New horizons in operator algebras: finite-dimensional approximations and quantized function theory

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The proposed research program aims to discover new structure in the theory of operator algebras. Classically, these algebras came into being as a mathematical counterpart to the theory of quantum physics. The link between operator algebras and natural sciences continues to grow today, as witnessed in quantum information theory or quantum field theory for instance. Simply put, the connection between operator algebras and quantum physics is realized by symmetric operators between Euclidean spaces. Much of the sought-after structure of these operators can be unlocked upon considering the operator algebras (or C*-algebras) that they generate. What sets this research proposal apart from such classical investigations is that it is concerned with algebras of operators that lack the usual symmetry of quantum mechanical observables. In some applications the relevant operators happen to be a perturbation of a usual observable, so the resulting operator algebras are not susceptible to the tools designed to study C*-algebras. To elucidate the structure of these non self-adjoint operator algebras, I will employ innovative techniques in quantized function theory to implement finite-dimensional approximations. If we consider matrices to be the most basic objects, we may subsequently try to use them as building blocks for more complex algebras. This is the basic idea of a finite-dimensional approximation, a paradigm that has led to recent spectacular developments in the classification theory of C*-algebras. Some of the crucial insight involved in these advances was brought to light via a powerful functional analogy: one can view non-commutative algebras as consisting of functions, and exploit the resulting intuition coming from classical topology or dynamical systems theory. In spite of the unquestionable success of such a "quantized" theory of functions for C*-algebras, the corresponding non self-adjoint version has received measurably less attention. It is the purpose of this proposal to fill this gap. I will develop new tools in quantized function theory, a vibrant and rapidly evolving field with applications to systems and control theory, free probability and real algebraic geometry. The passage from C*-algebras to non self-adjoint ones should be mirrored by the set of non-commutative continuous functions collapsing to those that are in fact holomorphic. Classical intuition then suggests that this should result in a significant loss of flexibility and in a wealth of new rigidity phenomena. Towards achieving the above objectives, I have formulated an extensive scaffolding of concrete steps. The resulting numerous sub-problems of varied difficulties form the basis of my training plan for a diverse group of highly qualified personnel that I will recruit at all levels. Collaborations and interactions within my research group will be encouraged, and will foster an inclusive training environment.