Operator Algebras of Product Systems

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In the 1930s, von Neumann suggested Operator Algebras as an efficient tool for the systematic approach to Quantum Mechanics that connects the viewpoints of Heisenberg and Schrödinger. His vision has evolved to a solid framework for studying reversible transformations by extending representation theory to infinite dimensions, and thus incorporating tools from both Algebra and Analysis. A central aspect in this endeavour is to identify invariants that can effectively classify the C*-algebra of a group action on a possibly noncommutative state space. Such constructs already appeared in the work of Murray and von Neumann for the classification of factors. Their rich structure has instigated research in its own respect and has inspired many directions in the general theory as they offer important experimental devices for identifying and testing key conjectures.In the past 30 years there has been a great effort to extend our understanding to irreversible transformations. In the one-variable case this has led to the production of operator algebras related to structures of great importance in Mathematics such as graphs, stochastic matrices or analytic varieties. However much less has been known for more involved semigroup transformations, and only recently the correct model has been identified through the work of many experts. The quantization here is more involved than just an extension of the group-case or the one-variable case. This is not a surprise as the class of semigroups is too vast. Thus new methods need to be invented for a thorough analysis at that level.In the current project we wish to work in the context of amenable (and amenably controlled) transformations. They form a rather broad class that covers a wide number of previous constructs, for example in relation to abelian lattices and higher-rank graphs. Our main goal is to study the properties of their related operator algebras and thus promote them to a key object in the C*-theory as a source for examples, counterexamples and applications.There are three fundamental questions at the basis of an effective analysis in relation to Arveson's Programme, Elliott's Programme and Laca-Neshveyev-Raeburn programme, which we plan to pursue here. Those will be investigated during four visits of the PI to international research centres over a period of 16 months.