Abstract. We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C ∗-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

Abstract not found

Sheaf theory for stacks in manifolds and twisted cohomology for

The Z2-graded Čech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven. In some special case we deduce an isomorphism between this noncommutative theory and the classical Z2-graded Čech cohomology theory. Keywords: Čech cohomology of a sheaf, cyclic theory

The Z2-graded Čech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven. In some special case we deduce an isomorphism between this noncommutative theory and the classical Z2-graded Čech cohomology theory. Keywords: Čech cohomology of a sheaf, cyclic theory

Abstract. A central result here is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C ∗-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M. 1.

For an orbifold X and α ∈ H 3 (X, Z), we introduce the twisted cohomology H ∗ c (X, α) and prove that the non-commutative Chern character of Connes-Karoubi establishes (X)⊗C and the twisted cohomology an isomorphism between the twisted K-groups K ∗ α H ∗ c (X, α). This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson’s theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.

cyclic homology of stable continuous trace algebras

Abstract. We compute the entire cyclic cohomology of the noncommutative 2-tori, which are isomorphic to their periodic ones. 1.

Abstract. Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2-tori. First of all, we clarify their algebraic structure of noncommutative 2-tori as a F ∗-algebra, according to the idea of Elliott-Evans. Actually, they are the inductive limit of subhomogeneous F ∗-algebras. Using such a result, we compute their entire cyclic cohomology, which is isomorphic to their periodic one as a complex vector space. 1.