1,720,974 research outputs found
Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products
Full text linkAnzai 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*-dynamical
systems
On distributional symmetries on the CAR algebra
No full textSpreadable, exchangeable, and rotatable states on the CAR algebra are shown to be the sam
Skew-product dynamical systems for crossed product C*-algebras and their ergodic properties
No full textStarting from a discrete -dynamical system , we define and study most of the main ergodic properties of the crossed product -dynamical system
, being the canonical conditional expectation of
onto , provided commute with the -automorphism up to a unitary . Here,
can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai for the product of two tori in the classical case
The Motzkin sub-product system
No full textWe introduce a sub-product system of finite-dimensional Hilbert spaces by using the Motzkin planar algebra and its Motzkin Jones-Wenzl idempotents, which generalizes the Temperley-Lieb sub-product system of Habbestad and Neshveyev. We provide a description of the corresponding Toeplitz and Cuntz-Pimsner C∗-algebras as universal C∗-algebras, defined in terms of generators and relations, and we highlight properties of their representation theory
De Finetti theorem on the infinte non-commutative torus
No full textThe set of spreadable states on an infinite non-commutative torus \mathbb{A}_\a^\bz is
determined for all values of the deformation parameter \a. If \frac{\a}{2\pi} is irrational, the canonical
trace is the only spreadable state. If \frac{\a}{2\pi} is rational, the set of all spreadable states is a Bauer simplex.
Moreover, its boundary is the set of all infinite products of a single state
on C(\bt), which is invariant under all rotations
by -th roots of unity, where , with
being for even and for odd, if is the representation of \frac{\a}{2\pi} in lowest terms.\\
Finally, the simplex of all stationary states on \mathbb{A}_\a^\bz is proved to be the Poulsen simplex for all
values of the deformation parameter \a
On truncated -free Fock spaces: spectrum of position operators and shift-invariant states
Full text linkThe ergodic properties of the shift on both full and -truncated -free
-algebras are analyzed. In particular, the shift is shown to be uniquely
ergodic with respect to the fixed-point algebra. In addition, for every , the invariant states of the shift acting on the -truncated -free
-algebra are shown to yield a -dimensional Choquet simplex, which
collapses to a segment in the full case. Finally, the spectrum of the position
operators on the -truncated -free Fock space is also determined.Comment: 15 page
On the monotone -algebra
Full text linkThe concrete monotone -algebra, that is the (unital) -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 -theory.Comment: 10 page
Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions
No full textIn this paper we extend the local iterative Lie-Schwinger block-diagonalization method – introduced in [8] for quantum lattice systems with bounded interactions in arbitrary dimension– to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t| independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive. As in [8], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [8] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentials, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators. We stress that no “large-field problems” appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest
Weakly-monotone C*-algebras as Exel-Laca algebras
Full text linkAn abstract characterization of weakly monotone -algebras, namely the
concrete -algebras generated by creators and annihilators acting on the
so-called weakly monotone Fock spaces, is given in terms of (quotient of)
suitable Exel-Laca algebras. The weakly monotone -algebra indexed by
is shown to be a type-I -algebra and its representation
theory is entirely determined, whereas the weakly monotone -algebra
indexed by is shown not to be of type
Quantum Operations on Conformal Nets
Full text linkOn a conformal net , one can consider collections of unital
completely positive maps on each local algebra , subject to
natural compatibility, vacuum preserving and conformal covariance conditions.
We call \emph{quantum operations} on the subset of extreme such
maps. The usual automorphisms of (the vacuum preserving
invertible unital *-algebra morphisms) are examples of quantum operations, and
we show that the fixed point subnet of under all quantum
operations is the Virasoro net generated by the stress-energy tensor of
. Furthermore, we show that every irreducible conformal subnet
is the fixed points under a subset of quantum
operations.
When is discrete (or with finite Jones
index), we show that the set of quantum operations on that leave
elementwise fixed has naturally the structure of a compact (or
finite) hypergroup, thus extending some results of [Bis17]. Under the same
assumptions, we provide a Galois correspondence between intermediate conformal
nets and closed subhypergroups. In particular, we show that intermediate
conformal nets are in one-to-one correspondence with intermediate subfactors,
extending a result of Longo in the finite index/completely rational conformal
net setting [Lon03].Comment: 22 page
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