260 research outputs found
Dbrane Phase Transitions and Monodromy in K-theory
Majumder and Sen have given an explicit construction of a first order phase
transition in a non-supersymmetric system of Dbranes that occurs when the B
field is varied. We show that the description of this transition in terms of
K-theory involves a bundle of K groups of non-commutative algebras over the
Kahler cone with nontrivial monodromy. Thus the study of monodromy in K groups
associated with quantized algebras can be used to predict the phase structure
of systems of (non-supersymmetric) Dbranes.Comment: 8 pages, RevTeX, 1 figur
Traces on cores of C*-algebras associated with self-similar maps
We completely classify the extreme tracial states onthe cores of the
C*-algebras associated with self-similar maps on compact metric spaces. We
present a complete list of them. The extreme tracial states are the union of
the discrete type tracial states given by measures supported on the finite
orbits of the branch points and a continuous type tracial state given by the
Hutchinson measure on the original self-similar set.Comment: LaTeX 2e, 25 page
Operator *-correspondences in analysis and geometry
An operator *-algebra is a non-selfadjoint operator algebra with completely
isometric involution. We show that any operator *-algebra admits a faithful
representation on a Hilbert space in such a way that the involution coincides
with the operator adjoint up to conjugation by a symmetry. We introduce
operator *-correspondences as a general class of inner product modules over
operator *-algebras and prove a similar representation theorem for them. From
this we derive the existence of linking operator *-algebras for operator
*-correspondences. We illustrate the relevance of this class of inner product
modules by providing numerous examples arising from noncommutative geometry.Comment: 31 pages. This work originated from the MFO workshop "Operator spaces
and noncommutative geometry in interaction
Exel's crossed product for non-unital C*-algebras
We consider a family of dynamical systems (A,alpha,L) in which alpha is an
endomorphism of a C*-algebra A and L is a transfer operator for \alpha. We
extend Exel's construction of a crossed product to cover non-unital algebras A,
and show that the C*-algebra of a locally finite graph can be realised as one
of these crossed products. When A is commutative, we find criteria for the
simplicity of the crossed product, and analyse the ideal structure of the
crossed product.Comment: 22 page
Levinson's theorem and higher degree traces for Aharonov-Bohm operators
We study Levinson type theorems for the family of Aharonov-Bohm models from
different perspectives. The first one is purely analytical involving the
explicit calculation of the wave-operators and allowing to determine precisely
the various contributions to the left hand side of Levinson's theorem, namely
those due to the scattering operator, the terms at 0-energy and at infinite
energy. The second one is based on non-commutative topology revealing the
topological nature of Levinson's theorem. We then include the parameters of the
family into the topological description obtaining a new type of Levinson's
theorem, a higher degree Levinson's theorem. In this context, the Chern number
of a bundle defined by a family of projections on bound states is explicitly
computed and related to the result of a 3-trace applied on the scattering part
of the model.Comment: 33 page
Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions
We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite
dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka
and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of
coideal subalgebras. We reduce the study to KD(n) by proving that the others
are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We
derive many examples of lattices of intermediate subfactors of the inclusions
of depth 2 associated to those Kac algebras, as well as the corresponding
principal graphs, which is the original motivation.
Along the way, we extend some general results on the Galois correspondence
for depth 2 inclusions, and develop some tools and algorithms for the study of
twisted group algebras and their lattices of coideal subalgebras. This research
was driven by heavy computer exploration, whose tools and methodology we
further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added
connections with Masuoka's families KA and KB, description of K3 in KD(n),
lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for
KD(6), misc improvements, accepted for publication in Journal of Algebra and
Its Application
The K-theory of free quantum groups
In this paper we study the K -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K -amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K -theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ -element and that γ=1 . As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting
Weak Riemannian manifolds from finite index subfactors
Let be a finite Jones' index inclusion of II factors, and
denote by their unitary groups. In this paper we study the
homogeneous space , which is a (infinite dimensional) differentiable
manifold, diffeomorphic to the orbit
of the Jones projection of the inclusion. We endow with a
Riemannian metric, by means of the trace on each tangent space. These are
pre-Hilbert spaces (the tangent spaces are not complete), therefore is a weak Riemannian manifold. We show that enjoys certain
properties similar to classic Hilbert-Riemann manifolds. Among them, metric
completeness of the geodesic distance, uniqueness of geodesics of the
Levi-Civita connection as minimal curves, and partial results on the existence
of minimal geodesics. For instance, around each point of ,
there is a ball (of uniform radius ) of
the usual norm of , such that any point in the ball is joined to
by a unique geodesic, which is shorter than any other piecewise smooth curve
lying inside this ball. We also give an intrinsic (algebraic) characterization
of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold
of the von Neumann algebra generated by and .Comment: 19 page
Wieler solenoids, Cuntz-Pimsner algebras and K-theory
We study irreducible Smale spaces with totally disconnected stable sets and their associated -theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one -theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The -theoretic invariants of Cuntz-Pimsner algebras are then studied using the Cuntz-Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples
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