3.7 Operator Algebras

Organizer R. Longo (Rome)

3.7.1 From Vertex Algebras to Local Nets of von Neuman Algebras

Sebastiano Carpi

Università di Chieti e Pescara carpi@gotham.sci.unich.it

We explain recent results on the connection between vertex algebras and local nets of von Neumann algebras.

(Joint work with Y. Kawahigashi, R. Longo and M. Weiner)

3.7.2 Non-Commutative Manifolds and Quantum Groups

Giovanni Landi

Università di Trieste landi@units.it

For quite sometime, it has been problematic to endow spaces coming from quantum groups with a noncommutative spin structure and such a possibility has eluded several approaches. We explicitly construct equivariant spectral triples on a variety of examples that include the manifold of quantum SU(2), families of quantum two-spheres, as well as higher dimensional quantum spheres.

3.7.3 L2 Invariants, Free Probability and Operator Algebras

Dimitri Shlyakhtenko

UCLA shlyakht@math.ucla.edu

The Cheeger-Gromov L2 Betti numbers of a discrete group are numerical invariants, going back to Atiyah’s work on the equivariant Atiyah-Singer index theorem. On the other hand, Voiculescu has introduced another discrete group invariant, coming from his free probability theory, called the free entropy dimension. Very roughly, this number measures the “asymptotic dimensions” of the sets of approximate embeddings of a group into unitary matrices. We describe our joint work with A. Connes and I. Mineyev that has provided a connection between these numbers.