Functional Analysis Seminar

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, Kasia Wyczesany will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. Whilst Mahler’s influential 1939 conjecture regarding the minimiser of the volume product will be mentioned, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions. Main new results are joint work with S. Artstein-Avidan and S. Sadovsky, and S. Artstein-Avidan and M. Fradelizi. 

AF-embeddability, i.e., the question whether a given C*-algebra can be realised as a subalgebra of an AF-algebra, has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then, many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. 

In this talk by Joachim Zacharias, we will consider a separable unital C*-algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT. We also consider some extensions of this inclusion and indicate some applications.

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