SDU Operator Algebras Seminar

Title: Spectral localizers in KK-theory

Abstract: In this talk we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.

Title: Noncommutative Choquet theory

Abstract: In this talk, I will introduce noncommutative convexity and present a brief overview of noncommutative Choquet theory, which provides a structural foundation for the study of noncommutative convex sets. Since the category of compact noncommutative convex sets is dual to the category of operator systems, which are unital self-adjoint subspaces of operators, noncommutative Choquet theory can potentially shed new light on problems of an operator-algebraic nature. I will discuss two recent examples of this. The first relates to the notion of self testing in quantum information theory, and the second provides a new characterization of operator systems satisfying an important approximation-theoretic property called hyperrigidity. This talk will feature joint work with Ken Davidson and Eli Shamovich.

Title: TBA

Abstract: TBA

Title: One-cross bundles  

Abstract: We present a framework for constructing a noncommutative geometry on a quantum homogeneous space. The resulting differential calculus—or quantum de Rham complex—exhibits a remarkably rich structure, encompassing a noncommutative Riemannian metric, a Levi-Civita connection, a complex structure, and associated holomorphic structures. As motivating examples, we discuss the irreducible quantum flag manifolds, and, time permitting, explore the analytic properties of their corresponding Dolbeault–Dirac operators.

Title: TBA

Abstract: TBA

Title: TBA

Abstract: TBA

Title: TBA

Abstract: TBA

Title: TBA

Abstract: TBA

Title: TBA

Abstract: TBA