This meeting of the Scandinavian noncommutative geometers and operator algebraists is going take place at Chalmers University Technology and University of Gothenburg on April 29th & 30th 2025.
Inductive limits are a central construction in operator algebras because they allow one to construct sophisticated objects using well-understood building blocks, particularly matrix algebras. However, if one wants to use matrix algebras as building blocks, then this construction can be quite restrictive, at least in the case of C*-algebras (closed *-algebras of bounded Hilbert space operators). In particular, the only C*-algebras that arise as inductive limits of matrix algebras are the almost finite dimensional (AF) algebras. This well-behaved class includes interesting examples such as the CAR algebra but falls far short of covering all amenable C*-algebras (a notably tractable class including noncommutative tori, and Cuntz and Toeplitz algebras). To realize this broader class as inductive limits of matrix algebras, one must consider inductive sequences in a broader category: the category of operator systems (*-subspaces of C*-algebras). In this talk, we will describe inductive sequences in the classical sense and then delve into the broader context of operator systems where we will investigate amenability and when one can still expect a C*-algebra at the limit.
Part of this is based on joint work with Niklas Galke, Lauritz van Luijk, Matt Kennedy, and Alexander Stottmeister.
In this talk we investigate the noncommutative metric geometry of the higher Vaksman-Soibelman quantum spheres. More precisely, we shall see how to endow a given quantum sphere with the structure of a compact quantum metric space by means of a seminorm arising from noncommutative differential geometric data. We view our quantum sphere as a noncommutative circle bundle over the corresponding quantum projective space. Using techniques from unbounded KK-theory this point of view allows us to construct vertical and horizontal differential geometric data on the quantum sphere in question. The vertical data comes from the generator of the circle action and the horizontal data comes from the unital spectral triple on quantum projective space introduced by D’Andrea and Dabrowski. An interesting feature of our setting is that the horizontal geometric data yields a twisted derivation on the coordinate algebra whereas the vertical geometric data produces a derivation in the usual sense. Nonetheless we can assemble these two (twisted) derivations into a single seminorm on our quantum sphere and show that the corresponding metric on the state space metrizes the weak*-topology.
The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.
The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.
The quantum Stein’s lemma gives the quantum relative entropy an operational interpretation in terms of hypothesis testing which is the task of distinguishing two quantum states. Composite hypothesis testing is a generalization of this task where one of the states is replaced by convex sets of quantum states, satisfying some natural axioms. The “generalized quantum Stein’s lemma” asserts that the relative entropy is still the relevant quantity in this context. Despite many attempts, this result was recently been proved in full generality. Some recent developments have also shown that this result has surprisingly deep connections with the theory of entanglement.
In this talk, I will show that the assertion of the generalized Stein’s lemma is true for the setting where the convex sets are the state spaces of finite- dimensional C∗-subalgebras. Our proof provides a different, algebraic approach for the case of subalgebras, which might be of independent interests.
No prior knowledge of quantum theory will be assumed. This is a joint work with Li Gao.
The classification of simple nuclear C*-algebras is usually split into two cases: the purely infinite and the stably finite case. Recently, in collaboration with Carrión, Schafhauser, Tikuisis, and White, we provided a new approach for classifying stably finite C*-algebras based on the so-called trace-kernel extension. I will outline a very similar new proof of the purely infinite case (the Kirchberg-Phillips theorem) using state-kernel extensions. Not only does this provide the framework for a unified approach for classification, this method has other huge advantages compared to previous proofs of the Kirchberg-Phillips theorem, such as implying Kirchberg's O_2-embedding theorem as a corollary, rather than using it as a crucial ingredient.
This is joint work with Ben Bouwen.
Lyudmila Turowsska
Søren Eilers
Magnus Goffeng [magnus dot goffeng (at) math dot lth dot se (responsible for the homepage)]
Kevin Aguyar Brix