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.