C*-algebras: structure, symmetries, and equilibrium states

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C*-dynamical systems were initially introduced to model quantum mechanical systems. The observable magnitudes are represented by elements of a C*-algebra, on which a time evolution (a group of automorphisms) is given. In addition to time evolutions, other group actions (e.g. symmetries) can also be modelled via a generalization of this idea. The dynamical or geometric information can be subsumed in an algebraic construction, called the crossed product algebra, which has proven relevant in a number of abstract situations beyond its original physical motivation. The proposed research aims to develop methods to analyze symmetries and equilibrium states of C*-dynamical systems, and to use these methods in the study of questions arising from various areas of mathematics like number theory, dynamical systems, and geometric group theory. There is evidence that constructions and techniques inspired by physical systems can shed new light onto long-standing problems and can generate relevant questions and new examples on which to test the existing knowledge. One such phenomenon is that of phase transition with spontaneous symmetry-breaking, in which a system can be in equilibrium in a variety of asymmetric configurations at low temperature, while having only one equilibrium state at high temperature. Everyday examples of this are the freezing of water into ice and the spontaneous magnetization of a ferromagnet at room temperature which is lost through heating.