Executive Summary : | Classical analysis relies on ambient space to define functions and perform analysis, but noncommutative analysis has revolutionized this process. Operator algebras have been identified as appropriate function algebras associated with quantum "spaces," leading to significant advances in areas such as geometry, number theory, and probability theory. By dropping the commutativity condition while retaining all other defining conditions, we can obtain genuinely non-commutative function algebras, transcending the classical setup to get quantum spaces. Extensive research has been done on C* algebras, von-Neumann algebras, free probability theory, and quantum groups. However, little is known about the structure of sets whose elements are precisely these operator algebras. The object of study for us are quantum super spaces, whose elements are quantum spaces represented by their non-commutative function algebras. Recent work by Effros, Marechal, and Haagerup-Winslow has shown remarkable connections between the topology of spaces of von-Neumann algebras with the Connes Embedding Problem. A comprehensive study of the topology of spaces of von-Neumann algebras has been carried out by myself, P. Fima, F. Le Maitre, and K. Mukherjee, focusing on subspaces like maximal abelian subalgebras and continuity of invariants like the Cowling-Haagerup invariant. The key objective of the current project is to define an invariant for non-hyperfinite von-Neumann algebras based on analysis of quantum super spaces and apply this invariant to probe isomorphism problems of von-Neumann algebras, focusing on the free group factor isomorphism problem. |