Results 1  10
of
19
CuntzKrieger algebras of directed graphs
, 1996
"... We associate to each rowfinite directed graph E a universal CuntzKrieger C  algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allow ..."
Abstract

Cited by 136 (31 self)
 Add to MetaCart
We associate to each rowfinite directed graph E a universal CuntzKrieger C  algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the CuntzKrieger uniqueness theorem and give a characterisation of when C (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C (E) is AF; if E has a loop, then C (E) is purely infinite.
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
Abstract

Cited by 25 (11 self)
 Add to MetaCart
We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
Group C ∗ algebras as compact quantum metric spaces
 Doc. Math
"... Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). We investigate whether the topology from this metric coincides with the weak ∗ topology (our definition of a “compact quantum metric space”). We give an affirmative answer for G = Zd when ℓ is a wordlength, or the restriction to Zd of a norm on Rd. This works for C ∗ r (G) twisted by a 2cocycle, and thus for noncommutative tori. Our approach involves Connes ’ cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays. The group C ∗algebras of discrete groups provide a muchstudied class of “compact noncommutative spaces ” (that is, unital C ∗algebras). In [11] Connes showed that the “Dirac ” operator of an unbounded
Quantization of Poisson algebras associated to Lie algebroids
 Contemp. Math
"... Abstract. We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C ∗algebra may be regarded as a result of a quantization procedure. The C ∗algebra of the tangent groupoid of a giv ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
Abstract. We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C ∗algebra may be regarded as a result of a quantization procedure. The C ∗algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebroid A(G)) is the C ∗algebra of a continuous field of C ∗algebras over R with fibers A0 = C ∗ (A(G)) ≃ C0(A ∗ (G)) and A � = C ∗ (G) for � = 0. The same is true for the corresponding reduced C ∗algebras. Our results have applications to, e.g., transformation group C ∗algebras, Ktheory, and index theory.
Quantum Hall Effect on the hyperbolic plane
 Commun. Math. Physics
, 1997
"... Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potentia ..."
Abstract

Cited by 20 (13 self)
 Add to MetaCart
Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.
Twisted higher index theory on good orbifolds and fractional quantum numbers
"... Abstract. In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related to general ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related to generalizations of the BetheSommerfeld conjecture, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher traces on Ktheory, which we then apply to compute the range of values of the Hall conductance in the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in this case is that the Hall conductance again has plateaus at all energy levels belonging to any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued invariant. Moreover the set of possible denominators is finite and has been explicitly determined. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers.
KMS states on C*algebras associated to expansive maps. Preprint: arXiv:math.OA/0305044
"... Abstract. Using Walters ’ version of the RuellePerronFrobenius Theorem we show the existence and uniqueness of KMS states for a certain oneparameter group of automorphisms on a C*algebra associated to a positively expansive map on a compact metric space. 1. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. Using Walters ’ version of the RuellePerronFrobenius Theorem we show the existence and uniqueness of KMS states for a certain oneparameter group of automorphisms on a C*algebra associated to a positively expansive map on a compact metric space. 1.
Cuntzlike algebras
 Proceedings of the 17th International Conference on Operator Theory (Timisoara 98), The Theta Fondation
, 2000
"... The usual crossed product construction which associates to the homeomorphism T of the locally compact space X the C ∗algebra C ∗ (X, T) is extended to the case of a partial local homeomorphism T. For example, the CuntzKrieger algebras are the C ∗algebras of the onesided Markov shifts. The genera ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The usual crossed product construction which associates to the homeomorphism T of the locally compact space X the C ∗algebra C ∗ (X, T) is extended to the case of a partial local homeomorphism T. For example, the CuntzKrieger algebras are the C ∗algebras of the onesided Markov shifts. The generalizations of the CuntzKrieger algebras (graph algebras, algebras OA where A is an infinite matrix) which have been introduced recently can also be described as C ∗algebras of Markov chains with countably many states. This is useful to obtain such properties of these algebras as nuclearity, simplicity or pure infiniteness. One also gives examples of strong Morita equivalences arising from dynamical systems equivalences. 1 1 Introduction. Let us recall (with no respect for history) two striking results pertaining the rich interplay between ergodic theory and von Neumann algebras (we refer the reader to the survey [19] and the references thereof for details; at that time, the theory
Quantum Hall effect and noncommutative geometry
"... Abstract. We study magnetic Schrödinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect (QHE) in which the hyperbolic geometry provides an effective Hamiltonian. In addition we add some refinements to earlier results. We deri ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. We study magnetic Schrödinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect (QHE) in which the hyperbolic geometry provides an effective Hamiltonian. In addition we add some refinements to earlier results. We derive an analogue of the ConnesKubo formula for the Hall conductance via the quantum adiabatic theorem, identifying it as a geometric invariant associated to an algebra of observables that turns out to be a crossed product algebra. We modify the Fredholm modules defined in [4] in order to prove the integrality of the Hall conductance in this case.