We develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a’spectral triple’, i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled ’Dirac operator’, encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the ’Connesdistance function’ and show ...

We consider the algebra A of bounded operators on L2(Rn) generated by quantizations isometric affine canonical transformations. The includes as subalgebras noncommutative tori all dimensions and toric orbifolds. define spectral triple (A,H,D) with H=L2(Rn,?(Rn)) Euler operator D, a first order differential index 1. show that this has simple dimension spectrum: For every B in ?(A,H,D) Shubin typ...

Alternative titles of this paper would have been “Index theory without index” or “The Baum–Connes conjecture without Baum.” In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields ofC∗-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. Th...

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.