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Radial multipliers and approximation properties for $q$-Araki Woods algebras Brannan, Michael


In the early 2000's, Fumio Hiai introduced the $q$-Araki Woods von Neumann algebras, which are generalizations of both Shlyakhtenko's free Araki-Woods factors and Bozejko-Speicher's $q$-Gaussian algebras.  In comparison to the free Araki-Woods and the $q$-Gaussian cases, very little is known about the structure of generic $q$-Araki Woods algebras.  In particular, their type classification, factoriality, (strong) solidity, and even injectivity is not fully understood.  The reason for this lack of progress can be chalked up to the fact that when dealing with $q$-Araki-Woods algebras, one simultaneously looses the ''nice'' properties of free independence and traciality.  In this talk, I will discuss the problem of computing the cb-norms of an important class of linear maps on $q$-Araki-Woods algebras called radial multipliers.  Although we are unable to obtain nice formulas for the cb norms of radial multipliers (as was achieved in the free case by Houdayer-Ricard), we are able to use non-commutative central limit and ultraproduct techniques to show that their cb-norms do not depend on the underlying orthogonal transformation group governing the non-traciality of the algebra.  As a consequence of this deformation-invariance property for radial multipliers, we establish that all $q$-Araki-Woods algebras have the complete metric approximation property (CMAP).  This talk is based on joint work with Mateusz Wasilewski (IMPAN).

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