Stefano Rossi

Università degli Studi di Bari Aldo Moro | Università di Bari · Dipartimento di Matematica

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are sh...

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of C∗\documentc...

The concrete monotone C⁎-algebra, that is the (unital) C⁎-algebra generated by monotone independent algebraic random variables of Bernoulli type, is characterized abstractly in terms of generators and relations and is shown to be UHF. Moreover, its Bratteli diagram is explicitly given, which allows for the computation of its K-theory.

The compact convex set of all spreadable states on the CAR algebra is shown to be a Choquet simplex of which the exchangeable states make up a proper face. Moreover, the set of rotatable states on the CAR algebra is seen to coincide with the set of exchangeable states.

Spreadability of a sequence of random variables is a distributional symmetry that is implemented by suitable actions of $\mathbb{J}_\mathbb{Z}$, the unital semigroup of strictly increasing maps on $\mathbb{Z}$ with cofinite range. We show that $\mathbb{J}_\mathbb{Z}$ is left amenable but not right amenable, although it does admit a right Folner seq...

Spreadability of a sequence of random variables is a distributional symmetry that is implemented by suitable actions of JZ, the unital semigroup of strictly increasing maps on Z with cofinite range. We show that JZ is left amenable but not right amenable, although it does admit a right Følner sequence. This enables us to prove that on the CAR algeb...

The ergodic properties of the shift on both full and $m$-truncated $t$-free $C^*$-algebras are analyzed. In particular, the shift is shown to be uniquely ergodic with respect to the fixed-point algebra. In addition, for every $m\geq 1$, the invariant states of the shift acting on the $m$-truncated $t$-free $C^*$-algebra are shown to yield a $m+1$-d...

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 z...

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 z...

The concrete monotone $C^*$-algebra, that is the (unital) $C^*$-algebra generated by monotone independent algebraic random variables of Bernoulli type, is characterized abstractly in terms of generators and relations and is shown to be UHF. Moreover, its Bratteli diagram is explicitly given, which allows for the computation of its $K$-theory.

In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consi...

We systematically investigate $C^*$-norms on the algebraic graded product of $\mathbb{Z}_2$-graded $C^*$-algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $C^*$-norms. To this...

Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on $\mathbb{N}$, on quasi-local algebras are defined and proved to be $\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are s...

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\CQ_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $\chi_k\in{\rm End}(\CQ_p)$ by setting $\chi_k(U):=U^k$ and $\chi_k(S_p):=S_p$. We then comp...

We present a broad selection of results on endomorphisms and automorphisms of the Cuntz algebras On that have been obtained in the last decades. A wide variety of open problems is also included.

We provide a new interpretation of the group of Bogolubov automorphisms of the Cuntz algebras $O_n$ and the group $Aut(O_n, F_n)$ of all automorphisms preserving the UHF subalgebra £Fn \subset O_n$ as the isometry groups coming from two distinct spectral triples on $O_n$.

We systematically investigate $C^*$-norms on the algebraic graded product of $\bz_2$-graded $C^*$-algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $C^*$-norms. To this end, we f...

In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital C*-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some gen...

In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital $C^*$-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some...

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $C^*$-dynami...

We investigate the structure of the fixed-point algebra of On under the action of the cyclic permutation of the generating isometries. We prove that it is *-isomorphic with On, thus generalizing a result of Choi and Latrémolière on O2. As an application of the technique employed, we also describe the fixed-point algebra of O2n under the exchange au...

We investigate the structure of the fixed-point algebra of On under the action of the cyclic permutation of the generating isometries. We prove that it is ⁎-isomorphic with On, thus generalizing a result of Choi and Latrémolière on O2. As an application of the technique employed, we also describe the fixed-point algebra of O2n under the exchange au...

Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, \theta, \omega_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_\alpha\mathbb{Z}, \Phi_{\theta, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_\alpha\mathbb{Z}\rightarrow\ga$ being the canonical conditional expe...

Starting from a discrete C⁎-dynamical system (A,θ,ωo), we define and study most of the main ergodic properties of the crossed product C⁎-dynamical system (A⋊αZ,Φθ,u,ωo∘E), E:A⋊αZ→A being the canonical conditional expectation of A⋊αZ onto A, provided α∈Aut(A) commute with the ⁎-automorphism θ up tu a unitary u∈A. Here, Φθ,u∈Aut(A⋊αZ) can be consider...

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $\chi_k\in{\rm End}(\mathcal{Q}_p)$ by setting $\chi_k(U):=U^k$ and $\chi_k(S_p):=S_p...

We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian consisting of a sum of on-site terms that do not couple the degrees of freedom located at different sites of the chain and have a strictly positive energy gap above their ground-state energy. For interactions that are form-bounded w.r.t. the...

In this paper the local iterative Lie-Schwinger block-diagonalization method, introduced in [FP], [DFPR1], and [DFPR2] for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation consisting o...

We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have...

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai ske...

A Fejér-type theorem is proved within the framework of C*-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

A Fejér-type theorem is proved within the framework of C*-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

We provide a systematic study of a noncommutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the noncommutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-...

We show that a natural notion of irreducibility implies connectedness in the Compact Quantum Group setting. We also investigate the converse implication and show it is related to Kaplansky's conjectures on group algebras.

We show that a natural notion of irreducibility implies connectedness in the Compact Quantum Group setting. We also investigate the converse implication and show it is related to Kaplansky's conjectures on group algebras.

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra D2 in the 2-adic ring C⁎-algebra Q2, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion O2⊂Q2 is proved not to be regular. Finally, countably many novel permutative endomorphisms of Q2 are exh...

We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove t...

We consider quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we have...

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally,...

We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal C$^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism ma...

The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to...

The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to...

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and...

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ poin...

In the $C^*$-algebraic setting the spectrum of any group-like element of a compact quantum group is shown to be a closed subgroup of the one-dimensional torus. A number of consequences of this fact are then illustrated, along with a loose connection with the so-called Kadison-Kaplansky conjecture.

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynom...

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in th...