Soft questions: $C^\ast$-dynamical systems

Question

I have read some papers about $C^\ast$-dynamical systems. But there are still some questions in my mind which I can not answer.

1. When is the $C^\ast$-dynamical system introduced?
2. Why is the $C^\ast$-dynamical system introduced?
3. By whom is the $C^\ast$-dynamical system introduced?

Thanks to any one who can help me!

Answer 1

As with many concepts, I think it's hard to provide a precise starting point because there are many different more-or-less closely related formulations at different times. Basically, the theory of C^* dynamical systems comes from Quantum Mechanics. The Hilbert space formulation of this was developed by von Neumann in the 1930's. But when you want to describe systems with infinitely many degrees of freedom (interacting quantum fields, or statistical mechanics in the thermodynamic limit) Hilbert space is not adequate. This leads to the more general algebraic approach. A major milestone in this is the GNS-construction (Gelfand & Naimark 1943, Segal 1947).