232 research outputs found
Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz,
are Leibniz quantum compact metric spaces and that they form a continuous
family over the group of invertible matrices with entries in the commutant of
the quantum tori in the regular representation, when this group is endowed with
a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic
An Obstruction to Quantization of the Sphere
In the standard example of strict deformation quantization of the symplectic
sphere , the set of allowed values of the quantization parameter
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including ) in which can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing is never connected.Comment: 23 page. v2: changed sign conventio
On the role of twisted statistics in the noncommutative degenerate electron gas
We consider the problem of a degenerate electron gas in the background of a
uniformly distributed positive charge, ensuring overall neutrality of the
system, in the presence of non-commutativity. In contrast to previous
calculations that did not include twisted statistics, we find corrections to
the ground state energy already at first order in perturbation theory when the
twisted statistics is taken into account. These corrections arise since the
interaction energy is sensitive to two particle correlations, which are
modified for twisted anti-commutation relations
Connes distance and optimal transport
We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states
A Gromov-Hausdorff distance between von Neumann algebras and an application to free quantum fields
A distance between von Neumann algebras is introduced, depending on a further norm inducing the w*-topology on bounded sets. Such notion is related both with the Gromov–Hausdorff distance for quantum metric spaces of Rieffel and with the Effros–Maréchal topology on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz–Wichmann nuclearity condition, showing the continuity of such algebras w.r.t. the mass parameter
Finitely-Generated Projective Modules over the Theta-deformed 4-sphere
We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and
C(S^{4}_{theta}), where theta is any real number. We show that all
finitely-generated projective modules over C(S^{3}_{theta}) are free, and that
C(S^{4}_{theta}) has the cancellation property. We classify and construct all
finitely-generated projective modules over C(S^{4}_{\theta}) up to isomorphism.
An interesting feature is that if theta is irrational then there are nontrivial
"rank-1" modules over C(S^{4}_{\theta}). In that case, every finitely-generated
projective module over C(S^{4}_{\theta}) is a sum of a rank-1 module and a free
module. If theta is rational, the situation mirrors that for the commutative
case theta=0.Comment: 34 page
BRST quantization of quasi-symplectic manifolds and beyond
We consider a class of \textit{factorizable} Poisson brackets which includes
almost all reasonable Poisson structures. A particular case of the factorizable
brackets are those associated with symplectic Lie algebroids. The BRST theory
is applied to describe the geometry underlying these brackets as well as to
develop a deformation quantization procedure in this particular case. This can
be viewed as an extension of the Fedosov deformation quantization to a wide
class of \textit{irregular} Poisson structures. In a more general case, the
factorizable Poisson brackets are shown to be closely connected with the notion
of -algebroid. A simple description is suggested for the geometry underlying
the factorizable Poisson brackets basing on construction of an odd Poisson
algebra bundle equipped with an abelian connection. It is shown that the
zero-curvature condition for this connection generates all the structure
relations for the -algebroid as well as a generalization of the Yang-Baxter
equation for the symplectic structure.Comment: Journal version, references and comments added, style improve
Leibniz Seminorms and Best Approximation from C*-subalgebras
We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a
bounded approximate identity for A, and if L is the pull-back to A of the
quotient norm on A/B, then L is strongly Leibniz. In connection with this
situation we study certain aspects of best approximation of elements of a
unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator
Algebras and Related Topics". v2: added a corollary to the main theorem, plus
several minor improvements v3: much simplified proof of a key lemma,
corollary to main theorem added v4: Many minor improvements. Section numbers
increased by
Extensions and degenerations of spectral triples
For a unital C*-algebra A, which is equipped with a spectral triple and an
extension T of A by the compacts, we construct a family of spectral triples
associated to T and depending on the two positive parameters (s,t).
Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact
quantum metric spaces it is possible to define a metric on this family of
spectral triples, and we show that the distance between a pair of spectral
triples varies continuously with respect to the parameters. It turns out that a
spectral triple associated to the unitarization of the algebra of compact
operators is obtained under the limit - in this metric - for (s,1) -> (0, 1),
while the basic spectral triple, associated to A, is obtained from this family
under a sort of a dual limiting process for (1, t) -> (1, 0).
We show that our constructions will provide families of spectral triples for
the unitarized compacts and for the Podles sphere. In the case of the compacts
we investigate to which extent our proposed spectral triple satisfies Connes' 7
axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s
sphere plus comments on the relations to matricial quantum metrics. In ver.3
the word "deformations" in the original title has changed to "degenerations"
and some illustrative remarks on this aspect are adde
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