_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})/2\\\
\lambda_{21}\bar{\lambda}_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})/2&(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})/2-1/4\end{bmatrix}.$

Note that, $T^{*}T$ turns out to be a scalar matrix with eigen value
$\lambda$, where

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\left(\frac{|\lambda_{11}|^{4}+|\lambda_{12}|^{4}}{2}-\frac{1}{4}\right)^{2}+\frac{|\lambda_{11}|^{2}|\lambda_{21}|^{2}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})^{2}}{4}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left(2(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})-1\right)^{2}+\frac{|\lambda_{11}|^{2}|\lambda_{21}|^{2}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})^{2}}{4}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left(2(|\lambda_{11}|^{4}+|\lambda_{12}|^{4})-(|\lambda_{11}|^{2}+|\lambda_{12}|^{2})^{2}\right)^{2}+\frac{|\lambda_{11}|^{2}|\lambda_{12}|^{2}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})^{2}}{4}$
(since $|\lambda_{11}|^{2}+|\lambda_{12}|^{2}=1$ and
$|\lambda_{21}|=|\lambda_{12}|$) $\displaystyle=$
$\displaystyle\left(\frac{|\lambda_{11}|^{2}-|\lambda_{12}|^{2}}{4}\right)^{2}+\frac{|\lambda_{11}|^{2}|\lambda_{12}|^{2}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})^{2}}{4}$
$\displaystyle=$
$\displaystyle\left(\frac{(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})}{4}\right)^{2}\left(1+4|\lambda_{11}|^{2}|\lambda_{12}|^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left((|\lambda_{11}|^{2}+|\lambda_{12}|^{2})^{2}-4|\lambda_{11}|^{2}|\lambda_{12}|^{2}\right)\left((1+4|\lambda_{11}|^{2}|\lambda_{12}|^{2})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{16}\left(1-(2|\lambda_{11}||\lambda_{12}|)^{4}\right).$

Thus,

$\|\langle
e_{\Delta}-e_{1}\;e_{D}-e_{1}\rangle_{A}\|=\|E_{1}(e_{\Delta}e_{D}-e_{1})\|=\|T\|=\left(\sqrt{1-(2|\lambda_{11}||\lambda_{12}|)^{4}}\right)/4.$

Finally, substituting the above values of $\|e_{\Delta}-e_{1}\|_{A_{1}}$,
$\|e_{D}-e_{1}\|_{A_{1}}$ and $\|\langle
e_{\Delta}-e_{1}\;e_{D}-e_{1}\rangle_{A}\|$ in (3.2), we obtain

$\cos(\alpha(\Delta,u\Delta
u^{*}))=\sqrt{1-(2|\lambda_{11}||\lambda_{12}|)^{4}}.$

∎

Recall that a unitary matrix whose all entries have the same modulus is called
a Hadamard matrix. Also, if $(B,C,D,A)$ is a quadruple of finite von Neumann
algebras (i.e., $B\subset C,D\subset A$) with a faithful normal tracial state
$\tau:A\to\mathbb{C}$, then $(B,C,D,A)$ is said to be a commuting square if
$E^{A}_{C}E^{A}_{D}=E^{A}_{B}=E^{A}_{D}E^{A}_{C}$, where $E^{A}_{X}:A\to X$
denotes the unique $\tau$-preserving conditional expectation from $A$ onto any
von Neumann subalgebra $X$ of $A$.

######  Corollary 4.5.

Let $u=\in U(2)$. Then,

  1. (1)

$\alpha(\Delta,u\Delta u^{*})=\frac{\pi}{2}$ if and only if $u$ is a Hadamard
matrix if and only if $(B,\Delta,u^{*}\Delta u,A)$ is a commuting square; and,

  2. (2)

if $u=[\lambda_{ij}]$, then $\alpha(\Delta,u\Delta u^{*})=0$ if and only if
either $u\in\Delta$ or $\lambda_{11}=0=\lambda_{22}$.
In particular, $\alpha(\Delta,u\Delta u^{*})=0$ if and only if $\Delta=u\Delta
u^{*}$.

###### Proof.

(1): From Theorem 4.4, we observe that

$\displaystyle\cos(\alpha(\Delta,u\Delta u^{*}))=0$
$\displaystyle\Leftrightarrow$
$\displaystyle\sqrt{1-(2|\lambda_{11}||\lambda_{12}|)^{4}}=0$
$\displaystyle\Leftrightarrow$
$\displaystyle|\lambda_{11}||\lambda_{12}|=\frac{1}{2}$
$\displaystyle\Leftrightarrow$
$\displaystyle|\lambda_{11}|=|\lambda_{12}|\qquad\text{(since
$|\lambda_{11}|^{2}+|\lambda_{12}|^{2}$ = 1)}$ $\displaystyle\Leftrightarrow$
$\displaystyle|\lambda_{11}|=|\lambda_{12}|=|\lambda_{21}|=|\lambda_{22}|$
$\displaystyle\Leftrightarrow$ $\displaystyle\text{ $u$ is Hadamard Matrix}.$

Note that $F:M_{2}\to\Delta$ and $F_{u}:M_{2}\to u\Delta u^{*}$ are the unique
trace preserving conditional expectations. And, it is a well known fact - see,
for instance, [8, $\S 5.2.2$] \- that $(B,\Delta,u^{*}\Delta u,A)$ is a
commuting square if an only if $u$ is a Hadamard matrix.

(2): Again, from Theorem 4.4,

$\displaystyle\cos(\alpha(\Delta,u\Delta u^{*}))=1$
$\displaystyle\Leftrightarrow$ $\displaystyle|\lambda_{11}||\lambda_{12}|=0$
$\displaystyle\Leftrightarrow$
$\displaystyle|\lambda_{11}|=0\quad\text{or}\quad|\lambda_{12}|=0$
$\displaystyle\Leftrightarrow$
$\displaystyle|\lambda_{11}|=0=|\lambda_{22}|\quad\text{or}\quad\text{ $u$ is
diagonal}.\qquad\text{(as $u$ is unitary)}$

∎

We can now deduce our assertion that the interior angle attains all values in
$[0,\pi/2]$.

######  Corollary 4.6.

$\\{\alpha(\Delta,u\Delta u^{*}):u\in U(2)\\}=\left[0,\frac{\pi}{2}\right].$

###### Proof.

Note that, for each $u=[\lambda_{ij}]\in U(2)$,
$0\leq(2|\lambda_{11}||\lambda_{12}|)^{4}\leq 1$. Thus, we can define a map
$\varphi:U(2)\to[0,1]$ given by

$\varphi\left([\lambda_{ij}]\right)=\sqrt{1-(2|\lambda_{11}||\lambda_{12}|)^{4}}.$

Clearly, $\varphi$ is a continuous function. Since $U(2)$ is connected, it
follows that $\varphi(U(2))$ is also connected. Note that, from Corollary 4.5,
we have $\varphi(u)=0$ for any complex Hadamard matrix $u\in U(2)$ and
$\varphi(I_{2})=1$. Hence, $\varphi(U(2))=[0,1]$. In particular, in view of
Theorem 4.4, we obtain

$\\{\alpha(\Delta,u\Delta u^{*}):u\in U(2)\\}=\left[0,\frac{\pi}{2}\right],$

as was desired. ∎

######  Corollary 4.7.

There exist $C,D\in\mathrm{IMS}(B,A,E)$ such that $e_{C}e_{D}\neq e_{D}e_{C}$.

###### Proof.

Fix a $u=[\lambda_{ij}]\in U(2)$ and let $C=\Delta$ and $D=u\Delta u^{*}$.
Then, both $C,D\in\mathrm{IMS}(A,B,E)$ and using the values of $e_{C}$ and
$e_{D}$ from Lemmas 4.2 and 4.3, we obtain

$e_{C}e_{D}=\begin{bmatrix}|\lambda_{11}|^{4}+|\lambda_{12}|^{4}&\lambda_{21}\bar{\lambda}_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})&\bar{\lambda}_{21}\lambda_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})&2|\lambda_{11}|^{2}|\lambda_{12}|^{2}\\\
0&0&0&0\\\ 0&0&0&0\\\
2|\lambda_{11}|^{2}|\lambda_{12}|^{2}&\lambda_{21}\bar{\lambda}_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})&\bar{\lambda}_{21}\lambda_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})&|\lambda_{11}|^{4}+|\lambda_{12}|^{4}\end{bmatrix}$

and

$e_{D}e_{C}=\begin{bmatrix}|\lambda_{11}|^{4}+|\lambda_{12}|^{4}&0&0&2|\lambda_{11}|^{2}|\lambda_{12}|^{2}\\\
\bar{\lambda}_{21}\lambda_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})&0&0&\bar{\lambda}_{21}\lambda_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})\\\
\lambda_{21}\bar{\lambda}_{11}(|\lambda_{11}|^{2}-|\lambda_{12}|^{2})&0&0&\lambda_{21}\bar{\lambda}_{11}(|\lambda_{12}|^{2}-|\lambda_{11}|^{2})\\\
2|\lambda_{11}|^{2}|\lambda_{12}|^{2}&0&0&|\lambda_{11}|^{4}+|\lambda_{12}|^{4}\end{bmatrix}.$

Thus, if $u$ is neither a diagonal matrix nor a Hadamard matrix nor
$\lambda_{11}=0=\lambda_{22}$, then we see that $C\neq D$ and $e_{C}e_{D}\neq
e_{D}e_{C}$. ∎

##  5\. Angles between intermediate crossed product subalgebras of crossed
product inclusions

Recall that if a countable discrete group $G$ acts on a unital $C^{*}$-algebra
$P$ via a map $\alpha:G\to\mathrm{Aut}(P)$, then the space $C_{c}(G,P)$
consisting of compactly supported $P$-valued functions on $G$ can be
identified with the space $\\{\sum_{\text{finite}}a_{g}g:a_{g}\in P,g\in G\\}$
of formal finite sums and is a unital $*$-algebra with respect to (the so
called twisted) multiplication given by the convolution operation

$\left(\sum_{s\in I}a_{s}s\right)\left(\sum_{t\in J}b_{t}t\right)=\sum_{s\in
I,t\in J}a_{s}\alpha_{s}(b_{t})st$

and involution given by

$\left(\sum_{s\in I}a_{s}s\right)^{*}=\sum_{s\in
I}\alpha_{s^{-1}}(a_{s}^{*})s^{-1}$

for any two finite sets $I$ and $J$ in $G$. Further, the reduced crossed
product $P\rtimes_{\alpha,r}G$ and the universal crossed product
$P\rtimes_{\alpha}G$ are defined, respectively, as the completions of
$C_{c}(G,P)$ with respect to the reduced norm

$\left\|\sum_{\text{finite}}a_{g}g\right\|_{r}:=\left\|\sum_{\text{finite}}\pi(a_{g})(1\otimes\lambda_{g})\right\|_{B(H\otimes\ell^{2}(G))},$

where $P\subset B(H)$ is a (equivalently, any) fixed faithful representation
of $A$, $\lambda:G\to B(\ell^{2}(G))$ is the left regular representation and
$\pi:A\to B(H\otimes\ell^{2}(G))$ is the representation satisfying
$\pi(a)(\xi\otimes\delta_{g})=\alpha_{g^{-1}}(a)(\xi)\otimes\delta_{g}$ for
all $\xi\in H$ and $g\in G$; and the universal norm

$\|x\|_{u}:=\sup_{\pi}\|\pi(x)\|\text{ for }x\in C_{c}(G,P),$

where the supremum runs over all (cyclic) $*$-homomorphisms $\pi:C_{c}(G,P)\to
B(H)$, respectively. We suggest the reader to see [4, 12] for more on crossed
products.

When $G$ is a finite group, then it is well known that the reduced and
universal norms coincide on $C_{c}(G,P)$ and $C_{c}(G,P)$ is complete with the
common norm; thus, $P\rtimes_{\alpha,r}G=C(G,P)=P\rtimes_{\alpha}G$ (as
$*$-algebras).

In this section, analogous to [2, Proposition 2.7], we derived a concrete
value for the interior angle between intermediate crossed product subalgebras
of an inclusion of crossed product algebras.

The following important observations are well known.

######  Proposition 5.1.

[5, 9] Let $G$ be a countable discrete group and $H$ be its subgroup. Let $P$
be a unital $C^{*}$-algebra such that $G$ acts on $P$ via a map
$\alpha:G\to\mathrm{Aut}(P)$. Let $A:=P\rtimes_{\alpha,r}G$ (resp.,
$A:=P\rtimes G$) and $B:=P\rtimes_{\alpha,r}H$ (resp., $B:=P\rtimes H$). Then,

  1. (1)

the canonical injective $*$-homomorphism

$C_{c}(H,P)\ni\sum_{\mathrm{finite}}a_{h}h\mapsto\sum_{\mathrm{finite}}a_{h}h\in
C_{c}(G,P)$

extends to an injective $*$-homomorphism from $B$ into $A$; and

  2. (2)

the natural map

$C_{c}(G,P)\ni\sum_{\mathrm{finite}}a_{g}g\mapsto\sum a_{h}h\in C_{c}(H,P)$

extends to a conditional expectation $E:A\to B$.

Moreover, $E$ has finite index if and only if $[G:H]<\infty$ and in that case
a quasi-basis for $E$ is given by $\\{g_{i}:1\leq i\leq[G:H]\\}$ for any set
$\\{g_{i}\\}$ of left coset representatives of $H$ in $G$ and $E$ has scalar
index equal to $[G:H]$.

###### Proof.

(1) follows from [5] (also see [9, Remark 3.2]) and the first part of the
proof of [9, Proposition 3.1], respectively.

(2): Consider the canonical $C_{c}(H,P)$-bilinear projection
$E_{0}:C_{c}(G,P)\to C_{c}(H,P)$ given by

$E_{0}\left(\sum_{\text{finite}}a_{g}g\right)=\sum a_{h}h.$

Then, from [5] (also see [9, Remark 3.2]) and [9, Proposition 3.1, Remark
3.2], respectively, it follows that $E_{0}$ extends to a conditional
expectation from $A$ onto $B$. Also, from [9, Theorem 3.4], it follows that
$E$ has finite index (with a quasi-basis as in the statement) if and only
$[G:H]<\infty$. ∎

######  Proposition 5.2.

Let $G,H,P,\alpha,A,B$ and $E$ be as in Proposition 5.1 with $[G:H]<\infty$
and $\\{g_{i}:1\leq i\leq[G:H]\\}$ be a set of left coset representatives of
$H$ in $G$. Let $K$ and $L$ be proper intermediate subgroups of $H\subset G$
and let $C:=P\rtimes_{\alpha,r}K$ (resp., $P\rtimes_{\alpha}K$) and
$D:=P\rtimes_{\alpha,r}L$ (resp., $P\rtimes_{\alpha}L$). Then,
$C,D\in\mathrm{IMS}(B,A,E)\setminus\\{A,B\\}$ and the interior angle between
them is given by

$\cos(\alpha(C,D))=\frac{[K\cap L:H]-1}{\sqrt{[K:H]-1}\sqrt{[L:H]-1}}.$

###### Proof.

Note that, $B\subset C,D\subset A$, by Proposition 5.1. Also, $[G:K]$ and
$[G:L]$ are both finite as $[G:H]$ is finite. So, $C,D\in\mathrm{IMS(B,A,E)}$
with respect to the natural finite-index conditional expectations guaranteed
by Proposition 5.1.

Fix left coset representatives $\\{k_{r}:1\leq r\leq[K:H]\\}$ and
$\\{{l_{s}}:1\leq s\leq[L:H]\\}$ of $H$ in $K$ and $L$, respectively. Then, it
is readily seen that $E_{\restriction_{C}}:C\to B$ and
$E_{\restriction_{D}}:D\to B$ have quasi-bases $\\{{k_{r}}:1\leq
r\leq[K:H]\\}$ and $\\{{l_{s}}:1\leq s\leq[L:H]\\}$, respectively. Then, from
(3.5), we obtain

$\displaystyle\cos(\alpha(C,D))$ $\displaystyle=$
$\displaystyle\frac{\|(\sum_{r,s}k_{r}E(k_{r}^{*}l_{s})l_{s}^{*})-1\|}{\|\sqrt{[K:H]-1}\|\|\sqrt{[L:H]-1}\|}$
$\displaystyle=$
$\displaystyle\frac{\|\left(\sum_{\\{r,s:(k_{r}H)\cap(l_{s}H)\neq\phi\\}}k_{r}k_{r}^{*}l_{s}l_{s}^{*}\right)-1\|}{\sqrt{[K:H]-1}\sqrt{[L:H]-1}}$
$\displaystyle=$ $\displaystyle\frac{[(K\cap
L):H]-1}{\sqrt{[K:H]-1}\sqrt{[L:H]-1}},$

where the last equality holds because the map

$\\{(r,s):k_{r}H\cap l_{s}H\neq\emptyset\\}\ni(r,s)\mapsto
k_{r}H=l_{s}H\in(K\cap L)/H$

is a bijection. ∎

######  Corollary 5.3.

Let the notations be as in Proposition 5.2. Then,

  1. (1)

$\alpha(C,D)=\frac{\pi}{2}$ if and only if $K\cap L=H$; and,

  2. (2)

$\alpha(C,D)=0$ if and only if $K=L$.

In particular, if $C_{g}:=P\rtimes_{\alpha}(g^{-1}Kg)$ (resp.,
$P\rtimes_{\alpha,r}(g^{-1}Kg)$) then $\alpha(C,C_{g})=0$ for all $g\in G$ if
and only if $K$ is normal in $G$.

###### Proof.

(1) is straight forward and, for (2), we just need to show the necessity.

Note that $\alpha(C,D)=0$ implies that $\frac{[(K\cap
L):H]-1}{\sqrt{[K:H]-1}\sqrt{[L:H]-1}}=1$, which then implies that

$\left(\sqrt{\frac{[(K\cap L):H]-1}{[K:H]-1}}\right)\left(\sqrt{\frac{[(K\cap
L):H]-1}{[L:H]-1}}\right)=1$

and that $[K\cap L:H]\neq 1$. Note that

$0<\frac{{[(K\cap L):H]-1}}{{[K:H]-1}},\frac{{[(K\cap L):H]-1}}{{[L:H]-1}}\leq
1;$

so, it follows that

$\frac{{[(K\cap L):H]-1}}{{[K:H]-1}}=1=\frac{{[(K\cap L):H]-1}}{{[L:H]-1}}.$

Hence, $K=K\cap L=L$. ∎

Recall that for a subgroup $H$ of a group $G$, its normalizer is given by

$\mathcal{N}_{G}(H)=\\{g\in G:g^{-1}Hg=H\\}.$

######  Corollary 5.4.

Let $G,H$ and $K$ be as in Proposition 5.2. If $g\in\mathcal{N}_{G}(H)$, then
$\alpha(C,C_{g})=0$ if and only if $g\in\mathcal{N}_{G}(K)$, where $C_{g}$ is
same as in Corollary 5.3

###### Proof.

Let $L:=g^{-1}Kg$. Since $[K:H]=[L:H]$, from Expression (5.1), we obtain

$\cos(\alpha(C,C_{g}))=\frac{[(K\cap L):H]-1}{[K:H]-1}.$

Thus, $\alpha(C,C_{g}))=0$ if and only if $K\cap(g^{-1}Kg)=K$ if and only if
$g\in\mathcal{N}_{G}(K)$. ∎

Note that, if $P=\mathbb{C}$ and $\alpha:G\to\mathrm{Aut}(\mathbb{C})$ is the
trivial representation, then we know that
$C^{*}_{r}(G)=\mathbb{C}\rtimes_{\alpha,r}G$ and
$C^{*}(G)=\mathbb{C}\rtimes_{\alpha}G$. Thus, we readily deduced the
following:

######  Corollary 5.5.

Let $G$ be a countable discrete group with subgroups $H,K$ and $L$ such that
$H\subseteq K\cap L$, $H\neq K,L$ and $[G:H]<\infty$. Let $A:=C^{*}_{r}(G)$
(resp., $C^{*}(G)$), $B:=C^{*}_{r}(H)$ (resp., $C^{*}(H)$), $C:=C^{*}_{r}(K)$
(resp., $C^{*}(K)$) and $D:=C^{*}_{r}(L)$ (resp., $C^{*}(L)$). Then,
$C,D\in\mathrm{IMS}(B,A,E)$ and

(5.1) $\cos(\alpha(C,D))=\frac{[K\cap L:H]-1}{\sqrt{[K:H]-1}\sqrt{[L:H]-1}},$

where $E:A\to B$ is the conditional expectation as in Proposition 5.1 with
$P=\mathbb{C}$.

######  Example 5.6.

Let
$G=\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$.
Consider its subgroups
$K=\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{5}\oplus(0)$ ,
$L=(0)\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{5}\oplus(0)$ and
$H=(0)\oplus(0)\oplus\mathbb{Z}_{5}\oplus(0)$. Then,

$\cos\left(\alpha(\mathbb{C}[K],\mathbb{C}[L])\right)=\frac{1}{2}.$

Thus, $\alpha(\mathbb{C}[K],\mathbb{C}[L])$ = $\pi/3$.

In particular, this illustrates that if $B\neq C\subsetneq D\subsetneq A$,
then $\alpha(C,D)$ need not be $0$.

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