2.10.1: de Broglie paradox in $B({\mathbb C}^2)$
Axiom1 (measurement in $\S$2.7) includes the paradox ( that is, so called de Broglie paradox :"there is something faster than light"). In what follows, we shall explain de Broglie paradox in $B({\mathbb C}^2)$, though the original idea is mentioned in $B(L^2({\mathbb R}))$ (cf. $\S$11.2). Also, it should be noted that the argument below is essentially the same as the Stern=Gerlach experiment.
Example 2.37 [de Broglie paradox in $B({\mathbb C}^2)$ ] Let $H$ be a two dimensional Hilbert space, i.e., $H={\mathbb C}^2$. Consider the quantum basic structure:
\begin{align*} [B({\mathbb C}^2)\subseteq B({\mathbb C}^2 ) \subseteq {B({\mathbb C}^2 )}] \end{align*} Now consider the situation in the following Figure 2.11.Thus, we have the state $\rho = u \rangle \langle u $ $(\in {\frak S}^p(B({\mathbb C}^2)))$. Let $U (\in B({\mathbb C}^2 ))$ be an unitary operator such that
\begin{align*} U = \left[\begin{array}{ll} 1 & 0 \\ 0 & e^{i\pi/2 } \end{array}\right] \quad \end{align*} and let $\Phi: B({\mathbb C}^2) \to B({\mathbb C}^2) $ be the homomorphism such that \begin{align*} \Phi(F) = U^* F U \qquad (\forall F \in B({\mathbb C}^2) ) \end{align*}Consider the observable ${\mathsf O}_f=(\{1,2\}, 2^{\{1,2\}}, F)$ in $B({\mathbb C}^2 )$ such that
\begin{align*} & F(\{1\}) = f_1 \rangle \langle f_1  , \quad F(\{ 2 \}) = f_2 \rangle \langle f_2  \end{align*} and thus, define the observable $\Phi {\mathsf O}_f=(\{1,2\}, 2^{\{1,2\}}, \Phi F)$ by \begin{align*} \Phi F (\Xi) = U^* F(\Xi ) U \qquad (\forall \Xi \subseteq \{1, 2\} ) \end{align*}Let us explain Figure 2.38. The photon P with the state $u=\frac{1}{\sqrt{2}}( f_1+f_2 )$ ( precisely, $u \rangle \langle u $ ) rushed into the halfmirror 1
$(A_1):$  the $f_1$ part in $u$ passes through halfmirror 1, and goes along course 1 to photon detector $D_1$. 
$(A_2):$  the $f_2$ part in $u$ rebounds on halfmirror 1 (and strictly saying, the $f_2$ changes to ${\sqrt{1}}f_2$, we are not concerned with it ), and goes along course 2 to photon detector $D_2$. 
$(B):$  The probability that a $\left[\begin{array} \mbox{\rm measured\;\; value }1 \\ \mbox{measured value }2 \end{array}\right]$ is obtained by the measurement $ {\mathsf M}_{B({\mathbb C}^2)} ( \Phi{\mathsf O}_f, S_{[\rho]} ) $ is given by \begin{align*} \left[\begin{array}{l} \mbox{Tr}(\rho \cdot \Phi F(\{1\}) ) \\ \mbox{Tr}(\rho \cdot \Phi F(\{2\}) ) \end{array}\right] = \left[\begin{array}{l} \langle u, \Phi F(\{1\})u \rangle \\ \langle u, \Phi F(\{2\}) u \rangle \end{array}\right] = \left[\begin{array}{l} \langle Uu, F(\{1\}) Uu \rangle \\ \langle Uu, F(\{2\}) U u \rangle \end{array}\right] = \left[\begin{array}{l}  \langle u, f_1 \rangle^2 \\  \langle u, f_2 \rangle^2 \end{array}\right] = \left[\begin{array}{l} \frac{1}{2} \\ \frac{1}{2} \end{array}\right] \end{align*} 
$(C):$  Assume that

$\qquad \qquad $Fig. 1.1; history of worlddescriptions 
Or
be bothered in physics D in Figure 1.1 ( in $\S$1.1) !
$\fbox{Note 2.8}$  The de Broglie paradox (i.e., there may be something faster than light ) always appears in quantum mechanics. For example, the readers should confirm that it appears in Example 2.36 (SchternGerlach experiment). I think that
