58,635 research outputs found
Commutants of von Neumann Correspondences and Duality of Eilenberg-Watts Theorems by Rieffel and by Blecher
The category of von Neumann correspondences from B to C (or von Neumann
B-C-modules) is dual to the category of von Neumann correspondences from C' to
B' via a functor that generalizes naturally the functor that sends a von
Neumann algebra to its commutant and back. We show that under this duality,
called commutant, Rieffel's Eilenberg-Watts theorem (on functors between the
categories of representations of two von Neumann algebras) switches into
Blecher's Eilenberg-Watts theorem (on functors between the categories of von
Neumann modules over two von Neumann algebras) and back.Comment: 20 page
Bicommutant categories from fusion categories
Bicommutant categories are higher categorical analogs of von Neumann algebras
that were recently introduced by the first author. In this article, we prove
that every unitary fusion category gives an example of a bicommutant category.
This theorem categorifies the well known result according to which a finite
dimensional *-algebra that can be faithfully represented on a Hilbert space is
in fact a von Neumann algebra.Comment: Updated to the published version + fixed some small typo
Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
Q-systems describe "extensions" of an infinite von Neumann factor , i.e.,
finite-index unital inclusions of into another von Neumann algebra .
They are (special cases of) Frobenius algebras in the C* tensor category of
endomorphisms of . We review the relation between Q-systems, their modules
and bimodules as structures in a category on one side, and homomorphisms
between von Neumann algebras on the other side. We then elaborate basic
operations with Q-systems (various decompositions in the general case, and the
centre, the full centre, and the braided product in braided categories), and
illuminate their meaning in the von Neumann algebra setting. The main
applications are in local quantum field theory, where Q-systems in the
subcategory of DHR endomorphisms of a local algebra encode extensions
of local nets. These applications, notably in conformal
quantum field theories with boundaries, are briefly exposed, and are discussed
in more detail in two separate papers [arXiv:1405.7863, 1410.8848].Comment: v1: 54 pages. v2: 90 pages. Title changed by request of editor,
numerous material added, especially an overview of applications in Quantum
Field Theory; some corrections. v3: minor corrections to match the published
version; Springer Briefs in Mathematical Physics, vol. 3, 201
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