Tail algebras for monotone and q-deformed exchangeable stochastic processes

Abstract

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti’s theorem for this type of processes. In addition, since the vacuum state on the q-deformed $C^*$-algebra is the only exchangeable state when $|q|<1$, we draw our attention to its tail algebra, which turns out to obey a zero-one law.

Appendix A

We provide here a result that allows for a full reconstruction of classical (real-valued) stochastic processes from the general setting of $C^*$-algebras and their representation theory. To this end, in the definition of a quantum stochastic process, the sample algebra ${{\mathfrak {A}}}$ must be chosen to be $C_0({{\mathbb {R}}})$, the $C^*$-algebra of all continuous functions on the real line vanishing at infinity. Furthermore, the algebra $\bigvee _{j\in {{\mathbb {Z}}}}\iota _j (C_0({{\mathbb {R}}}))$ needs to be assumed commutative as well. The result is actually more or less known. Nevertheless, we state it in a possibly novel form, which is more suited to the language of Quantum Probability. We keep the notation we established in Sect. 2.

Proposition A.1

Let $(C_0({{\mathbb {R}}}), {{\mathcal {H}}}, \{\iota _j\}_{j\in {{\mathbb {Z}}}}, \xi )$ be a stochastic process such that the von Neumann algebra $\bigvee _{j\in {{\mathbb {Z}}}}\iota _j (C_0({{\mathbb {R}}}))$ is commutative. Then there exists a commuting family $\{A_j: j\in {{\mathbb {Z}}}\}$ of (possibly unbounded) self-adjoint operators on ${{\mathcal {H}}}$ such that for every $j\in {{\mathbb {Z}}}$ one has $\iota _j(f)=f(A_j)$, $f\in C_0({{\mathbb {R}}})$.

In addition there exist a probability measure $\mu$ on $({{\mathbb {R}}}^{{\mathbb {Z}}}, {\mathfrak {C}})$ and a unitary $U: {{\mathcal {H}}}\rightarrow L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$ such that:

• (1) $U\xi =[1]_\mu$

• (2) $UA_jU^*=M_j$, for every $j\in {{\mathbb {Z}}}$, where $M_j$ is the operator acting on $L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$ as the multiplication by $X_j$.

Proof

Let ${{\mathcal {B}}}_b({{\mathbb {R}}})$ be the $C^*$-algebra of all bounded Borel functions on ${{\mathbb {R}}}$. As is known, each $\iota _j$ can be extended to a $*$-representation of ${{\mathcal {B}}}_b({{\mathbb {R}}})$, which we denote by $\widetilde{\iota _j}$. By a standard approximation argument, one easily sees that the ranges of the extended representations continue to commute with each other.

For any fixed j, set $E_j(\Delta ):=\widetilde{\iota _j}(\chi _\Delta )$, where $\Delta$ is a Borel subset of the real line. For every $j\in {{\mathbb {Z}}}$, the family of orthogonal projections $\{E_j(\Delta ): \Delta \subset {{\mathbb {R}}}\, \text{ is a Borel subset}\}$ is a resolution of the identity, and as such it defines a self-adjoint operator $A_j:=\int _{{{\mathbb {R}}}}\lambda \mathrm{d}E_j(\lambda )$. Note that the operators $A_j$, $j\in {{\mathbb {Z}}}$, commute with one another because their spectral projections do.

Fix $n\in {{\mathbb {N}}}$ and $j_1, j_2, \ldots , j_n\in {{\mathbb {Z}}}$. Consider the bounded linear functional $\varphi : C_0({{\mathbb {R}}}^n)\rightarrow {{\mathbb {C}}}$ given by $\varphi (f)=\langle f(A_{j_1}, A_{j_2}, \ldots , A_{j_n})\xi , \xi \rangle$ for any $f\in C_0({{\mathbb {R}}}^n)$. By the Riesz-Markov theorem there exists a Borel probability measure $\mu _{j_1, j_2, \ldots , j_n}$ on ${{\mathbb {R}}}^n$ such that

$$\begin{aligned} \langle f(A_{j_1}, A_{j_2}, \ldots , A_{j_n})\xi , \xi \rangle = \int _{{{\mathbb {R}}}^n} f\mathrm{d}\mu _{j_1, j_2, \ldots , j_n}, \, f\in C_0({{\mathbb {R}}}^n). \end{aligned}$$

(A.1)

Observe that the family $\{\mu _{j_1, j_2, \ldots , j_n}: n\in {{\mathbb {N}}}, j_1, j_2, \ldots , j_n\in {{\mathbb {Z}}}\}$ satisfies the consistency conditions of Kolmogorov’s theorem. Therefore, there exists a probability measure $\mu$ on $({{\mathbb {R}}}^{{\mathbb {Z}}}, {\mathfrak {C}})$ having the family above as its finite-dimensional distributions.

Our next aim is to define a unitary U from ${{\mathcal {H}}}$ to $L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$. To this end, first note that the linear subspace

$$\begin{aligned} {\mathcal {D}}:=\{f(A_{j_1}, A_{j_2}, \ldots , A_{j_n})\xi : n\in {{\mathbb {N}}}, f\in C_0({{\mathbb {R}}}^n), j_1, j_2, \ldots , j_n\in {{\mathbb {Z}}}\} \end{aligned}$$

is dense in ${{\mathcal {H}}}$ because $\xi$ is a cyclic vector for $\bigvee _{j\in {{\mathbb {Z}}}}\iota _j(C_0({{\mathbb {R}}}))$. We define $U_0: {\mathcal {D}}\rightarrow L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$ by

$$\begin{aligned} U_0f(A_{j_1}, A_{j_2}, \ldots , A_{j_n})\xi := [f]_\mu \,, \end{aligned}$$

(A.2)

where $[f]_\mu$ is the $\mu$-equivalence class of the function $f(X_{j_1}, X_{j_2}, \ldots , X_{j_n})$, and $X_j: {{\mathbb {R}}}^{{\mathbb {Z}}}\rightarrow {{\mathbb {R}}}$ is for every j the j-th coordinate function, that is $X_j(x)=x_j$, $x=(x_j)\in {{\mathbb {R}}}^{{\mathbb {Z}}}$. Thanks to (A.1) one sees at once that $U_0$ is an isometry. By density of ${\mathcal {D}}$ in ${{\mathcal {H}}}$, $U_0$ can be extended to an isometry U defined on the whole ${{\mathcal {H}}}$. By standard approximation arguments the range of U is seen to be dense in $L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$, hence U is a unitary. In order to prove the equality $U\xi = [1]_\mu$, we consider a bijection $g:{{\mathbb {N}}}\rightarrow {{\mathbb {Z}}}$ and a sequence of cylinders sets $C_n=\{x\in {{\mathbb {R}}}^{{\mathbb {Z}}}: x_{g(1)}\in B_1, \ldots , x_{g(n)}\in B_n \}$ such that $\mu (C_n)\ge 1-\frac{1}{n}$ with $B_i\subset {{\mathbb {R}}}$ being suitable bounded Borel subsets. Let now be $\{h_n\}_{n\in {{\mathbb {N}}}}$ be a sequence of functions such that $h_n\in C_c({{\mathbb {R}}}^n)$ with $0\le h_n\le 1$ and $h_n(x_1, \ldots , x_n)=1$ for all $(x_1, \ldots , x_n)\in B_1\times \cdots \times B_n$, for every n. By evaluating (A.2) on $h_n(A_{g(1)}, \ldots , A_{g(n)})\xi$, we get

$$\begin{aligned} U_0h_n(A_{g(1)}, \ldots , A_{g(n)})\xi = [h_n]_\mu \,. \end{aligned}$$

By construction the sequence $\{k_n\}_{n\in {{\mathbb {N}}}}\subset L^2({{\mathbb {R}}}^{{\mathbb {Z}}}, \mu )$ given by $k_n:= h_n(X_{g(1)}, X_{g(2)}, \ldots , X_{g(n)})$ converges to $[1]_\mu$ in the $\Vert \cdot \Vert _{L^2}$-norm. Therefore, one can take the limit as $n\rightarrow \infty$ of the above equality and gets $U\xi =[1]_\mu$.

All is left to do is prove the equality $UA_jU^*=M_j$, for every $j\in {{\mathbb {Z}}}$. Fix $j\in {{\mathbb {Z}}}$ and define

$$\begin{aligned} {\mathcal {D}}_j:=\{f(A_j, A_{j_1}, \ldots , A_{j_n})\xi : n\ge 0, j_1, \ldots . j_n\in {{\mathbb {Z}}}, f\in C_c({{\mathbb {R}}}^{n+1})\}. \end{aligned}$$

We aim to show that ${\mathcal {D}}_j$ is a core per $A_j$. We first note that ${\mathcal {D}}_j$ is contained in the domain of $A_j$. Indeed, if $x=f(A_j, A_{j_1}, \ldots , A_{j_n})\xi$, where f is in $C_c({{\mathbb {R}}}^{n+1})$ one has $\int _{{\mathbb {R}}}\lambda _j^2 \mathrm{d}(E(\lambda _j)x, x)<\infty$ because the measure $\mathrm{d}(E(\lambda _j)x, x)$ is by construction compactly supported. Second, the linear subspace ${\mathcal {D}}_j$ is easily seen to be dense in ${{\mathcal {H}}}$. Finally, ${\mathcal {D}}_j$ is invariant for the one-parameter group $U_j(t):=e^{itA_j}$ since for any $f\in C_0({{\mathbb {R}}}^{n+1})$ the function ${{\mathbb {R}}}^{n+1}\ni (\lambda _j, \lambda _{j_1}, \ldots , \lambda _{j_n})\rightarrow e^{it\lambda _j}f(\lambda _j, \lambda _{j_1}, \ldots , \lambda _{j_n})\in {{\mathbb {C}}}$ still has compact support. From Theorem VIII.11 in [23] it follows that ${\mathcal {D}}_j$ is a core for $A_j$.

The equality $UA_j=M_jU$ is trivially satisfied on ${\mathcal {D}}_j$. Let now $\varphi$ be in ${\mathcal {D}}(A_j)$. Then there exists a sequence $\{\varphi _n\}_{n\in {{\mathbb {N}}}}$ in ${\mathcal {D}}_j$ such that $\varphi _n\rightarrow \varphi$ and $A_j\varphi _n\rightarrow A_j\varphi$. From the equality $UA_j\varphi _n=M_jU\varphi _n$, $n\in {{\mathbb {N}}}$, we see that the sequence $M_jU\varphi _n$ converges to $UA_j\varphi$. Because $M_j$ is closed, we must have that $U\varphi$ is in ${\mathcal {D}}(M_j)$ and $UA_j\varphi =M_jU\varphi$. In other words, we have $A_j\subset U^*M_jU$, and so $A_j=U^*M_jU$ as $A_j$ and $U^*X_jU$ are both self-adjoint. $\square$

Remark A.2

Under the general hypotheses we are working with, a common core for all operators $A_j$ may fail to exist, for the intersection $\bigcap _{j\in {{\mathbb {Z}}}} {\mathcal {D}}_j$ is not even necessarily dense in ${{\mathcal {H}}}$.