83 research outputs found
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
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
A generalized Fourier inversion theorem
In this work we define operator-valued Fourier transforms for suitable
integrable elements with respect to the Plancherel weight of a (not necessarily
Abelian) locally compact group. Our main result is a generalized version of the
Fourier inversion Theorem for strictly-unconditionally integrable Fourier
transforms. Our results generalize and improve those previously obtained by Ruy
Exel in the case of Abelian groups.Comment: 15 pages; some typos correcte
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
Metric Properties of the Fuzzy Sphere
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics
which we describe in detail. We show that the Bloch coherent states, with these
spectral distances, form a sequence of metric spaces that converge to the round
sphere in the high-spin limit.Comment: Slightly shortened version, no major changes, two new references,
version to appear on Letters in Mathematical Physic
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
Nonassociative strict deformation quantization of C*-algebras and nonassociative torus bundles
In this paper, we initiate the study of nonassociative strict deformation
quantization of C*-algebras with a torus action. We shall also present a
definition of nonassociative principal torus bundles, and give a classification
of these as nonassociative strict deformation quantization of ordinary
principal torus bundles. We then relate this to T-duality of principal torus
bundles with -flux. We also show that the Octonions fit nicely into our
theory.Comment: 15 pages, latex2e, exposition improved, to appear in LM
Gravity coupled with matter and foundation of non-commutative geometry
We first exhibit in the commutative case the simple algebraic relations
between the algebra of functions on a manifold and its infinitesimal length
element . Its unitary representations correspond to Riemannian metrics and
Spin structure while is the Dirac propagator ds = \ts \!\!---\!\! \ts =
D^{-1} where is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution . We then write a
spectral action, the trace of a function of the length element in Planck units,
which when applied to the non commutative geometry of the Standard Model will
be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian
coupled to gravity. The internal fluctuations of the non commutative geometry
are trivial in the commutative case but yield the full bosonic sector of SM
with all correct quantum numbers in the slightly non commutative case. The
group of local gauge transformations appears spontaneously as a normal subgroup
of the diffeomorphism group.Comment: 30 pages, Plain Te
A C*-Algebraic Model for Locally Noncommutative Spacetimes
Locally noncommutative spacetimes provide a refined notion of noncommutative
spacetimes where the noncommutativity is present only for small distances. Here
we discuss a non-perturbative approach based on Rieffel's strict deformation
quantization. To this end, we extend the usual C*-algebraic results to a
pro-C*-algebraic framework.Comment: 13 pages, LaTeX 2e, no figure
The bass and topological stable ranks of the Bohl algebra are infinite
The Bohl algebra B is the ring of linear combinations of functions t k e λt on the real line, where k is any nonnegative integer, and λ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of B (where we use the topology of uniform convergence) are infinite
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