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 >>>

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 >>>