Algebraic reformulation of Connes embedding problem and the free group algebra

A computability-theoretic reformulation of the Connes Embedding Problem

The Connes Embedding Problem (CEP) asks whether every separable II_1 factor embeds into an ultrapower of the hyperfinite II_1 factor. We show that the CEP is equivalent to the computability of the universal theory of every type II_1 von Neumann algebra. We also derive some further computability-theoretic consequences of the CEP.

An asymptotic property of factorizable completely positive maps and the Connes embedding problem

We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo-Werner channels W_n^- are factorizable, for all odd integers n different from 3. Furthermore, we investigate factorizability of convex combinations of W_3^+ and W_3^-, a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.

An algebraic approach to the radius of comparison

The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert…

康尼斯-基尔克堡问题与量子信息理论中的无限维现象

The Connes-Kirchberg Problem and infinite-dimensional phenomena in quantum information theory

本文探讨了康尼斯-基尔克堡问题与量子信息理论中无限维现象的联系，提出了可分解通道的概念，并给出了新的研究成果。

About the Connes Embedding Conjecture---Algebraic approaches---

This is an expanded lecture note for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5-9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture. It contains new proofs of equivalence of the Connes Embedding Conjecture, Positivstellensatze for trace positive polynomials, Kirchberg's Conjecture, and Tsirelson's Problem.

A C*-algebraic approach to interacting quantum field theories

A novel C*-algebraic framework is presented for relativistic quantum field theories, fixed by a Lagrangean. It combines the postulates of local quantum physics, encoded in the Haag-Kastler axioms, with insights gained in the perturbative approach to quantum field theory. Key ingredients are an appropriate version of Bogolubov's relative $S$-operators and a reformulation of the Schwinger-Dyson equations. These are used to define for any classical relativistic Lagrangean of a scalar field a non-trivial local net of C*-algebras, encoding the resulting interactions at the quantum level. The constr…