We consider the norm closure A of the algebra of all operators of order and class zero in Boutet ... more We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case ∂X has torsion free K-theory, we get K i (A/K) ≃ K i (C(X)) ⊕ K 1−i (C 0 (T * Ẋ)), i = 0, 1, with K denoting the compact ideal, and T * Ẋ denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis; and show, moreover, that K 1 (A) ≃ K 1 (C(X)) ⊕ ker χ, with χ : K 0 (T * Ẋ) → Z denoting the topological index. For the case of orientable, two-dimensional X, K 0 (A) ≃ Z 2g+m and K 1 (A) ≃ Z 2g+m−1 , where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ K ⊂ G ⊂ A, with A/G commutative and G/K isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L 2 (R +).
We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous fu... more We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic Ktheory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in K-theory.
We study the theory of projective representations for a compact quantum group G, i.e. actions of ... more We study the theory of projective representations for a compact quantum group G, i.e. actions of G on BpHq for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators KpHq, if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of p G in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C˚-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.
We find multipullback quantum odd-dimensional spheres equipped with natural U (1)-actions that yi... more We find multipullback quantum odd-dimensional spheres equipped with natural U (1)-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the K-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these K-groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces. Contents 1. Background 3 2. Twisted multipullback quantum odd spheres 7 3. Twisted multipullback quantum complex projective spaces 13 4. The K-groups of twisted multipullback quantum odd spheres and complex projective spaces 20 5. Noncommutative line bundles over multipullback quantum complex projective spaces 27 References 33
For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C ... more For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C * r (T − X) of C *-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.
For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C ... more For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C * r (T − X) of C *-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.
Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type z... more Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal L 1,∞ (H) for the Hilbert space H = L 2 (X, E) ⊕ L 2 (∂X, F) of L 2 sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form P u = f ; T u = 0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.
We consider the norm closure A of the algebra of all operators of order and class zero in Boutet ... more We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get K i (A/K) ≃ K i (C(X))⊕ K 1−i (C 0 (T * Ẋ)), i = 0, 1, with K denoting the compact ideal, and T * Ẋ denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, K 0 (A) ≃ Z 2g+m and K 1 (A) ≃ Z 2g+m−1 , where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ K ⊂ G ⊂ A, with A/G commutative and G/K isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L 2 (R +).
The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheim... more The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer in the seventies and since then, many developments occurred. Deformation quantization is based on such a philosophy in order to provide a mathematical procedure to pass from classical mechanics to quantum mechanics. Basically, it consists in deforming the pointwise product of functions to get a non-commutative one, which encodes the quantum mechanics behaviour. In formal deformation quantization, the non-commutative product (also said, star product) is given by a formal deformation of the pointwise product, i.e. by a formal power series in the deformation parameter which physically play the role of Planck's constant ̵ h. From a physical point of view this is clearly not sufficient to provide a reasonable quantum mechanical description and hence one needs to overcome the formal power series aspects in some way. One option is strict deformation quantization, which produces quantum algebras not just in the space of formal power series but in terms of C *-algebras, as suggested by Rieffel, with e.g. a continuous dependence on ̵ h. There are several other options in between continuous and formal dependence on ̵ h like analytic or smooth deformations. The Oberwolfach workshop Deformation quantization: between formal to strict consolidated, continued, and extended these research activities with a focus on the study of the connection between formal and strict deformation quantization in their various flavours and their applications in particular those in quantum physics and non-commutative geometry. It brought together specialists in differential geometry, operator algebras, non-commutative geometry, and quantum field theory with research interests in the mentioned quantization procedures. The aim of the workshop was to develop a coherent viewpoint of the many recent diverse developments in the field and to initiate new lines of research.
We show that Connes' metric on the state space associated with a spectral triple is nowhere infin... more We show that Connes' metric on the state space associated with a spectral triple is nowhere infinite exactly when it is globally bounded. Moreover, we produce a family of simple examples showing that this is not automatically the case.
In this paper we consider deformations of an algebroid stack on anétale groupoid. We construct a ... more In this paper we consider deformations of an algebroid stack on anétale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.
We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous fu... more We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic Ktheory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in K-theory.
Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. ... more Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C *-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in [9, 13] as special cases. Next we use the Araki-Woods III λ representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.
Uploads
Papers by Ryszard Nest