Banach algebras, operator spaces and their applications to locally compact quantum groups

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Fourier analysis is named after (and was initiated by) 17th and 19th century French mathematician and physicist Jean-Baptiste Fourier who studied heat transfer and vibration. Its starting point is the Fourier series, i.e., a way to express a giving periodic function through overlaying sine and cosine waves. The Fourier transform is, to this day, an important tool to solve differential equations. In the 20th century, it became clear that locally compact abelian (LCA) groups are the appropriate setting to develop Fourier analysis. It allows to define a general Fourier transform that encompasses both Fourier series as well as the classical Fourier transform. The crucial concept here is Pontryagin duality: every LCA group G has a dual group G^. For instance, the dual group of the real line is the real line again, and the dual group of the integers is the unit circle. Moreover, G^^ = G always holds. The study of (not necessarily abelian) locally compact groups is called abstract harmonic analysis, a discipline that has been traditionally strong in Canada since the mid 20th century. A key approach in abstract harmonic analysis is to study not the groups themselves, but the various Banach algebras and spaces associated with them. Since the 1960s - in particular, during the past quarter of a century -, quantization has become more and more important: this refers to replacing commutative objects, such as spaces and algebras of functions, by non-commutative ones, i.e., spaces and algebras of operators. Since the turn of the century, locally compact quantum groups have gained significance: unlike non-abelian locally compact groups, they allow for a Pontryagin style duality that extends the one for LCA groups. The proposed research focuses on three main topics: 1. Locally compact quantum groups: amenability properties and duality. This project is intended to deepen our understanding between the various notions of amenability that exist for locally compact quantum groups. 2. Quantizing functional analysis. The theory of operator spaces, i.e., of spaces of bounded linear operators on Hilbert space, is often referred to as quantized functional analysis. Many concepts and results of classical functional analysis have quantized analogs, but still there is a lot still unclear. We hope to contribute to a further understanding of quantization in functional analysis. 3. Amenability properties of quantized Banach algebras. In the past (and ongoing), my research has been concerned with the amenability properties of quantized Banach algebras. The proposed research will continue along these lines. Overall, the proposed research will continue my work over the past five to ten years and contribute to a deeper understanding of quantization in functional and abstract harmonic analysis.