Abstract. We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.

We study the index problem for the d-bar operators subject to Atiyah-Patodi-Singer boundary conditions on noncommutative disk and annulus.

In this paper we study decompositions and classification problems for unbounded derivations in Bunce-Deddens-Toeplitz and Bunce-Deddens algebras. We also look at implementations of these derivations on associated GNS Hilbert spaces.

Abstract. We construct a spectral triple for the C * -algebra of continuous functions on the space of p-adic integers by using a rooted tree obtained from coarse-grained approximation of the space, and the forward derivative on the tree. Additionally, we verify that our spectral triple satisfies the properties of a compact spectral metric space, and we show that the metric on the space of p-adic integers induced by the spectral triple is equivalent to the usual p-adic metric.

We define and study smooth subalgebras of Bunce-Deddens C * -algebras. We discuss various aspects of noncommutative geometry of Bunce-Deddens algebras including derivations on smooth subalgebras, as well as K-Theory and K-Homology.