At the present, there is in Italy an internationally recognized and expanding research group in the field of operator algebras. The research is directed to structural problems of C*-algebras and von Neumann algebras, index theory for subfactors, applications to quantum field theory and connections with non-commutative geometry. The present state of the art in this field of research and the international standing of the present research group is documented in the proceedings of the international congress on "Operator Algebras and Quantum Field Theory" held in 1996 at the Accademia Nazionale dei Lincei, a congress with nearly 200 participants organized by members of the group [DLRZ], or the proceedings of the Congress "Mathematical Physics in Mathematics and in Physics. Quantum and operator algebraic aspects", Siena, June 2000 [L4].
The group's first fertile line of research concerns the theory of subfactors initiated by V.Jones but developed here in Italy from a point of view motivated by algebraic quantum field theory (sectors, endomorphisms, tensor categories), and the modular theory of Tomita-Takesaki. This approach has produced such results of intrinsic interest as duality for finite dimensional Hopf algebras, a Galois correspondence for compact automorphism groups (or actions of compact Kac algebras), restrictions on the range of the index in the presence of braid group symmetry and a theory of dimension for tensor C*-categories, as well as notable applications to quantum field theory (see [BCL], [CC], [CDR], [CF], [FI1], [FI2], [GLRV], [GLW], [ILP], [KLM], [L1]).
Another important line of research, motivated by superselection structure and later connected to the preceding line, concerns tensor C*-categories. This has led to a duality theory for compact groups going beyond the classical theory of Tannaka and Krein and allowing the construction of a net of fields associated to a net of von Neumann algebras of local observables. Later developments regard Hilbert modules, multiplicative unitaries and amenability (see [DPR], [DR], [KPW], [LR], [RT]).
A third rapidly developing line of research concerns the methods of analytic functions with values in Banach spaces applied to operator algebras. These methods allow a natural approach to the foundation of the modular theory of Tomita-Takesaki. Recently, in response to current needs of quantum field theory (relativistic KMS condition, field theory on manifolds) results have been obtained for problems of analytic continuation of functions of several variables (see [DZ1], [DZ3], [GL]).
Finally, the physically motivated quantum spacetime model considered in [DFR] calls for a Quantum Field Theory and related geometric structures on non-commutative manifolds, close to A.Connes' Noncommutative Geometry.
Operator algebraic methods have been used to tackle and resolve geometric problems on differentiable manifolds. In particular, invariants of Novikov-Shubin type (originally introduced on compact manifolds) have been defined and studied for arbitrary open amenable manifolds (see [GI1], [GI2], [GI3]).