The current project aims to investigate geometric properties of operator spaces, with the emphasis on two topics: the existence of operator spaces with prescribed properties, and the structure of the Fermion algebra and spaces related to it. The problem of constructing spaces with given properties was first posed by Grothendieck, and later attracted attention of many accomplished mathematicians, such as Gowers and Maurey. However, many questions in this area remain open, chief among them the so-called ``square-cube problem'': the existence of a space which is not isomorphic to its square, but such that its square is isomorphic to its cube. We plan to approach this class of problems using operator space ``building blocks''. In investigating the Fermion algebra, we plan to discover what properties it shares with spaces of functions. In particular, we will investigate the complemented and completely complemented subspaces of this algebra.
One of the key advances in physics over the past century was the creation and development of quantum mechanics. The main mathematical tool of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Initially, the mathematicians and physicists investigated only single operators, but later the need to consider whole sets of operators became apparent. Research by von Neumann, Gelfand, and others led to the development of the theory of C*-algebras. More recently, operator spaces (also called ``non-commutative'' or ``quantized Banach spaces'') arose from the study of maps on C*-algebras. It turns out that operator spaces provide an appropriate framework for studying algebras of operators. In fact, several long-standing problems in operator theory have been solved using operator space techniques. The current project deals with two topics: the existence of operator spaces with prescribed properties; and an investigation of the operator space structure of classical spaces, such as the Fermion algebra (related to the behavior of sub-atomic particles such as protons or neutrons). If successful, this research will advance our understanding of operator spaces, and potentially, enhance our knowledge of physical phenomena.
