Fermi Markov states

We investigate the structure of the Markov states on general Fermion algebras. The situation treated in the present paper covers, beyond the d--Markov states on the CAR algebra on Z (i.e. when there are d--annihilators and creators on each site), also the non homogeneous case (i.e. when the numbers of generators depends on the localization). The present analysis provides the first necessary step for the study of the general properties, and the construction of nontrivial examples of Fermi Markov states on the d--standard lattice, that is the Fermi Markov fields. Natural connections with the KMS boundary condition and entropy of Fermi Markov states are studied in detail. Apart from a class of Markov states quite similar to those arising in the tensor product algebras (called "strongly even" in the sequel), other interesting examples of Fermi Markov states naturally appear. Contrarily to the strongly even examples, the latter are highly entangled and it is expected that they describe interactions which are not "commuting nearest neighbor". Therefore, the non strongly even Markov states, in addition to the natural applications to quantum statistical mechanics, might be of interest for the information theory as well.Comment: 32 pages. Journal of Operator Theory, to appea

A note on Boolean stochastic processes

For the quantum stochastic processes generated by the Boolean Commutation Relations, we prove the following version of De Finetti Theorem: each of such Boolean process is exchangeable if and only if it is independent and identically distributed with respect to the tail algebra.Comment: 9 page

Some operator ideals in non-commutative functional analysis

We characterize classes of linear maps between operator spaces $E$, $F$ which factorize through maps arising in a natural manner via the Pisier vector-valued non-commutative $L^p$ spaces $S_p[E^*]$ based on the Schatten classes on the separable Hilbert space $l^2$. These classes of maps can be viewed as quasi-normed operator ideals in the category of operator spaces, that is in non-commutative (quantized) functional analysis. The case $p=2$ provides a Banach operator ideal and allows us to characterize the split property for inclusions of $W^*$-algebras by the 2-factorable maps. The various characterizations of the split property have interesting applications in Quantum Field Theory.Comment: 23 pages, LaTe

Infinite dimensional entangled Markov chains

We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state--space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on $\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}$, $F$ being any infinite dimensional type $I$ factor, $J$ a finite interval of $Z$, and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a r\^ole in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type $II_1$ von Neumann factors.Comment: 16 page

New topics in ergodic theory

The entangled ergodic theorem concerns the study of the convergence in the strong, or merely weak operator topology, of the multiple Cesaro mean \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} , where $U$ is a unitary operator acting on the Hilbert space $H$, \a:\{1,..., m\}\mapsto\{1,..., k\} is a partition of the set made of $m$ elements in $k$ parts, and finally $A_{1},...,A_{2k-1}$ are bounded operators acting on $H$. While reviewing recent results about the entangled ergodic theorem, we provide some natural applications to dynamical systems based on compact operators. Namely, let $(\mathfrak A,\alpha)$ be a $C^{*}$--dynamical system, where $\mathfrak A=K(H)$, and $\alpha=ad(U)$ is an automorphism implemented by the unitary $U$. We show that $$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\alpha^{n}=E ,$$ pointwise in the weak topology of \K(H). Here, $E$ is a conditional expectation projecting onto the $C^{*}$--subalgebra $$\bigg(\bigoplus_{z\in\sigma_{\mathop{\rm pp}}(U)} E_{z}B(H)E_{z}\bigg)\bigcap K(H) .$$ If in addition $U$ is weakly mixing with $\Omega\in H$ the unique up to a phase, invariant vector under $U$ and $\omega=$, we have the following recurrence result. If $A\in K(H)$ fulfils $\omega(A)>0$, and $0<m_{1}<m_{2}<...<m_{l}$ are natural numbers kept fixed, then there exists an $N_{0}$ such that $$\frac{1}{N}\sum_{n=0}^{N-1}\omega(A\alpha^{nm_{1}}(A)\alpha^{nm_{2}}(A)... \alpha^{nm_{l}}(A))>0$$ for each $N>N_{0}$.Comment: 18 page