Conversations on mind, matter, and mathematics

Conclusion  The debate between Jean-Pierre Changeux and Alain Connes is one of the most, interesting to take place in recent years. It re-frames in a very up-to-date context a whole series of traditional and difficult questions from the standpoint of the knowledge and experience of two of the leading protagonists of contemporary science. To the choice presented by the neurobiologist between a Platonist ontology and a neurocognitive psychology of mathematical activity, the mathematician replies with a conception that is objective (neither ontological nor psychological) of the thoroughly consistent universe of mathematical idealities. It is indeed in this three-sided arena that the major difficulties play themselves out. One of the great virtues of the book is to cast a spotlight on this confrontation.     

Hopf Algebras, Renormalization and Noncommutative Geometry

We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations. Received: 14 August 1998/ Accepted: 5 October 1998     

On the spatial theory of von Neumann algebras

If M is a von Neumann algebra in $\text{H}$, each faithful weight ψ on M′ defines an operator-valued weight ψ^?1 of $\text{L(H)}$ on M. For each weight ? on M the positive unbounded operator $\text{d?}\text{dψ}\text{= ? ° ψ}{}^{\mathrm{?1}}$ satisfies all the usual properties of a Radon-Nikodym derivative.     

Measure space automorphisms,the normalizers of their full groups,and approximate finiteness

We deal with the normalizer $\text{N}$[T] of the full group [T] of a nonsingular transformation T of a Lebesgue measure space in the group of all nonsingular transformations. We solve the conjugacy problem in $\text{N}$[T]/[T] for a measure preserving and ergodic T. Our results show that a locally finite extension of a solvable group is approximately finite.     

Almost periodic states and factors of type III1

We construct a factor of type III₁ which has no almost-periodic state (or weight). We exhibit a factor N of type II_∞ and two automorphisms θ₁θ₂ of N which are not in the same conjugacy class in Out $\text{N =}\text{Aut}\text{N}\text{Int}\text{N}$ though τθ₁ = λ_τ, τθ₂ = λ_τ with λ? ]0, 1[, τ = Trace on N. We introduce and study two invariants Sd and τ for factors of type III₁. We relate the closedness of Int M in Aut M to the absence of central sequences in the von Neumann algebra M.     

Homogeneity of the state space of factors of type III1

A factor M is of type III₁ if and only if the action of its unitary group on its state space by inner automorphisms is topologically transitive in the norm topology.