Poisson-Pinsker factor and infinite measure preserving group actions

Take $\alpha,\beta>0$ and consider the direct product

of Poisson suspensions. Thanks to the classical formula $\left(\alpha\mu\right)^{*}*\left(\beta\mu\right)^{*}=\left(\alpha+\beta\right)\mu^{*}$, the map $\varphi:\,\left(\nu_{1},\nu_{2}\right)\mapsto\nu_{1}+\nu_{2}$ induces a factor map from

to

Denote by $\Pi_{\alpha}$ and $\Pi_{\beta}$ and $\Pi_{\alpha+\beta}$ the Pinsker algebra of the systems $\left(X^{*},\mathcal{A}^{*},\left(\alpha\mu\right)^{*},T_{*}^{g}\right)$, $\left(X^{*},\mathcal{A}^{*},\left(\beta\mu\right)^{*},T_{*}^{g}\right)$ and $\left(X^{*},\mathcal{A}^{*},\left(\left(\alpha+\beta\right)\mu\right)^{*},T_{*}^{g}\right)$ respectively and set $\mathcal{B}:=\varphi^{-1}\mathcal{A}^{*}$ and $\widetilde{\Pi_{\alpha+\beta}}:=\varphi^{-1}\Pi_{\alpha+\beta}$. Thanks to a classical result (generalized to countable amenable group actions in Theorem 4 in [6]), the Pinsker algebra of the product $\left(X^{*}\times X^{*},\mathcal{A}^{*}\otimes\mathcal{A}^{*},\left(\alpha\mu\right)^{*}\otimes\left(\beta\mu\right)^{*},T_{*}^{g}\times T_{*}^{g}\right)$ is $\Pi_{\alpha}\otimes\Pi_{\beta}$. The extension $\mathcal{B}\to\widetilde{\Pi_{\alpha+\beta}}$ is a cpe extension and $\Pi_{\alpha}\otimes\Pi_{\beta}\to\widetilde{\Pi_{\alpha+\beta}}$ is a zero entropy extension. Therefore, by Lemma 3 in [12] (once again generalized to countable amenable group actions in Theorem 1 in [6]), they are relatively disjoint over $\widetilde{\Pi_{\alpha+\beta}}$. As a consequence, $L^{2}\left(\mathcal{B}\right)\ominus L^{2}\left(\widetilde{\Pi_{\alpha+\beta}}\right)$ and $L^{2}\left(\Pi_{\alpha}\otimes\Pi_{\beta}\right)\ominus L^{2}\left(\widetilde{\Pi_{\alpha+\beta}}\right)$ are orthogonal in $L^{2}\left(\mathcal{A}^{*}\otimes\mathcal{A}^{*}\right)$. Indeed if $f\in L^{2}\left(\mathcal{B}\right)\ominus L^{2}\left(\widetilde{\Pi_{\alpha+\beta}}\right)$ and $g\in L^{2}\left(\Pi_{\alpha}\otimes\Pi_{\beta}\right)\ominus L^{2}\left(\widetilde{\Pi_{\alpha+\beta}}\right)$, we have:

We can therefore decompose $L^{2}\left(\mathcal{A}^{*}\otimes\mathcal{A}^{*}\right)$ into the following orthogonal sum:

where $H$ is the orthocomplement of everything else. Now write $f\in L^{2}\left(\mathcal{A}^{*}\otimes\mathcal{A}^{*}\right)$ as $f=f_{1}+f_{2}+f_{3}+f_{4}$ according to the decomposition, we have:

thus $\mathbb{E}\left[f\mid\widetilde{\Pi_{\alpha+\beta}}\right]=\mathbb{E}\left[\mathbb{E}\left[f\mid\mathcal{B}\right]\mid\Pi_{\alpha}\otimes\Pi_{\beta}\right]$.

Now form the relatively independent joining of

over $\Pi_{\alpha}\otimes\Pi_{\beta}$ and remark that it is just the direct product of the relatively independent joinings of $\left(X^{*},\mathcal{A}^{*},\left(\alpha\mu\right)^{*},T_{*}^{g}\right)$ and $\left(X^{*},\mathcal{A}^{*},\left(\beta\mu\right)^{*},T_{*}^{g}\right)$ over their respective Pinsker factors $\Pi_{\alpha}$ and $\Pi_{\beta}$. Let’s now compute the distribution of the self-joining of $\left(X^{*},\mathcal{A}^{*},\mu^{*},T_{*}^{g}\right)$ induced by the preceding joining, through the factor map $\varphi\times\varphi$. Take $A$ and $B$ in $\mathcal{B}$ and compute:

Therefore, the joining induced is nothing else than the relatively independent joining over its Pinsker factor. We have just proved that the image measure of

by the sum application $\varphi\times\varphi$ is the measure $\mu^{*}\otimes_{\Pi_{\alpha+\beta}}\mu^{*}$ , that is, we have proved the following formula:

and we can deduce, for any integer $k$:

This means that the distribution of this relatively independent joining is infinitely divisible, i.e. it is a Poissonian joining. But according to Proposition 2, $\Pi_{1}=\Pi$ is a Poissonian factor.∎

If the Pinsker factor of $\left(X^{*},\mathcal{A}^{*},\mu^{*},T_{*}^{g}\right)$ is trivial, the system has complete positive entropy, therefore, for any $T^{g}$-invariant set $B\subset X$ of positive measure, $\left(B,\mathcal{A}_{\mid B},\mu_{\mid B},T^{g}\right)$ has totally positive Poisson entropy since any factor $\mathcal{B}_{\mid B}$ corresponds to the non-trivial Poissonian factor $\left(\mathcal{B}_{\mid B}\right)^{*}$ on which the Poisson suspension has positive entropy. If the Pinsker factor $\Pi$ of $\left(X^{*},\mathcal{A}^{*},\mu^{*},T_{*}^{g}\right)$ is not trivial, from Proposition 3, there exists a $T^{g}$-invariant set $A\subset X$ of positive measure and a factor, say $\mathcal{P}_{\mid A}$, of the restricted system $\left(A,\mathcal{A}_{\mid A},\mu_{\mid A},T^{g}\right)$ such that $\left(\mathcal{P}_{\mid A}\right)^{*}=\Pi$. $\mathcal{P}_{\mid A}$ is clearly the Poisson-Pinsker factor of the system $\left(A,\mathcal{A}_{\mid A},\mu_{\mid A},T^{g}\right)$. Indeed, if $\mathcal{C}_{\mid A}\subset\mathcal{A}_{\mid A}$ is a factor with zero Poisson entropy, $\left(\mathcal{C}_{\mid A}\right)^{*}\subset\left(\mathcal{P}_{\mid A}\right)^{*}$ as the latter is the Pinsker factor of the suspension, and this implies $\mathcal{C}_{\mid A}\subset\mathcal{P}_{\mid A}$ (the fact that a factor $\mathcal{R}$ is the Poisson-Pinsker factor if the associated Poissonian factor $\mathcal{R}^{*}$ is the Pinsker factor of the suspension was already observed in [7]).

If $A^{c}$ has positive measure, the Poisson suspension $\left(X^{*},\mathcal{A}^{*},\mu^{*},T_{*}^{g}\right)$ splits into the direct product

which implies that the Pinsker factor $\Pi$ also splits accordingly. But as $\left(\mathcal{P}_{\mid A}\right)^{*}=\Pi$, the Pinsker factor is concentrated in the first side of the product, that is, $\left(\left(A^{c}\right)^{*},\left(\mathcal{A}_{\mid A^{c}}\right)^{*},\left(\mu_{\mid A^{c}}\right)^{*},T_{*}^{g}\right)$ has complete positive entropy and we conclude as in the first part of the proof. ∎

The first statement is obvious and the proof of the second is identical to the $\mathbb{Z}$-action case which can be found in [7]. ∎

From Proposition 3 $\left(X^{*},\mathcal{A}^{*},\mu^{*},T_{*}^{g}\right)$ has complete positive entropy and is therefore mixing. But, thanks to the classical isometry formula

we have

which goes to zero as $g$ tends to infinity.∎

Since, thanks to Proposition 3, $\Pi$ has the structure of a Poissonian factor, its associated $L^{2}$-space is a Fock space compatible with the underlying one, in particular $\mathfrak{C}=\left(L^{2}\left(\Pi\right)^{\perp}\cap\mathfrak{C}\right)\begin{array}[b]{c}\perp\\ \oplus\end{array}\left(L^{2}\left(\Pi\right)\cap\mathfrak{C}\right)$. Since, by assumption $L^{2}\left(\Pi\right)\neq L^{2}\left(\mathcal{A}^{*}\right)$, $\mathfrak{C}\not\subset L^{2}\left(\Pi\right)$ as $\sigma\left(\mathfrak{C}\right)=\mathcal{A}^{*}$. Therefore $\left(L^{2}\left(\Pi\right)^{\perp}\cap\mathfrak{C}\right)$ is not empty and we get the result, as the maximal spectral type on $L^{2}\left(\Pi\right)^{\perp}$ is Lebesgue by a Theorem proved in [5] and independently by Thouvenot (unpublished). ∎

This follows directly from Proposition 8 and from the unitary isomorphism between the first chaos of $L^{2}\left(\mathcal{A}^{*}\right)$ and $L^{2}\left(\mathcal{A}\right)$. ∎

The first statement follows from Proposition 10.2 in [7], combined with Proposition 9. The second can also be deduced from an adaptation of Proposition 10.2 in [7] but is also a direct application of Theorem 3.2 in [2] combined with the fact that conditional Poisson entropy coincides with conditional Krengel entropy as proved in [7]. ∎