Noncommutative function theory in operator algebras and operator spaces

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The principal investigator proposes two main lines of research, focused around new directions, and around foundational problems, in the relatively new field of operator spaces. These lines are: 1) 'noncommutative functional analysis and noncommutative Banach function theory', which in part continues the investigators introduction, and application, of powerful tools from classical functional analysis and C*-algebra and von Neumann algebra theory to 'noncommutative functional analysis', 2) the general theory and structure of operator algebras, for example extending to general operator algebras important new notions of equivalence that are attracting widespread attention in the field. The investigator and collaborators are also proposing to investigate applications of the above.The study of operator algebras originally grew out of quantum mechanics. It is often of crucial importance to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. Because operator variables do not commute, this is often called 'noncommutative mathematics' It is out of such a process that the theory of operator spaces and completely bounded maps emerged. This theory, which the investigator helped to found, is a novel and compelling approach to problems involving linear analysis which arise in 'noncommutative mathematics'. The investigator's research focuses partly on transferring important ideas and tools from classical subfields of linear analysis, via such 'quantization', to solve significant problems in 'noncommutative' mathematics. Most of the projects involve a deep unification of ideas from several different research areas; and often require extremely deep and technically demanding analysis. Successful findings would bring several subjects closer together, attract scientists with disparate backgrounds, and lead to cross-fertilization between these disciplines.