Operator Modules, Quantum Groups and Quantum Information

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In 1925 our view of the physical world drastically changed with the advent of Heisenberg's matrix mechanics. He showed that we may accurately describe quantum phenomena by interpreting time dependent variables as non-commuting matrices rather than functions. This "quantization" of functions, which underlies the theoretical development of quantum mechanics, has motivated mathematicians to quantize other areas of mathematics, including functional analysis, harmonic analysis, and information theory. The resulting areas: operator spaces, topological quantum groups, and quantum information theory, are, to this day, prominent world-wide research areas at the forefront of modern analysis and mathematical physics. They have all been shown to have a profound structure theory, and deep mathematical connections between them continue to emerge. My research program lies at the confluence of these three areas.