We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The
second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.
We consider a measurement of the gravitational acceleration of antimatter, gbar, using muonium. A monoenergetic, low-velocity, horizontal muonium beam will be formed from a surface-muon beam using a novel technique and directed at an atom interferome
ter. The measurement requires a precision three-grating interferometer: the first grating pair creates an interference pattern which is analyzed by scanning the third grating vertically using piezo actuators. State-of-the-art nanofabrication can produce the needed membrane grating structure in silicon nitride or ultrananoscrystalline diamond. With 100 nm grating pitch, a 10% measurement of gbar can be made using some months of surface-muon beam time. This will be the first gravitational measurement of leptonic matter, of 2nd-generation matter and, possibly, the first measurement of the gravitational acceleration of antimatter.
This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (fo
rmulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Arakis notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds.
We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary cond
itions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras.
