RICERCA ITALIANA     

Operator Algebras and Applications

Università degli Studi di Roma "Tor Vergata"

Abstract

The research programme can be classified in more detail as follows:

a) Conformal nets of local algebras
b) Quantum fields on curved spacetime
c) Algebraic formulation of the renormalization group
d) Field theory on noncommutative manifolds
e) Thermodynamics and Statistical mechanics
f) C*-tensor categories - Quantum groups
g) Noncommutative geometry
h) Free probability and factors of type II_1
i) Probability and quantum statistics
l) Noncommutative dynamical systems
m) Connes-Moscovici local index theory
n) Cohomological invariant for combinatorial manifolds
o) Localization and microlocalization of Hochschild and cyclic (co)homology. <<<

Principal Investigator

Roberto LONGO Università degli Studi di ROMA "Tor Vergata"

Research Objectives

The theory of Operator Algebras has developed particularly rapidly in the last thirty years. The content has been much enriched and deep interrelations with other mathematical disciplines have become apparent so that it now provides a unified language allowing a higher level of comprehension.
From the start, the theory developed in close relation with the theory of operators, ergodic theory, harmonic analysis, the theory of group representations and quantum physics. More recently, its domain has broadened and new connections with other branches of mathematics have emerged. It is enough to recall the non-commutative geometry of A. Connes and the polynomial invariants for topological knots of V. Jones.
The applications of operator algebras to quantum physics have always provided an important motivation and have continued to yield important contributions and reveal unexpected connections. The relation between the modular structure of von Neumann algebras and the KMS equilibrium condition in statistical mechanics, the quantum Noether theorem and split inclusions of von Neumann algebras, the structure of superselection sectors and its links with Jones index theory and the construction of the field algebra and the abstract duality theory of compact groups testify to this.
This project, which includes almost all Italian experts in Operator Algebras and Noncommutative Geometry, has as its primary aim to develop in an integrated way the investigations of the individual research units by concentrating the existing competences toward the solution of relevant problems. To this end, we intend to strengthen the collaborations among different units already under way and to stimulate new international exchanges. We also intend to organize one or more international mini-workshops on operator algebras and their natural connections with other areas of mathematics and theoretical physics. <<<

Timescale

24 months

National and international background

From the start, the theory of operator algebras developed in close relation with the theory of operators, ergodic theory, harmonic analysis, the theory of group representations and quantum physics. More recently, its domain has broadened and new connections with other branches of mathematics have emerged. It is enough to recall the non-commutative geometry of A.Connes and the polynomial invariants for topological knots of V.Jones.
Our group has always been particularly active in some problems concerning the applications of operator algebras to quantum physics, in particular the relation between the modular structure of von Neumann algebras and the KMS equilibrium condition in statistical mechanics, the quantum Noether theorem and split inclusions of von Neumann algebras, the structure of superselection sectors and its links with Jones index theory and the construction of the field algebra and the abstract duality theory of compact groups.
The research is directed to structural problems of C*-algebras and von Neumann algebras, index theory for subfactors, applications to quantum field theory and statistical mechanics, 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 Rome 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 Kac 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 R.Longo: Von Neumann Algebras and Quantum Field Theory, in Proc. International Congress of Mathematicians (Zürich, 1994), Birkhäuser, Basel, 1995, 1281-1291 and recent papers [ILP], [LR], [BCL], [CF], [CP], [FI1], [FI2], [GL], [GLW], [BuDMRS], [FG].)
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 S.Doplicher, J.E.Roberts: A New Duality Theory for Compact Groups, Invent.math. 98 (1989), 157-218, S.Doplicher, J.E.Roberts: Why there is a Field Algebra with a Compact Group describing the Superselection Structure in Partcle Physics, Commun.Math.Phys. 131 (1990), 51-107 and recent papers [PR],[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 I.Cioranescu, L.Zsidó: Analytic Generators for One-Parameter Groups, Tôhoku Math.J. 28 (1976), 327-362, L.Zsidó: A Proof of Tomita's Fundamental Theorem in the Theory of Standard von Neumann Algebras, Revue Roum.Math.Pures Appl. 20 (1975), 609-619 and recent papers [DZ1], [GL].)
Finally, 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], [GI4].) <<<