Taken from Condition Expectation in Quantum Probabilty by Denes Petz.

In quantum probability there are a number of fundamental questions that ask how faithfully can one quantise classical probability. Suppose that $(\Omega,\mathcal{S},P)$ is a (classical) probability space and $\mathcal{G}\subset\mathcal{A}$ a sub-$\sigma$-algebra. The conditional expectation of some integrable function $f$ (with respect to some $L$-space) relative to $\mathcal{G}$ is the orthogonal projection onto the closed subspace $L(\mathcal{G})$:

$\mathbb{E}^{\mathcal{G}}:L(\mathcal{A})\rightarrow L(\mathcal{G})$$f\mapsto \mathbb{E}(f|\mathcal{G})$.

Suppose now that $(A,\rho)$ is a quantum probability space and that $B$ is some C*-subalgebra of $A$. Can we always define a conditional expectation with respect to $B$? The answer turns out to be not always, although this paper gives sufficient conditions for the existence of such a projection. Briefly, things work the other way around. Distinguished states give rise to quantum conditional expectations — and these conditional expectations define a subalgebra. We can’t necessarily start with a subalgebra and find the state which gives rise to it — Theorem 2 gives necessary conditions in which this approach does work.

Let $(\Omega,\mathcal{S},P)$ be a probability space and $\mathcal{S}_0$ a sub-$\sigma$-algebra of $\mathcal{S}$. In 1954 Moy observed that the property

$\mathbb{E}(f\mathbb{E}(g))=\mathbb{E}(f)\mathbb{E}(g)$   ($f,\,g$ are bounded)

plays a crucial role in the characterisation of the conditional expectation operator

$\mathbb{E}:L^1(\mathcal{S},P)\rightarrow L^1(\mathcal{S}_0,P)$.

More or less at the same time Nakamura, Turumaru and Umegaki created the algebraic form of the conditional expectation (i.e. with respect to quantum probability). Here we deal with states on a von Neumann algebra $A$. A state $\rho$ is called tracial if $\rho(ab)=\rho(ba)$ for all $a,\,b\in A$. When $A$ is finite dimensional (and so it is a direct sum of some matrix algebras), then it always possesses a tracial state $\tau$. In this case every state $\rho$ has the form $\rho(a)=\tau(x_\rho a)$ with a unique $x_\rho\in A^+$, called the density of $\rho$. Due to the lack of unbounded operators the finite dimensional case is rather easily computable.

Let $\rho$ be a fixed state on $A$ and $E:A\rightarrow A$ a linear mapping such that

1. $E(a)\geq 0$ if $a\geq 0$.
2. $E(I_A)=I_A$.
3. $\rho(E(a))=\rho(a)$ for all $a\in A$.
4. $E(a E(b))=E(a)E(b)$ for every $a,\,b\in A$.

Property 3. implies that $E$ is weak continuous (I can’t seem to show this) and hence $A_0=\{a\in A:E(a)=a\}$ is weak closed (as the kernel of the continuous map $\rho\circ (E-I_A)$). It also follows easily that $A_0$ is an algebra ($\checkmark$) and the range of $E$

This just implies that $E$ is a projection — it’s nice to know we don’t have to assume this. I can prove that $M_0\in\text{ran }E$ easily. Suppose now that $a\in\text{ran }E$. We can assume that $a$ is positive because everything in sight is linear. By assumption there exists a $b\in A$ such that $E(b)=a$. By linearity, $b$ must be positive also. Now, using property 3.

$\rho(a)=\rho(E(b))=\rho(b)$.

As $\rho$ is faithful this implies that $a=b$ so that $E(a)=a$ as required.

Such an $E$ is called a $\rho$preserving conditional expectation onto $A_0$.

## Theorem 1

Let $\tau$ be a tracial state on $A$ and von Neuman subalgebra of $A$. Then there exists a unique conditional expectation preserving $\tau$ and mapping $A$ onto $A_0$.

Proof : The reference proffered is Umegaki although I can’t find the proof of this theorem in it $\bullet$

$\langle a,b\rangle=\tau(b^*a)$ defines an inner product on $A$ (the Hilbert-Schmidt inner product). To avoid the completion process assume that $A$ is finite dimensional and so it is a Hilbert space.  $A_0$ is a closed subspace of $A$  and we consider the the orthogonal projection $E_{A_0}:A\rightarrow A_0$. The properties 1. to 4. are verified straightforwardly.

$A_0$ is defined in terms of the conditional expectation associated to $\tau$. What Petz does here is prove that $E=E_{A_0}$.

The case of tracial $\tau$ reminds very much of the commutative situation. From this point of view the following example may be surprising. If

$\displaystyle A=M_2(\mathbb{C})\otimes M_2(\mathbb{C})\cong M_4(\mathbb{C})$

and $A_0=M_2(\mathbb{C})\otimes\mathbb{C}\cong M_2(\mathbb{C})$, then for a state $\rho$ on $A$ the conditional expectation preserving $\rho$, $E:A\rightarrow A_0$ exists if and only if $\rho=\rho_1\otimes\rho_2$. The Takesaki theorem clarifies the existence of the conditional expectation by means of the modular group of the state.

## Theorem 2 (Takesaki)

Let $A_0$ be a von Neumann subalgebra of $A$ and $\rho$ a state on $A$. Then the $\rho$-preserving conditional expectation from $A$ to $A_0$ exists if and only if $A_0$ is stable under the modular group of $\rho$. If the conditional expectation exists then it is unique.

### Modular Groups

It would probably be wise to find out about these objects before I say any more. This is taken from here and has zero rigour so we are just getting a flavour and then we’ll move on.

Suppose that $A$ is a von Neumann algebra acting on a Hilbert space $H$, and $x$ is a separating, cyclic vector of $H$ of norm $1$. We write $\rho$ for the state $\rho(a)=\langle ax,x\rangle$ on $A$, so that $H$ is constructed from $\rho$ via the GNS construction. We can define an antilinear operator $T_0$ on $H$ with domain $Ax$ by setting

$T_0(ax)=a^*x$ for all $a\in A$,

and similarly we can define an antilinear operator $S_0$ on $H$ with domain $A'$ by setting

$S_0(a)=a^*x$ for all $a\in A'$.

These operators are closable, and we denote their closures by $T$ and $S$. They have polar decompositions

$T=J|T|=J\Delta^{1/2}=\Delta^{-1/2}J$,

$S=J|S|=J\Delta^{-1/2}=\Delta^{1/2}J$.

where $J=J^{-1}=J^*$ is an antilinear isometry called the modular conjugation and $\Delta=T^*T=ST$ is a positive self adjoint operator called the modular operator. The main result of Tomita–Takesaki theory states that

$JAJ=A'$, the commutant of $A$, and

There is a 1-parameter family of modular automorphisms $\sigma^\rho_t$ of $A$ associated to the state $\rho$, defined by

$\sigma^\rho_t(a)=\Delta^{it}a\Delta^{-it}$.

### Proof of Theorem 2

See Takesaki (no access) $\bullet$

If we try to apply the argument (Umegaki’s to Theorem 1) which leads to success for a tracial state then we arrive at troubles with the positivity condition 1. Accardi and Cecchini have defined a mapping $E_\rho:A\rightarrow A_0$, which is completely positive (need to look at this — I have no idea why this might jeapardise condition 4.), always exists and therefore possesses only a weaker form of condition 4.  $E_\rho$, called $\rho$-conditional expectation, does not map $A$ onto $A_0$. In fact, they did not consider postulates like 1. to 4. but extended to von Neumann algebras the construction of measure-theoretic conditional expectation. If $g$ is a bounded $\mathcal{S}$-measurable function, then $I(f)=\int fg\,dP$ is a functional on $L^1(\mathcal{S},P)$, and its restriction determines an $\mathcal{S}_0$-measurable function $E(g)$ such that

$\displaystyle E(g)h\,dP=\int gh\,dP$ for all $h\in L^1(\mathcal{S},P)$.

So in this construction is on the duality $(L^1)^*=L^\infty$.

The following result is standard in Tomita-Takesaki theory

### Lemma 3

If $\rho$ is a state on $A$ and $a,\,b\in A$, then there exists a function

$\displaystyle F:\{z\in\mathbb{C}:0\leq\text{Im }z\leq1\}\rightarrow \mathbb{C}$

so that

1. $F$ is bounded, continuous and is analytic on the interior of its domain.
2. $F(t)=\rho(\sigma^\rho_t(b)a)$ for all $t\in\mathbb{R}$.
3. $F(t+i)=\rho(a\sigma^\rho_t(b))$ for all $t\in\mathbb{R}$.
We will use this lemma to abuse notation slightly and write
$F(z,a,b)=F(z)=\rho(\sigma_z(b)a)$.

## Theorem 4

Let $A_0\subset A$ be von Neumann algebras and $rho$ a state on $A$. If $\sigma_t$ ($\tilde{\sigma_t}$) is the modular group of $\rho$ ($\tilde{\rho}=\rho_{|A_0}$) then the formula

$\rho\left(\sigma_{i/2}(b_0)a\right)=\rho\left(\tilde{\rho}_{i/2}E_\rho(a)\right)$

determines a completely positive mapping $E_\rho:A\rightarrow A_0$. For $a_0\in A_0$

$E_\rho(aa_0)=E_\rho(a)a_0$ for every $latex a\in A$

holds if and only if $\sigma_t(a_0)\in A_0$ for every $t\in \mathbb{R}$.

I better stop at this point and go and find out more about completely positive maps. The modular group of $\rho$ I imagine is a departure from the classical case.

Advertisement