By Toke M. Carlsen, Nadia S. Larsen, Sergey Neshveyev, Christian Skau

Like the 1st Abel Symposium, held in 2004, the Abel Symposium 2015 fascinated about operator algebras. it truly is attention-grabbing to determine the striking advances which were made in operator algebras over those years, which strikingly illustrate the energy of the sphere. a complete of 26 talks got on the symposium on numerous issues, all highlighting the richness of the subject.

The box of operator algebras was once created within the Nineteen Thirties and was once influenced through difficulties of quantum mechanics. It has to that end constructed way past its preliminary meant realm of purposes and increased into such various components of arithmetic as illustration idea, dynamical platforms, differential geometry, quantity concept and quantum algebra. One department, referred to as “noncommutative geometry”, has develop into a strong instrument for learning phenomena which are past the achieve of classical analysis.

This quantity comprises examine papers that current new effects, surveys that debate the improvement of a selected line of study, and articles that provide a mix of survey and learn. those contributions supply a multifaceted portrait of gorgeous arithmetic that either newbies to the sphere of operator algebras and professional researchers alike will appreciate.

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Commun. Math. , to appear. 07390. 1007/s00220-016-2642-7 18. S. Neshveyev and M. Yamashita, A few remarks on the tube algebra of a monoidal category. Preprint. 06332 19. A. Ocneanu, Chirality for operator algebras. In Subfactors (Kyuzeso, 1993), World Sci. , River Edge, 1994, pp. 39–63. 20. N. Ozawa, Examples of groups which are not weakly amenable. Kyoto J. Math. 52 (2012), 333–344. 21. S. Popa, Symmetric enveloping algebras, amenability and AFD properties for subfactors. Math. Res. Lett. 1 (1994), 409–425.

A weak . In order to prove the reverse implication it will suffice to prove a -weak version for the canonical inclusion ÃW A ,! A . Namely, we prove the following proposition in the remainder of this section. A/. c. maps i i A ! Fi ! P. Brown et al. such that 1. a/ ! a/ in the -weak topology for every a 2 A; 2. i is an order zero map; 3. b/k ! 0 for every a; b 2 A. 3]. 1 of [16] to see that for every i there is a net of contractive order zero maps . i; W Fi ! x/ for every x 2 Fi . We may therefore assume that the image of i is contained in A for every i.

So, G has the Haagerup property. G// such that n ! ıe weakly and k n ı Ad x n k ! 0 uniformly on compact sets of x 2 G. Conversely assume that G has the Haagerup property and that n is such a sequenceR of probability measures. By restricting n to K, normalizing and integrating K . n ı Ad k/ dk, we may assume that the probability measures n are supported on K and are Ad K-invariant. Fix a strictly positive right K-invariant R function w W G ! g/ dg D 1. g/ n ı Ad g dg : G Note that Q n is still Ad K-invariant.