Geometric realization and K-theoretic decomposition of C*-algebras

Suppose that A is a separable C*-algebra and that G_* is a (graded) subgroup of K_*(A). Then there is a natural short exact sequence 0 \to G_* \to K_*(A) \to K_*(A)/G_* \to 0. In this note we demonstrate how to geometrically realize this sequence at the level of C*-algebras. As a result, we KK-theoretically decompose A as 0 \to A\otimes \Cal K \to A_f \to SA_t \to 0 where K_*(A_t) is the torsion subgroup of K_*(A) and K_*(A_f) is its torsionfree quotient. Then we further decompose A_t : it is KK-equivalent to \oplus_p A_p where K_*(A_p) is the p-primary subgroup of the torsion subgroup of K_*(A). We then apply this realization to study the Kasparov group K^*(A) and related objects.Comment: 9 pages.To appear in International J. Mat

Spaces of sections of Banach algebra bundles

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta : P \to X$ is a standard principal $G$-bundle, and $A_\zeta = \Gamma (X, P \times_G B)$ is the associated algebra of sections. We produce a spectral sequence which converges to $\pi_*(GL_o A_\zeta)$ with [E^2_{-p,q} \cong \check{H}^p(X ; \pi_q(GL_o B)).] A related spectral sequence converging to \K_{*+1}(A_\zeta) (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if \pi_*(GL_o B) \to \K_{*+1}(B) is an isomorphism for all $*>0$) then so is $A_\zeta$.Comment: 15 pages. Results generalized to include both real and complex K-theory. To appear in J. K-Theor

The fine structure of the Kasparov groups I: continuity of the KK-pairing

In this paper it is demonstrated that the Kasparov pairing is continuous with respect to the natural topology on the Kasparov groups, so that a KK-equivalence is an isomorphism of topological groups. In addition, we demonstrate that the groups have a natural pseudopolonais structure, and we prove that various KK-structural maps are continuous.Comment: 35 pages. To appear in J. Functional Analysi

Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem

The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by \v{C}ech cohomology of the tiling space) and the spectral properties (of Hamiltonians defined on such tilings) defined by $K$-theory, and to show their equivalence in dimensions $\leq 3$. A theorem makes precise the conditions for this relationship to hold. It can be viewed as an extension of the "Bloch Theorem" to a large class of aperiodic tilings. The idea underlying this result is based on the relationship between cohomology and $K$-theory traces and their equivalence in low dimensions.Comment: 36 pages, 4 figures, 1 tabl